Related dramatic dialogue from FUBAR —
Hero — I guess I'll take the pill, and get it over with. (Dramatic music playing.)
Villain — This will be fun. (Music intensifies.) Cheers … Nothing's happening.
Hero — Come to think of it, I might have taken the antidote.
Read more at: https://tvshowtranscripts.ourboard.org/… .
Related synchronology check —
Source data for an Instagram story this evening —
An image from the story itself —
Related reading — "Flame Alphabet" in this journal.
From that link . . .
See also . . .
Underlying Structure of the Design —
Schoolgirl Problem —
… Smart Set .
Related detective story . . .
A Cosmic Nod
Related aesthetic puzzle —
What's black and white and red all over?
Last updated at 22:46 PM ET on 1 February 2023.
Click for a designer's obituary.
Paraphrase for a road-sign collector:
See as well … Today's New York Times obituary
of the Harvard Business School Publishing
Director of Intellectual Property.
From a Log24 post of Feb. 5, 2009 —
An online logo today —
See also Harry Potter and the Lightning Bolt.
“Danes have been called the happiest people.
I wonder how they measure this.”
— Copenhagen designer in today's online New York Times .
A version of this article is to appear in print on March 26, 2017,
in T Magazine with the headline: "Gray Matters."
See also last night's quarter-to-three post as well as
the webpage "Grids, You Say?" by Norwegian artist Josefine Lyche.
The Fano Plane —
"A balanced incomplete block design , or BIBD
with parameters b , v , r , k , and λ is an arrangement
of b blocks, taken from a set of v objects (known
for historical reasons as varieties ), such that every
variety appears in exactly r blocks, every block
contains exactly k varieties, and every pair of
varieties appears together in exactly λ blocks.
Such an arrangement is also called a
(b , v , r , k , λ ) design. Thus, (7, 3, 1) [the Fano plane]
is a (7, 7, 3, 3, 1) design."
— Ezra Brown, "The Many Names of (7, 3, 1),"
Mathematics Magazine , Vol. 75, No. 2, April 2002
W. Cherowitzo uses the notation (v, b, r, k, λ) instead of
Brown's (b , v , r , k , λ ). Cherowitzo has described,
without mentioning its close connection with the
Fano-plane design, the following —
"the (8,14,7,4,3)-design on the set
X = {1,2,3,4,5,6,7,8} with blocks:
{1,3,7,8} {1,2,4,8} {2,3,5,8} {3,4,6,8} {4,5,7,8}
{1,5,6,8} {2,6,7,8} {1,2,3,6} {1,2,5,7} {1,3,4,5}
{1,4,6,7} {2,3,4,7} {2,4,5,6} {3,5,6,7}."
We can arrange these 14 blocks in complementary pairs:
{1,2,3,6} {4,5,7,8}
{1,2,4,8} {3,5,6,7}
{1,2,5,7} {3,4,6,8}
{1,3,4,5} {2,6,7,8}
{1,3,7,8} {2,4,5,6}
{1,4,6,7} {2,3,5,8}
{1,5,6,8} {2,3,4,7}.
These pairs correspond to the seven natural slicings
of the following eightfold cube —
Another representation of these seven natural slicings —
These seven slicings represent the seven
planes through the origin in the vector
3-space over the two-element field GF(2).
In a standard construction, these seven
planes provide one way of defining the
seven projective lines of the Fano plane.
A more colorful illustration —
In memory of set designer Robin Wagner, winner of three
Tony awards, who reportedly died at 89 on May 29 —
|
From Log24 on May 29
"When the men on the chessboard Don't take the Brown acid!
Sunday, May 28, 2023
|
See as well Chess Set and Efficient Packing.
The New York Times on a set designer who
reportedly died at 83 on Monday (Feb. 6, 2023) —
"Adrian Hall, the founding artistic director,
brought him in as resident designer.
(Mr. Hall died on Feb. 4 in Van, Texas.)"
Hall was the founding artistic director of
Trinity Repertory Company, Providence, R.I.
Not-so-holy writ ….
Panthers — "Dimensions," Log24, Feb. 5, 2023.
Beast Belly — Tonight's previous post, "Gutter Mathematics."
The previous post displayed part of a page from
a newspaper published the day Olivia Newton-John
turned 21 — Friday, September 26, 1969.
A meditation, with apologies to Coleridge:
In Xanadu did Newton-John
A stately pleasure-square decree
Where Aleph the sacred symbol ran
Through subsquares measureless to man.
A related video —
Beware, beware, her flashing eyes, her floating hair:
Set design —
As opposed to block design —
See the signature link in last night's post for a representation of Madison Avenue.
For a representation by Madison Avenue, see today's New York Times—
"As a movement Pop Art came and went in a flash, but it was the kind of flash that left everything changed. The art public was now a different public— larger, to be sure, but less serious, less introspective, less willing or able to distinguish between achievement and its trashy simulacrum. Moreover, everything connected with the life of art— everything, anyway, that might have been expected to offer some resistance to this wholesale vulgarization and demoralization— was now cheapened and corrupted. The museums began their rapid descent into show biz and the retail trade. Their exhibitions were now mounted like Broadway shows, complete with set designers and lighting consultants, and their directors pressed into service as hucksters, promoting their wares in radio and television spots and selling their facilities for cocktail parties and other entertainments, while their so-called education programs likewise degenerated into sundry forms of entertainment and promotion. The critics were co-opted, the art magazines commercialized, and the academy, which had once taken a certain pride in remaining aloof from the blandishments of the cultural marketplace, now proved eager to join the crowd— for there was no longer any standard in the name of which a sellout could be rejected. When the boundary separating art and fashion was breached, so was the dividing line between high art and popular culture, and upon all those institutions and professions which had been painstakingly created to preserve high art from the corruptions of popular culture. The effect was devastating. Some surrendered their standards with greater alacrity than others, but the drift was unmistakable and all in the same direction— and the momentum has only accelerated with the passage of time."
— Hilton Kramer, The Triumph of Modernism: The Art World, 1985-2005 , publ. by Ivan R. Dee on Oct. 26, 2006, pp. 146-147
Related material— Rubik in this journal, Exorcist in this journal, and For the Class of '11.
Best Set Design, Vegas ACM Awards, Sunday Night—
Related literature— Knoxville: Summer of 1915—
"The stars are wide and alive, they seem each like a smile of great sweetness, and they seem very near."
For Mary Gaitskill,
continued from
June 21, 2008:
This minimal art
is the basis of the
chess set image
from Tuesday:
Related images:
“The key is the
cocktail that begins
the proceedings.”
— Brian Harley,
Mate in Two Moves
|
https://www.instagram.com/p/CEZ9ISzDkun/ marcelanow … East Village #gymlife #sunset #bbg #fitnessmotivation #shootfilm #socialdistancing #lookbook #fashiondesign #modellife #homeworkout #fashionblogger #fitnesslife #kaylaitsines #aloyoga #thelightsofbeauty #sportlovers #disposablecamera #femalemodel #polishgirl #editorial #blondegirl #manhattangirl #yogapractice #newyork #quarantine August 27, 2020 |
* Early title of the Nabokov novel Laughter in the Dark.
|
by Google Gemini AI Monday, March 4, 2024
Absolutely! Here's an academic paper about the Cullinane diamond theorem. Please note that while I provide citations, in a fully formatted paper you'd need to adhere to your chosen academic style guide for proper formatting:
The Cullinane Diamond Theorem: Abstract This paper examines the Cullinane diamond theorem, a principle in finite projective geometry that offers unexpected insights into the symmetries found in graphic designs. The theorem, initially formulated by Steven H. Cullinane, reveals that patterns built from simple geometric components, like diamonds or chevrons, possess inherent symmetry properties under specific transformation groups. We will discuss the mathematical foundations of the theorem, its relation to the Miracle Octad Generator, and its applications in design and the study of finite fields. Introduction Symmetry plays a vital role in art and design, contributing to a sense of balance, harmony, and aesthetic appeal. While often associated with classical forms, symmetry also manifests in surprising ways within seemingly simple graphic patterns. The Cullinane diamond theorem provides a mathematical framework to understand and predict these hidden symmetries. Mathematical Background Finite Projective Geometries: The Cullinane diamond theorem has its roots in finite projective geometry. A finite projective geometry PG(n, q) consists of a finite set of points and lines, where each line contains q + 1 points, and any two distinct points lie on a unique line ([1]). Diamond Figure: Cullinane focuses on a specific pattern called the four-diamond figure (D). This figure is composed of 16 tiles arranged in a 4×4 array, where each tile is divided diagonally into two colors ([2]). Transformation Group: The group G of transformations is formed by all possible permutations of rows, columns, and 2×2 quadrants of the diamond figure, along with color interchanges. The size of this group is 322,560 ([2]). The Cullinane Diamond Theorem
Theorem:
Proof: Connections to the Miracle Octad Generator The Cullinane diamond theorem shares a relationship with R.T. Curtis's Miracle Octad Generator (MOG). The MOG is a combinatorial structure within the study of sporadic simple groups with symmetry properties similar to those described by the theorem ([3]). Applications Graphic Design: The theorem informs artistic compositions employing diamond and chevron-like shapes. Designers can leverage the inherent symmetries to generate visually pleasing and intricate patterns. Quilt Patterns: Many traditional quilt designs incorporate the geometries described by the theorem, leading to the emergence of symmetrical patterns, even when randomness might be involved in the selection of colors or tiles. Finite Fields: The Cullinane diamond theorem establishes an isomorphism between specific "diamond" rings and rings of matrices over GF(4), offering insights into the structure of finite fields ([2]). Conclusion The Cullinane diamond theorem elegantly connects abstract mathematics with the world of graphic design. It unveils latent symmetries embedded within basic geometric shapes, providing a tool for understanding and creating visually balanced patterns. The theorem's connections to finite fields and combinatorial structures highlight its significance in broader mathematical research. Citations [1] Weisstein, E. W. “Finite Projective Geometry.” From MathWorld–A Wolfram Web Resource. [invalid URL removed] [2] Cullinane, S. H. "Cullinane Diamond Theorem." Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem [3] Curtis, R. T. (1976). A New Combinatorial Approach to M₂₄. Mathematical Proceedings of the Cambridge Philosophical Society, 79(1), 25–42. doi:10.1017/S0305004100052075 Let me know if you'd like me to expand on any particular aspect or provide additional citations!
1. www.hellenicaworld.com/Science/Mathematics/en/MathieuGroup.html
|
|
Bringing Up Bing : A Tutorial for Bing on Monday, December 18, 2023 Prompt: How is the Cullinane Diamond Theorem related to the Curtis Miracle Octad Generator?
Response: Searching for: Generating answers for you… The Cullinane Diamond Theorem and the Curtis Miracle Octad Generator (MOG) are both mathematical concepts that are deeply intertwined with finite projective geometry1,2. The Cullinane Diamond Theorem explains the surprising symmetry properties of some simple graphic designs and underlies the structure of the 35 square patterns in R. T. Curtis’s MOG1,2. The theorem uses a 4×4 array of two-color diagonally-divided square tiles, and considers the group of permutations of these 16 tiles generated by mixing random permutations of rows, columns, and the four 2×2 quadrants1,2. The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry1,2. The 35 structures of the 840 images of the 4×4 array are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), a finite field with two elements1,2. This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets1,2. The MOG is a 4×6 array of combinations describing any point in 24-dimensional space1. It preserves all of the symmetries and maximal subgroups of the Mathieu group M241. The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice2. In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG1,2. Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry1,2. Learn more Reply to Bing's Response — Actually, the Cullinane diamond theorem, by itself, does NOT explain the patterns in the MOG. The MOG is a natural correspondence, or pairing, of the 35 diamond-theorem structures with the 35 partitions of an eight-element set into two four-element sets. That correspondence is explained in a 1910 paper by G. M. Conwell, "The 3-Space PG(3,2) and Its Group." The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24=16+8 elements. Response to the Above Reply — Generating answers for you… Thank you for the clarification. You’re correct that the Cullinane Diamond Theorem, by itself, does not explain the patterns in the Miracle Octad Generator (MOG). The MOG is indeed a natural correspondence of the 35 structures from the Diamond Theorem with the 35 partitions of an eight-element set into two four-element sets. This correspondence was explained by G. M. Conwell in his 1910 paper “The 3-Space PG(3,2) and Its Group.” The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24 elements, which can be partitioned into a set of 16 and a set of 8. I appreciate your input and the opportunity to learn from it. If you have any more questions or need further clarification, feel free to ask!
|
"Unsocratic Dialogue" with Google DeepMind's Gemini AI continues . . .
|
Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now?
|
|
What is the Cullinane diamond theorem?
|
Flashback from a Log24 search for Caprica —
|
Monday, July 3, 2017
|
From Log24 on July 2, 2017 —
"For times like these, the Reading View feature in Microsoft Edge
acts as your own personal horse blinders, stripping unwanted
distractions and rendering just what you want to see."
— Scott Orgera at lifewire.com, January 12, 2017
"Feeling of belonging to the virtual environment" —
Marcela Nowak, May 10, 2017
Not always a good thing .
Related reading from a Google search today —
This post was suggested by a "matrix rotator" photo dated 1953-03-02.
Some context — Kracauer's notion of "mass ornament."
Further combinatorial properties* of 24261120 may
be investigated with the aid of a 9×9 square grid, and
perhaps (eventually) also with its triangular counterpart —
.
* Cap sets, gerechte designs, etc.
"You want an original design, tailored to your needs?"
"So, a needle pulling thread…La, a note to follow So…"
— Song lyric
From a recent Log24 Talent Search post —
|
The text of https://marcelanowak.com/home/ — About — We are a full-service creative agency based in LA.
Graphic and web design is our business. |
Like "La La Land" — A talent so nice they named it twice!
"It's going to be accomplished in steps,
this establishment of the Talented
in the scheme of things."
— To Ride Pegasus ,
by Anne McCaffrey (Radcliffe '47)
"Having skipped me, the talent for math concentrated extravagantly
in one of my nieces, Amie Wilkinson, a professor at the University
of Chicago, and I figured she could teach me."
— Wilkinson, Alec. A Divine Language (p. 5).
Farrar, Straus and Giroux, 12 July 2022, Kindle Edition.
Career Moves: Compare and Contrast —
"Amie Wilkinson is an explorer. Instead of seeking uncharted land,
she’s after undiscovered mathematical worlds — complex systems
of motion that unfold in unexpected ways. As a professor at the
University of Chicago, she’s known for discovering unique types
of these 'dynamical systems' that had been only conjectured to exist.
Wilkinson’s career has proceeded in a dynamical way, too. She now
approaches her roles as a researcher, teacher and mentor very
differently than she did when she was starting out."
— Kevin Hartnett in Quanta Magazine , June 13, 2019
|
The text of https://marcelanowak.com/home/ — About — We are a full-service creative agency based in LA.
Graphic and web design is our business. |
Like "La La Land" — A talent so nice they named it twice!
An essay from . . .
The Shape of Things: A Philosophy of Design Wittgenstein’s Architecture The universe of texts can be seen as a landscape. In it one can make out mountains and valleys, rivers and lakes, castles, farmyards and inner-city slums. On the horizon of the scene visualized in this way, the Bible and Homer appear as gigantic ice-covered mountains. The vast, tranquil lake of Aristotle’s texts, where fishermen idly throw their nets and philologists row their boats, occupies a part of the valley bottom. There, the tumbling waterfall of Nietzsche is captured by the broad river of modern pragmatism. Towering above everything, the Gothic cathedral of St Thomas Aquinas’s Summae dominates the cathedral square of the city, in which the roofs and gables of Baroque speculations jostle one another. In the suburbs of this city, one catches sight of the Romantic, Realist and Modernist housing-blocks and factories of more recent litera¬ ture; somewhat apart from all these stands a small, apparently insignificant house resembling scaffolding more than a finished building: Wittgenstein’s building. This little house is called the Tractatus. This name isn’t the product of a one-track mind. For when one enters the house, one notices immediately that this is not a place that has lost track of things. Quite the opposite: It is a place of mirror- images. The house stands on six foundation pillars which support one another by means of cross-beams organized in a hierarchy. In the middle, however, there rises a seventh pillar whose function it is to cut through the building and free it from the ground. So the house with all its corners, angles and joints is protected, armoured and impregnable. And yet, and for that very reason, it is threatened with collapse and disappearance without trace – condemned in advance and from the outset. The building is set out: It consists of propositions. Every proposition presupposes all the preceding ones and is itself the 76 presupposition of all the following propositions. Proposition by proposition, anyone who enters progresses through the prescribed rooms, and his step is supported by consistencies. Suddenly, with one proposition, one single proposition, the ground gives way beneath his feet. He falls head first into the abyss. Wittgenstein’s house is situated in a suburb of that city whose cathedral square is dominated by the towers of Thomas Aquinas’s cathedral. The small, modest pillars of Wittgenstein’s house support one another according to the same logico- philosophical method as the pillars of the cathedral support one another. But there appears to be a world of difference between the cathedral and the little house: The cathedral is a ship pointing in the direction of heaven, and the little house is a trap-door pointing in the direction of a bottomless abyss. But be careful: May Thomas Aquinas not have been right in saying after his revelation that everything he had written before was like straw? May not the heaven above the cathedral be the same black hole as the abyss beneath the little house? May not Wittgenstein’s little house be the cathedral of today? And those mirrors whose images simultaneously mirror one another, may they not be our equivalent of stained-glass windows? The landscape portrayed in this essay, it goes without saying, is a metaphor. Is it possible to identify it as Vienna? And is it possible for anyone entering Wittgenstein’s little house in that unlikely place to make out a hint of the unsayable? What we cannot speak about we must pass over in silence. 77 |
Click the above image to enlarge.
See as well . . .
Update of 2:40 AM May 7, 2022 —
Flusser's seven "pillars" appear to be the main sections of the Tractatus
— numbered 1 through 7, with many intermediate numbered passages.
For a more geometric meditation on "the shape of things," see other
posts tagged "Shape Constant" in this journal.
*Byron Gogol is a tech magnate in the HBO series "Made for Love."
* Song lyric, soundtrack album of
“Midnight in the Garden of Good and Evil“
Browsing related to the graphic design theory described in the previous post
yielded a four-color diamond illustrating design at Microsoft —
For some related mathematics see . . .
The Four-Color Diamond’s 2007 Source —
See also Log24 posts from August 2007 now tagged The Four-Color Ring.
| Maria Shriver, a contributor for NBC’s “TODAY,” remembered her aunt as an “extraordinary woman.”
Smith “had a great career on behalf of this country as ambassador to Ireland promoting peace there and also started very special arts for people with intellectual disabilities,” Shriver said on the 3rd hour of “TODAY.” “So I take solace in the fact that she is joining every other member of her family up in heaven. So it’s nice for her,” she added. Smith was born on Feb. 20, 1928, in Boston, Massachusetts to Rose and Joseph Kennedy. |
Related graphic design:
Feb. 20 square and June 17 Circle.
Related entertainment: “The Foreigner” (2017 film) and . . .
The Boston Globe on the dead architect of the previous post —
"Mr. McKinnell, who was a fellow of the American Institute of Architects
and the American Academy of Arts and Sciences, and a member of the
Royal Institute of British Architects, taught for many years at the
Harvard Graduate School of Design and the Massachusetts Institute of
Technology School of Architecture and Planning."
Some ugly rhetoric to go with the ugly architecture —
There are many approaches to constructing the Mathieu
group M24. The exercise below sketches an approach that
may or may not be new.
Exercise:
It is well-known that …
There are 56 triangles in an 8-set.
There are 56 spreads in PG(3,2).
The alternating group An is generated by 3-cycles.
The alternating group A8 is isomorphic to GL(4,2).
Use the above facts, along with the correspondence
described below, to construct M24.
Some background —
A Log24 post of May 19, 2013, cites …
Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.
See also a Google search for “56 triangles” “56 spreads” Mathieu.
Update of October 31, 2019 — A related illustration —
Update of November 2, 2019 —
See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel (Academic Press, 1991).
That page is from a paper published in 1970.
Update of December 20, 2019 —
|
arXiv.org > quant-ph > arXiv:1905.06914 Quantum Physics Placing Kirkman's Schoolgirls and Quantum Spin Pairs on the Fano Plane: A Rainbow of Four Primary Colors, A Harmony of Fifteen Tones J. P. Marceaux, A. R. P. Rau (Submitted on 14 May 2019) A recreational problem from nearly two centuries ago has featured prominently in recent times in the mathematics of designs, codes, and signal processing. The number 15 that is central to the problem coincidentally features in areas of physics, especially in today's field of quantum information, as the number of basic operators of two quantum spins ("qubits"). This affords a 1:1 correspondence that we exploit to use the well-known Pauli spin or Lie-Clifford algebra of those fifteen operators to provide specific constructions as posed in the recreational problem. An algorithm is set up that, working with four basic objects, generates alternative solutions or designs. The choice of four base colors or four basic chords can thus lead to color diagrams or acoustic patterns that correspond to realizations of each design. The Fano Plane of finite projective geometry involving seven points and lines and the tetrahedral three-dimensional simplex of 15 points are key objects that feature in this study. Comments:16 pages, 10 figures Subjects:Quantum Physics (quant-ph) Cite as:arXiv:1905.06914 [quant-ph] (or arXiv:1905.06914v1 [quant-ph] for this version) Submission history
From: A. R. P. Rau [view email] |
See also other posts tagged Tetrahedron vs. Square.
This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .
Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193-194, Feb. 1979.
Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —
Revision history accounting for the above change from yesterday —
The jargon "rm OR" means "remove original research."
The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square representation
of the 35 points and lines.
* The 35 squares, each consisting of four 4-element subsets, appeared earlier
in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
They were not at that time presented as constituting a finite geometry,
either affine (AG(4,2)) or projective (PG(3,2)).
WISC = Wechsler Intelligence Scale for Children
RISC = Reduced Instruction Set Computer or
Rust Inventory of Schizotypal Cognitions
See related material in earlier WISC RISC posts.
See also . . .
"Many parents ask us about the Block Design section
on the WISC and hope to purchase blocks and exercises
like those used on the WISC test. We explain that doing that
has the potential to invalidate their child's test results.
These Froebel Color Cubes will give you a tool to work with
your child on the skills tested for in the Block Design section
of the WISC in an ethical and appropriate way. These same
skills are applicable to any test of non-verbal reasoning like
the NNAT, Raven's or non-verbal sections of the CogAT or OLSAT. "
For a webpage that is perhaps un ethical and in appropriate,
see Block Designs in Art and Mathematics.
From a Log24 post of March 4, 2008 —
|
SINGER, ISAAC:
"Sets forth his own aims in writing for children and laments
— An Annotated Listing of Criticism
"She returned the smile, then looked across the room to
— A Swiftly Tilting Planet,
For "the dimension of time," see A Fold in Time, Time Fold,
A Swiftly Tilting Planet is a fantasy for children |
Ibid. —
The pen's point:
John Trever, Albuquerque Journal, 2/29/08
Note the figure on the cover of National Review above —
A related figure from Pentagram Design —
See, more generally, Isaac Singer in this journal.
From The New York Times this morning —
Where the Journey
is the Destination
A writer finds emotional solace on some of
Norway’s scenic remote roads, which have been
transformed into architectural wonders.
By ONDINE COHANE OCT. 16, 2017
. . . .
"… another project conceived along these routes is
the Juvet Landscape Hotel, designed by the architects
Jensen & Skodvin, and the creepy, if incredibly appropriate
aesthetically, setting for the 2015 film 'Ex Machina.' "
<meta property="article:published"
itemprop="datePublished"
content="2017-10-16T00:01:38-04:00" />
Cover design by Jarrod Taylor.
Book published on July 14, 2015.
For this journal on that date, see posts tagged Perspective.
See also Log24 posts related to "Go Set a Structure"
as well as "New Haven" + Grid.
Commentary —
|
"The close relationships between group theory and structural combinatorics go back well over a century. Given a combinatorial object, it is natural to consider its automorphism group. Conversely, given a group, there may be a nice object upon which it acts. If the group is given as a group of permutations of some set, it is natural to try to regard the elements of that set as the points of some structure which can be at least partially visualized. For example, in 1861 Mathieu… discovered five multiply transitive permutation groups. These were constructed as groups of permutations of 11, 12, 22, 23 or 24 points, by means of detailed calculations. In a little-known 1931 paper of Carmichael [5], they were first observed to be automorphism groups of exquisite finite geometries. This fact was rediscovered soon afterwards by Witt [11], who provided direct constructions for the groups and then the geometries. It is now more customary to construct first the designs, and then the groups…."
5. R. D. Carmichael, Tactical configurations of rank two,
11. E. Witt, Die 5-fach transitiven Gruppen von Mathieu,
— William M. Kantor, book review (pdf), |
The title refers to the previous post, which quotes a
remark by a poetry critic in the current New Yorker .
Scholia —
From the post Structure and Sense of June 6, 2016 —
Structure
Sense
From the post Design Cube of July 23, 2015 —
Broken Symmetries in Diamond Space
A version of the I Ching’s Hexagram 19:
From Katherine Neville's The Eight , a book on the significance
of the date April 4 — the author's birthday —
|
The Eight by Katherine Neville —
“What does this have to do with why we’re here?” |
Related material: Posts now tagged Hourglass Code.
See also the hourglass in a search for Pilgrim's Progress Illustration.
Symbol —
Monday, November 7, 2011
|
Images related to the previous post —
Detail of the 1697 Leibniz medal
|
“And so that I won’t come entirely empty-handed this time, I enclose a design of that which I had the pleasure of discussing with you recently. It is in the form of a memorial coin or medallion; and though the design is mediocre and can be improved in accordance with your judgment, the thing is such, that it would be worth showing in silver now and unto future generations, if it were struck at your Highness’s command. Because one of the main points of the Christian Faith, and among those points that have penetrated least into the minds of the worldly-wise and that are difficult to make with the heathen is the creation of all things out of nothing through God’s omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing. And it would be difficult to find a better illustration of this secret in nature or philosophy; hence I have set on the medallion design IMAGO CREATIONIS [in the image of creation]. It is no less remarkable that there appears therefrom, not only that God made everything from nothing, but also that everything that He made was good; as we can see here, with our own eyes, in this image of creation. Because instead of there appearing no particular order or pattern, as in the common representation of numbers, there appears here in contrast a wonderful order and harmony which cannot be improved upon…. Such harmonious order and beauty can be seen in the small table on the medallion up to 16 or 17; since for a larger table, say to 32, there is not enough room. One can further see that the disorder, which one imagines in the work of God, is but apparent; that if one looks at the matter with the proper perspective, there appears symmetry, which encourages one more and more to love and praise the wisdom, goodness, and beauty of the highest good, from which all goodness and beauty has flowed.” |
See also some related posts in this journal.
Software writer Richard P. Gabriel describes some work of design
philosopher Christopher Alexander in the 1960's at Harvard:
A more interesting account of these 35 structures:
"It is commonly known that there is a bijection between
the 35 unordered triples of a 7-set [i.e., the 35 partitions
of an 8-set into two 4-sets] and the 35 lines of PG(3,2)
such that lines intersect if and only if the corresponding
triples have exactly one element in common."
— "Generalized Polygons and Semipartial Geometries,"
by F. De Clerck, J. A. Thas, and H. Van Maldeghem,
April 1996 minicourse, example 5 on page 6.
For some context, see Eightfold Geometry by Steven H. Cullinane.
The latest Visual Insight post at the American Mathematical
Society website discusses group actions on the McGee graph,
pictured as 24 points arranged in a circle that are connected
by 36 symmetrically arranged edges.
Wikipedia remarks that …
"The automorphism group of the McGee graph
is of order 32 and doesn't act transitively upon
its vertices: there are two vertex orbits of lengths
8 and 16."
The partition into 8 and 16 points suggests, for those familiar
with the Miracle Octad Generator and the Mathieu group M24,
the following exercise:
Arrange the 24 points of the projective line
over GF(23) in a circle in the natural cyclic order
( ∞, 1, 2, 3, … , 22, 0 ). Can the McGee graph be
modeled by constructing edges in any natural way?
In other words, if the above set of edges has no
"natural" connection with the 24 points of the
projective line over GF(23), does some other
set of edges in an isomorphic McGee graph
have such a connection?
Update of 9:20 PM ET Sept. 20, 2015:
Backstory: A related question by John Baez
at Math Overflow on August 20.
The black rectangle at the end of Example 1.4
is known as the "end-of-proof symbol," "Halmos,"
or "tombstone."
Omega is a Greek letter, Ω , used in mathematics to denote
a set on which a group acts.
For instance, the affine group AGL(3,2) is a group of 1,344
actions on the eight elements of the vector 3-space over the
two-element Galois field GF(2), or, if you prefer, on the Galois
field Ω = GF(8).
Related fiction: The Eight , by Katherine Neville.
Related non-fiction: A remark by Werner Heisenberg
in this journal on Saturday, June 6, 2015, the eightfold cube ,
and the illustrations below —
|
Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
"The Brit Awards are… the British equivalent
of the American Grammy Awards." — Wikipedia
Detail of an image from yesterday's 5:30 PM ET post:
Related material:
From a review: "Imagine 'Raiders of the Lost Ark'
set in 20th-century London, and then imagine it
written by a man steeped not in Hollywood movies
but in Dante and the things of the spirit, and you
might begin to get a picture of Charles Williams's
novel Many Dimensions ."
See also Solomon's Seal (July 26, 2012).
The previous post displayed a set of
24 unit-square “points” within a rectangular array.
These are the points of the
Miracle Octad Generator of R. T. Curtis.
The array was labeled Ω
because that is the usual designation for
a set acted upon by a group:
* The title is an allusion to Point Omega , a novel by
Don DeLillo published on Groundhog Day 2010.
See “Point Omega” in this journal.
“What happens when you mix the brilliant wit of Noel Coward
with the intricate plotting of Agatha Christie? Set during a
weekend in an English country manor in 1932, Death by Design
is a delightful and mysterious ‘mash-up’ of two of the greatest
English writers of all time. Edward Bennett, a playwright, and
his wife Sorel Bennett, an actress, flee London and head to
Cookham after a disastrous opening night. But various guests
arrive unexpectedly….”
— Samuel French (theatrical publisher) on a play that
opened in Houston on September 9, 2011.
Related material:
The American Mathematical Society yesterday:
Harvey Cohn (1923-2014)
Wednesday September 10th 2014
Cohn, an AMS Fellow and a Putnam Fellow (1942), died May 16 at the age of 90. He served in the Navy in World War II and following the war received his PhD from Harvard University in 1948 under the direction of Lars Ahlfors. He was a member of the faculty at Wayne State University, Stanford University, Washington University in St. Louis, the University of Arizona, and at City College of New York, where he was a distinguished professor. After retiring from teaching, he also worked for the NSA. Cohn was an AMS member since 1942.
Paid death notice from The New York Times , July 27, 2014:
COHN–Harvey. Fellow of the American Mathematical Society and member of the Society since 1942, died on May 16 at the age of 90. He was a brilliant Mathematician, an adoring husband, father and grandfather, and faithful friend and mentor to his colleagues and students. Born in New York City in 1923, Cohn received his B.S. degree (Mathematics and Physics) from CCNY in 1942. He received his M.S. degree from NYU (1943), and his Ph.D. from Harvard (1948) after service in the Navy (Electronic Technicians Mate, 1944-46). He was a member of Phi Beta Kappa (Sigma Chi), won the William Lowell Putnam Prize in 1942, and was awarded the Townsend Harris Medal in 1972. A pioneer in the intensive use of computers in an innovative way in a large number of classical mathematical problems, Harvey Cohn held faculty positions at Wayne State University, Stanford, Washington University Saint Louis (first Director of the Computing Center 1956-58), University of Arizona (Chairman 1958-1967), University of Copenhagen, and CCNY (Distinguished Professor of Mathematics). After his retirement from teaching, he worked in a variety of capacities for the National Security Agency and its research arm, IDA Center for Computing Sciences. He is survived by his wife of 63 years, Bernice, of Laguna Woods, California and Ft. Lauderdale, FL, his son Anthony, daughter Susan Cohn Boros, three grandchildren and one great-granddaughter.
— Published in The New York Times on July 27, 2014
See also an autobiographical essay found on the web.
None of the above sources mention the following book, which is apparently by this same Harvey Cohn. (It is dedicated to "Tony and Susan.")
Advanced Number Theory, by Harvey Cohn
Courier Dover Publications, 1980 – 276 pages
(First published by Wiley in 1962 as A Second Course in Number Theory )
Publisher's description:
" 'A very stimulating book … in a class by itself.'— American Mathematical Monthly
Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.
The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups.
Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.
In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory."
User Review –
"In a nutshell, the book serves as an introduction to Gauss' theory of quadratic forms and their composition laws (the cornerstone of his Disquisitiones Arithmeticae) from the modern point of view (ideals in quadratic number fields). I strongly recommend it as a gentle introduction to algebraic number theory (with exclusive emphasis on quadratic number fields and binary quadratic forms). As a bonus, the book includes material on Dirichlet L-functions as well as proofs of Dirichlet's class number formula and Dirichlet's theorem in primes in arithmetic progressions (of course this material requires the reader to have the background of a one-semester course in real analysis; on the other hand, this material is largely independent of the subsequent algebraic developments).
Better titles for this book would be 'A Second Course in Number Theory' or 'Introduction to quadratic forms and quadratic fields'. It is not a very advanced book in the sense that required background is only a one-semester course in number theory. It does not assume prior familiarity with abstract algebra. While exercises are included, they are not particularly interesting or challenging (if probably adequate to keep the reader engaged).
While the exposition is *slightly* dated, it feels fresh enough and is particularly suitable for self-study (I'd be less likely to recommend the book as a formal textbook). Students with a background in abstract algebra might find the pace a bit slow, with a bit too much time spent on algebraic preliminaries (the entire Part I—about 90 pages); however, these preliminaries are essential to paving the road towards Parts II (ideal theory in quadratic fields) and III (applications of ideal theory).
It is almost inevitable to compare this book to Borevich-Shafarevich 'Number Theory'. The latter is a fantastic book which covers a large superset of the material in Cohn's book. Borevich-Shafarevich is, however, a much more demanding read and it is out of print. For gentle self-study (and perhaps as a preparation to later read Borevich-Shafarevich), Cohn's book is a fine read."
In "Notes on Finite Group Theory"
by Peter J. Cameron (October 2013),
http://www.maths.qmul.ac.uk/~pjc/notes/gt.pdf,
some parts are particularly related to the mathematics of
the 4×4 square (viewable in various ways as four quartets)—
Cameron is the author of Parallelisms of Complete Designs ,
a book notable in part for its chapter epigraphs from T.S. Eliot's
Four Quartets . These epigraphs, if not the text proper, seem
appropriate for All Saints' Day.
But note also Log24 posts tagged Not Theology.
Found this morning in a search:
A logline is a one-sentence summary of your script.
www.scriptologist.com/Magazine/Tips/Logline/logline.html
It's the short blurb in TV guides that tells you what a movie
is about and helps you decide if you're interested …
The search was suggested by a screenwriting weblog post,
"Loglines: WHAT are you doing?".
What is your story about?
No, seriously, WHAT are you writing about?
Who are the characters? What happens to them?
Where does it take place? What’s the theme?
What’s the style? There are nearly a million
little questions to answer when you set out
to tell a story. But it all starts with one
super, overarching question.
What are you writing about? This is the first
big idea that we pull out of the ether, sometimes
before we even have any characters.
What is your story about?
The screenwriting post was found in an earlier search for
the highlighted phrase.
The screenwriting post was dated December 15, 2009.
What I am doing now is checking for synchronicity.
This weblog on December 15, 2009, had a post
titled A Christmas Carol. That post referred to my 1976
monograph titled Diamond Theory .
I guess the script I'm summarizing right now is about
the heart of that theory, a group of 322,560 permutations
that preserve the symmetry of a family of graphic designs.
For that group in action, see the Diamond 16 Puzzle.
The "super overarching" phrase was used to describe
this same group in a different context:
This is from "Mathieu Moonshine," a webpage by Anne Taormina.
A logline summarizing my approach to that group:
Finite projective geometry explains
the surprising symmetry properties
of some simple graphic designs—
found, for instance, in quilts.
The story thus summarized is perhaps not destined for movie greatness.
For little Colva … The Mother Ship :
. . . .
For more light, see "Merton College" + Cameron
in this journal, as well as …
An Education
Leonardo DiCaprio and Carey Mulligan in Baz Luhrmann's
new version of The Great Gatsby :
We're going to Disney World! —
(For a more up-to-date version of little Colva,
see Primitive Groups and Maximal Subgroups.)
"Hans Castorp is a searcher after the Holy Grail.
You would never have thought it when you read
his story—if I did myself, it was both more and
less than thinking. Perhaps you will read the
book again from this point of view. And perhaps
you will find out what the Grail is: the knowledge
and the wisdom, the consecration, the highest
reward, for which not only the foolish hero but
the book itself is seeking. You will find it in the
chapter called 'Snow'…."
— Thomas Mann, "The Making of
The Magic Mountain "
In related entertainment news…
Click image for some backstory.
Mann's tale is set in Davos, Switzerland.
See also Mayer at Davos.
(Continued from 2 PM ET Tuesday)
“… the object sets up a kind of frame or space or field
within which there can be epiphany.”
— Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477)
"The absolute consonance is a state of chromatic plenitude."
"… the nearest precedent might be found in Becky Sharp .
The opening of the Duchess of Richmond's ball,
with its organization of strong contrasts and
display of chromatic plenitude, presents a schema…."
— Scott Higgins, Harnessing the Technicolor Rainbow:
Color Design in The 1930s , University of Texas Press,
2007, page 142
Note the pattern on the dance floor.
(Click for wider image.)
"At the still point…" — Four Quartets
For Mitt …
See "A Deskful of Girls" in Fritz Leiber's Selected Stories .
See also the Feast of St. Mary Magdalene in 2009.
… And for Clint—
From "Deskful":
I quickly settled myself in the chair, not to be gingerly
about it. It was rather incredibly comfortable, almost
as if it had adjusted its dimensions a bit at the last
instant to conform to mine. The back was narrow at
the base but widened and then curled in and over to
almost a canopy around my head and shoulders.
The seat too widened a lot toward the front, where
the stubby legs were far apart. The bulky arms
sprang unsupported from the back and took my own
just right, though curving inwards with the barest
suggestion of a hug. The leather or unfamiliar plastic
was as firm and cool as young flesh and its texture
as mat under my fingertips.
"An historic chair," the Doctor observed, "designed
and built for me by von Helmholtz of the Bauhaus…."
The Harvard Crimson on last night's Ig Nobel Prize ceremony:
"The theme of the evening was 'The Universe,' a catchword
that had the audience cheering any time it was mentioned
throughout the night. Throughout the ceremony, a mini opera
entitled 'The Intelligent Designer and the Universe'* premiered
in four acts.
The opera’s final line was “This is how the Universe decays
into insanity.”
* An opera "about an insane wealthy man who bequeaths his
fortune to have someone design a beautiful dress for the
universe." —Mark Pratt, Associated Press
In related news…
"Most mysteries begin in confusion and end in certainty;
Pynchon likes to change this trajectory, so that what begins
a mystery ends as pure chaos. (Well aware how frustrating
some readers find this, Pynchon sets up a running gag in
Inherent Vice about a class action suit brought against MGM
by audiences who don't like the way its stories end.)"
— Sarah Churchwell in The Guardian , Sunday, July 26, 2009
( Continued from yesterday's post FLT )
Context Part I —
"In 1957, George Miller initiated a research programme at Harvard University to investigate rule-learning, in situations where participants are exposed to stimuli generated by rules, but are not told about those rules. The research program was designed to understand how, given exposure to some finite subset of stimuli, a participant could 'induce' a set of rules that would allow them to recognize novel members of the broader set. The stimuli in question could be meaningless strings of letters, spoken syllables or other sounds, or structured images. Conceived broadly, the project was a seminal first attempt to understand how observers, exposed to a set of stimuli, could come up with a set of principles, patterns, rules or hypotheses that generalized over their observations. Such abstract principles, patterns, rules or hypotheses then allow the observer to recognize not just the previously seen stimuli, but a wide range of other stimuli consistent with them. Miller termed this approach 'pattern conception ' (as opposed to 'pattern perception'), because the abstract patterns in question were too abstract to be 'truly perceptual.'….
…. the 'grammatical rules' in such a system are drawn from the discipline of formal language theory (FLT)…."
— W. Tecumseh Fitch, Angela D. Friederici, and Peter Hagoort, "Pattern Perception and Computational Complexity: Introduction to the Special Issue," Phil. Trans. R. Soc. B (2012) 367, 1925-1932
Context Part II —
Context Part III —
A four-color theorem describes the mathematics of
general structures, not just symbol-strings, formed from
four kinds of things— for instance, from the four elements
of the finite Galois field GF(4), or the four bases of DNA.
Context Part IV —
A quotation from William P. Thurston, a mathematician
who died on Aug. 21, 2012—
"It may sound almost circular to say that
what mathematicians are accomplishing
is to advance human understanding of mathematics.
I will not try to resolve this
by discussing what mathematics is,
because it would take us far afield.
Mathematicians generally feel that they know
what mathematics is, but find it difficult
to give a good direct definition.
It is interesting to try. For me,
'the theory of formal patterns'
has come the closest, but to discuss this
would be a whole essay in itself."
Related material from a literate source—
"So we moved, and they, in a formal pattern"
Formal Patterns—
Not formal language theory but rather
finite projective geometry provides a graphic grammar
of abstract design—
See also, elsewhere in this journal,
Crimson Easter Egg and Formal Pattern.
The "FLT" of the above title is not Fermat's Last Theorem,
but Formal Language Theory (see image below).
In memory of George A. Miller, Harvard cognitive psychologist, who
reportedly died at 92 on July 22, 2012, the first page of a tribute
published shortly before his death—
The complete introduction is available online. It ends by saying—
"In conclusion, the research discussed in this issue
breathes new life into a set of issues that were raised,
but never resolved, by Miller 60 years ago…."
Related material: Symmetry and Hierarchy (a post of 9/11), and
Notes on Groups and Geometry, 1978-1986 .
Yesterday's post in memory of Octavio Paz—
|
… the free-standing, two-sided “Life-Death Figure,” |
An earlier post yesterday, Fashion Notes, linked to a Sting video—
|
From "Loo Ree," by Zenna Henderson "It's so hard to explain–" "Oh, foof!" I cried defiantly, taking off my glasses and, smearing the tears across both lenses with a tattered Kleenex. "So I'm a dope, a moron! If I can explain protective coloration to my six-year-olds and the interdependence of man and animals, you can tell me something of what the score is!" I scrubbed the back of my hand across my blurry eyes. "If you have to, start out 'Once upon a time."' I sat down– hard. Loo Ree smiled and sat down, too. "Don't cry, teacher. Teachers aren't supposed to have tears." "I know it," I sniffed. "A little less than human-that's us." "A little more than human, sometimes." Loo Ree corrected gently. "Well then, you must understand that I'll have to simplify. You will have to dress the bare bones of the explanation according to your capabilities. "Once upon a time there was a classroom. Oh, cosmic in size, but so like yours that you would smile in recognition if you could see it all. And somewhere in the classroom something was wrong. Not the whispering and murmuring– that's usual. Not the pinching and poking and tattling that goes on until you get so you don't even hear it." I nodded. How well I knew. "It wasn't even the sudden blow across the aisle or the unexpected wrestling match in the back of the room. That happens often, too. But something else was wrong. It was an undercurrent, a stealthy, sly sort of thing that has to be caught early or it disrupts the whole classroom and tarnishes the children with a darkness that will never quite rub off. "The teacher could feel it –as all good teachers can– and she spoke to the principal. He, being a good principal, immediately saw the urgency of the matter and also saw that it was beyond him, so he called in an Expert." "You?" I asked, feeling quite bright because I had followed the analogy so far. Loo Ree smiled. "Well, I'm part of the Expert." |
"If you have to, start out 'Once upon a time.'"
Yesterday's Paz post was at 6:48 PM EDT.
For the autistic, here is some related mathematics.
Yesterday's Fashion Notes post was at 1:06 PM EDT.
A related chronological note from Rolling Stone yesterday—
"Levon Helm, singer and drummer for the Band,
died on April 19th in New York of throat cancer.
He was 71.
"He passed away peacefully at 1:30 this afternoon…."
Helm and The Band performing "The Weight"—
"I pulled into Nazareth, I was a-feelin' 'bout half past dead…"
For Octavio Paz, who died on this date in 1998
|
… the free-standing, two-sided “Life-Death Figure,” — Holland Cotter, New York Times online today |
This journal on June 18, 2008—
The Wechsler Cubes story continues with a paper from December 2009…
"Learning effects were assessed for the block design (BD) task,
on the basis of variation in 2 stimulus parameters:
perceptual cohesiveness (PC) and set size uncertainty (U)." —
(Click image for some background.)
The real intelligence test is, of course, the one Wechsler flunked—
investigating the properties of designs made with sixteen
of his cubes instead of nine.
A search tonight for material related to the four-color
decomposition theorem yielded the Wikipedia article
Functional decomposition.
The article, of more philosophical than mathematical
interest, is largely due to one David Fass at Rutgers.
(See the article's revision history for mid-August 2007.)
Fass's interest in function decomposition may or may not
be related to the above-mentioned theorem, which
originated in the investigation of functions into the
four-element Galois field from a 4×4 square domain.
Some related material involving Fass and 4×4 squares—
A 2003 paper he wrote with Jacob Feldman—
"Design is how it works." — Steve Jobs
An assignment for Jobs in the afterlife—
Discuss the Fass-Feldman approach to "categorization under
complexity" in the context of the Wikipedia article's
philosophical remarks on "reductionist tradition."
The Fass-Feldman paper was assigned in an MIT course
for a class on Walpurgisnacht 2003.
"Design is how it works." — Steve Jobs
From a commercial test-prep firm in New York City—
From the date of the above uploading—
|
|
From a New Year's Day, 2012, weblog post in New Zealand—
From Arthur C. Clarke, an early version of his 2001 monolith—
"So they left a sentinel, one of millions they have scattered
throughout the Universe, watching over all worlds with the
promise of life. It was a beacon that down the ages has been
patiently signaling the fact that no one had discovered it.
Perhaps you understand now why that crystal pyramid was set…."
The numerical (not crystal) pyramid above is related to a sort of
mathematical block design known as a Steiner system.
For its relationship to the graphic block design shown above,
see the webpages Block Designs and The Diamond Theorem
as well as The Galois Tesseract and R. T. Curtis's classic paper
"A New Combinatorial Approach to M24," which contains the following
version of the above numerical pyramid—
For graphic block designs, I prefer the blocks (and the parents)
of Grand Rapids to those of New York City.
For the barbed tail of Clarke's "Angel" story, see the New Zealand post
of New Year's Day mentioned above.
A chess set previously mentioned in this journal—
These chessmen appeared in the weblog Minimalissimo
on Sept. 20, 2010. In Log24 on that date, the issue was
not so much the chessmen as the underlying board.
See "The Unfolding." See also the following from
the Occupy Space gallery in Limerick today—
| C A V E S – Anthony Murphy Solo Exhibition Opening 7 pm Thursday 1st Dec Exhibition 2nd – 22nd Dec 2011 Plato's allegory of the cave describes prisoners, inhabiting the cave since childhood, immobile, facing an interior wall. A large fire burns behind the prisoners, and as people pass this fire their shadows are cast upon the cave's wall, and these shadows of the activity being played out behind the prisoner become the only version of reality that the prisoner knows. C A V E S is an exhibition of three large scale works, each designed to immerse the viewer, and then to confront the audience with a question regarding how far they, as privileged viewers of the shadows and reflections being played out upon the walls, are willing to allow themselves to believe what they know to be a false reality. The works are based on explorations of simple 2D shapes; regular polygons are exploded to create fractured pattern, or layered upon one another until intricate forms emerge, upon which the projections can begin to draw out a third dimension. |
“Design is how it works.” — Steven Jobs (See Symmetry and Design.)
“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer
The name Carmichael is not to be found in Booher’s thesis. A book he does cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 1937. It should be dated 1931, as the following quotation shows—
From Log24 on Feb. 20, 2010—
“The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24.”
– R. D. Carmichael, “Tactical Configurations of Rank Two,” in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240
Epigraph from Ch. 4 of Design Theory , Vol. I:
“Es is eine alte Geschichte,
doch bleibt sie immer neu ”
—Heine (Lyrisches Intermezzo XXXIX)
See also “Do you like apples?“
Thomas N. Armstrong III, a former director of the Whitney Museum of American Art, died at 78 on Monday in Manhattan.
William Grimes in this morning's New York Times—
"… Mr. Armstrong set about strengthening the museum’s permanent collection, buying Frank Stella’s 1959 black painting “Die Fahne Hoch!” for $75,000 in 1977…."
See also "Fahne Hoch" in this journal and the following from the date of Armstrong's death—
"Sunrise — Hast thou a Flag for me?" — Emily Dickinson
Related material: Piracy Project and, from Flag Day,
"Dawn's Early Light" and "Expressionistic Depth."
An Adamantine View of "The [Philosophers'] Stone"
The New York Times column "The Stone" on Sunday, Nov. 21 had this—
"Wittgenstein was formally presenting his Tractatus Logico-Philosophicus , an already well-known work he had written in 1921, as his doctoral thesis. Russell and Moore were respectfully suggesting that they didn’t quite understand proposition 5.4541 when they were abruptly cut off by the irritable Wittgenstein. 'I don’t expect you to understand!' (I am relying on local legend here….)"
Proposition 5.4541*—
Related material, found during a further search—
A commentary on "simplex sigillum veri" leads to the phrase "adamantine crystalline structure of logic"—
For related metaphors, see The Diamond Cube, Design Cube 2x2x2, and A Simple Reflection Group of Order 168.
Here Łukasiewicz's phrase "the hardest of materials" apparently suggested the commentators' adjective "adamantine." The word "diamond" in the links above refers of course not to a material, but to a geometric form, the equiangular rhombus. For a connection of this sort of geometry with logic, see The Diamond Theorem and The Geometry of Logic.
For more about God, a Stone, logic, and cubes, see Tale (Nov. 23).
* 5.4541 in the German original—
Die Lösungen der logischen Probleme müssen einfach sein,
denn sie setzen den Standard der Einfachheit.
Die Menschen haben immer geahnt, dass es
ein Gebiet von Fragen geben müsse, deren Antworten—
a priori—symmetrisch, und zu einem abgeschlossenen,
regelmäßigen Gebilde vereint liegen.
Ein Gebiet, in dem der Satz gilt: simplex sigillum veri.
Here "einfach" means "simple," not "neat," and "Gebiet" means
"area, region, field, realm," not (except metaphorically) "sphere."
The Story of N
Roberta Smith in the New York Times of July 7, 2006—
Art Review
"… The show has an endgame, end-time mood, as if we are looking at the end of the end of the end of Pop, hyperrealism and appropriation art. The techniques of replication and copying have become so meticulous that they are beside the point. This is truly magic realism: the kind you can't see, that has to be explained. It is also a time when artists cultivate hybridism and multiplicity and disdain stylistic coherence, in keeping with the fashionable interest in collectivity, lack of ego, the fluidity of individual identity. But too often these avoidance tactics eliminate the thread of a personal sensibility or focus.
I would call all these strategies fear of form, which can be parsed as fear of materials, of working with the hands in an overt way and of originality. Most of all originality. Can we just say it? This far from Andy Warhol and Duchamp, the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.
Fear of form above all means fear of compression— of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible. With a few exceptions, forms of collage and assemblage dominate this show: the putting together (or simply putting side by side) of existing images and objects prevails. The consistency of this technique in two and three dimensions should have been a red flag for the curators. Collage has driven much art since the late 1970's. Lately, and especially in this exhibition, it often seems to have become so distended and pulled apart that its components have become virtually autonomous and unrelated, which brings us back to square one. This is most obvious in the large installations of graphic works whose individual parts gain impact and meaning from juxtaposition but are in fact considered distinct artworks."
Margaret Atwood on art and the trickster—
"The pleasures of fabulation, the charming and playful lie— this line of thought leads Hyde* to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning 'to join,' 'to fit,' and 'to make.' If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart."
* Lewis Hyde, Trickster Makes This World: Mischief, Myth, and Art, Farrar Straus & Giroux, January 1998
Smith mentions "an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."
Atwood mentions "a seamless whole."
For some related remarks, see "A Study in Art Education" and the central figure pictured above. (There "N" can stand for "number," "nine," or "narrative.")
Yesterday's excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.
That approach will appeal to few mathematicians, so here is another.
Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace is a book by Leonard Mlodinow published in 2002.
More recently, Mlodinow is the co-author, with Stephen Hawking, of The Grand Design (published on September 7, 2010).
A review of Mlodinow's book on geometry—
"This is a shallow book on deep matters, about which the author knows next to nothing."
— Robert P. Langlands, Notices of the American Mathematical Society, May 2002
The Langlands remark is an apt introduction to Mlodinow's more recent work.
It also applies to Martin Gardner's comments on Galois in 2007 and, posthumously, in 2010.
For the latter, see a Google search done this morning—
Here, for future reference, is a copy of the current Google cache of this journal's "paged=4" page.
Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron's web journal. Following the link, we find…
For n=4, there is only one factorisation, which we can write concisely as 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S4, and acts as S3 on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.
This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.
See also, in this journal, Window and Window, continued (July 5 and 6, 2010).
Gardner scoffs at the importance of Galois's last letter —
"Galois had written several articles on group theory, and was
merely annotating and correcting those earlier published papers."
— Last Recreations, page 156
For refutations, see the Bulletin of the American Mathematical Society in March 1899 and February 1909.
There is a remarkable correspondence between the 35 partitions of an eight-element set H into two four-element sets and the 35 partitions of the affine 4-space L over GF(2) into four parallel four-point planes. Under this correspondence, two of the H-partitions have a common refinement into 2-sets if and only if the same is true of the corresponding L-partitions (Peter J. Cameron, Parallelisms of Complete Designs, Cambridge U. Press, 1976, p. 60). The correspondence underlies the isomorphism* of the group A8 with the projective general linear group PGL(4,2) and plays an important role in the structure of the large Mathieu group M24.
A 1954 paper by W.L. Edge suggests the correspondence should be named after E.H. Moore. Hence the title of this note.
Edge says that
It is natural to ask what, if any, are the 8 objects which undergo
permutation. This question was discussed at length by Moore…**.
But, while there is no thought either of controverting Moore's claim to
have answered it or of disputing his priority, the question is primarily
a geometrical one….
Excerpts from the Edge paper—
Excerpts from the Moore paper—
Pages 432, 433, 434, and 435, as well as the section mentioned above by Edge— pp. 438 and 439
* J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford U. Press, 1985, p. 72
** Edge cited "E.H. Moore, Math. Annalen, 51 (1899), 417-44." A more complete citation from "The Scientific Work of Eliakim Hastings Moore," by G.A. Bliss, Bull. Amer. Math. Soc. Volume 40, Number 7 (1934), 501-514— E.H. Moore, "Concerning the General Equations of the Seventh and Eighth Degrees," Annalen, vol. 51 (1899), pp. 417-444.
From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—
Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—
Peterson associates the four-diamond figure
with the Pythagorean theorem.
A more relevant association is the
four-diamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—
This figure is relevant because of a
tesseract sculpture by Peter Forakis—
This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008-January 18, 2009.
The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
Non-Euclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).
For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).
"There is such a thing
as a tesseract."
— A Wrinkle in TIme
The Principle of Sufficient Reason
from "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940
EXCERPT 1—
My primary purpose will be to show how a properly formulated
Principle of Sufficient Reason plays a fundamental
role in scientific thought and, furthermore, is to be regarded
as of the greatest suggestiveness from the philosophic point
of view.2
In the preceding lecture I pointed out that three branches
of philosophy, namely Logic, Aesthetics, and Ethics, fall
more and more under the sway of mathematical methods.
Today I would make a similar claim that the other great
branch of philosophy, Metaphysics, in so far as it possesses
a substantial core, is likely to undergo a similar fate. My
basis for this claim will be that metaphysical reasoning always
relies on the Principle of Sufficient Reason, and that
the true meaning of this Principle is to be found in the
“Theory of Ambiguity” and in the associated mathematical
“Theory of Groups.”
If I were a Leibnizian mystic, believing in his “preestablished
harmony,” and the “best possible world” so
satirized by Voltaire in “Candide,” I would say that the
metaphysical importance of the Principle of Sufficient Reason
and the cognate Theory of Groups arises from the fact that
God thinks multi-dimensionally3 whereas men can only
think in linear syllogistic series, and the Theory of Groups is
2 As far as I am aware, only Scholastic Philosophy has fully recognized and ex-
ploited this principle as one of basic importance for philosophic thought
3 That is, uses multi-dimensional symbols beyond our grasp.
______________________________________________________________________
the appropriate instrument of thought to remedy our deficiency
in this respect.
The founder of the Theory of Groups was the mathematician
Evariste Galois. At the end of a long letter written in
1832 on the eve of a fatal duel, to his friend Auguste
Chevalier, the youthful Galois said in summarizing his
mathematical work,4 “You know, my dear Auguste, that
these subjects are not the only ones which I have explored.
My chief meditations for a considerable time have been
directed towards the application to transcendental Analysis
of the theory of ambiguity. . . . But I have not the time, and
my ideas are not yet well developed in this field, which is
immense.” This passage shows how in Galois’s mind the
Theory of Groups and the Theory of Ambiguity were
interrelated.5
Unfortunately later students of the Theory of Groups
have all too frequently forgotten that, philosophically
speaking, the subject remains neither more nor less than the
Theory of Ambiguity. In the limits of this lecture it is only
possible to elucidate by an elementary example the idea of a
group and of the associated ambiguity.
Consider a uniform square tile which is placed over a
marked equal square on a table. Evidently it is then impossible
to determine without further inspection which one
of four positions the tile occupies. In fact, if we designate
its vertices in order by A, B, C, D, and mark the corresponding
positions on the table, the four possibilities are for the
corners A, B, C, D of the tile to appear respectively in the
positions A, B, C, D; B, C, D, A; C, D, A, B; and D, A, B, C.
These are obtained respectively from the first position by a
4 My translation.
5 It is of interest to recall that Leibniz was interested in ambiguity to the extent
of using a special notation v (Latin, vel ) for “or.” Thus the ambiguously defined
roots 1, 5 of x2-6x+5=0 would be written x = l v 5 by him.
______________________________________________________________________
null rotation ( I ), by a rotation through 90° (R), by a rotation
through 180° (S), and by a rotation through 270° (T).
Furthermore the combination of any two of these rotations
in succession gives another such rotation. Thus a rotation R
through 90° followed by a rotation S through 180° is equivalent
to a single rotation T through 270°, Le., RS = T. Consequently,
the "group" of four operations I, R, S, T has
the "multiplication table" shown here:
This table fully characterizes the group, and shows the exact
nature of the underlying ambiguity of position.
More generally, any collection of operations such that
the resultant of any two performed in succession is one of
them, while there is always some operation which undoes
what any operation does, forms a "group."
__________________________________________________
EXCERPT 2—
Up to the present point my aim has been to consider a
variety of applications of the Principle of Sufficient Reason,
without attempting any precise formulation of the Principle
itself. With these applications in mind I will venture to
formulate the Principle and a related Heuristic Conjecture
in quasi-mathematical form as follows:
PRINCIPLE OF SUFFICIENT REASON. If there appears
in any theory T a set of ambiguously determined ( i e .
symmetrically entering) variables, then these variables can themselves
be determined only to the extent allowed by the corresponding
group G. Consequently any problem concerning these variables
which has a uniquely determined solution, must itself be
formulated so as to be unchanged by the operations of the group
G ( i e . must involve the variables symmetrically).
HEURISTIC CONJECTURE. The final form of any
scientific theory T is: (1) based on a few simple postulates; and
(2) contains an extensive ambiguity, associated symmetry, and
underlying group G, in such wise that, if the language and laws
of the theory of groups be taken for granted, the whole theory T
appears as nearly self-evident in virtue of the above Principle.
The Principle of Sufficient Reason and the Heuristic Conjecture,
as just formulated, have the advantage of not involving
excessively subjective ideas, while at the same time
retaining the essential kernel of the matter.
In my opinion it is essentially this principle and this
conjecture which are destined always to operate as the basic
criteria for the scientist in extending our knowledge and
understanding of the world.
It is also my belief that, in so far as there is anything
definite in the realm of Metaphysics, it will consist in further
applications of the same general type. This general conclu-
sion may be given the following suggestive symbolic form:
While the skillful metaphysical use of the Principle must
always be regarded as of dubious logical status, nevertheless
I believe it will remain the most important weapon of the
philosopher.
___________________________________________________________________________
A more recent lecture on the same subject —
by Jean-Pierre Ramis (Johann Bernoulli Lecture at U. of Groningen, March 2005)
It is well known that the seven
Similarly, recent posts* have noted that the thirteen
These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)
A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."
Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).
The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…
See also Geometry of the I Ching and a search in this journal for
* February 27 and March 13
** G20160 in Mitchell 1910, LF(3,22) in Edge 1965
— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
of the Finite Projective Plane PG(2,22),"
Princeton Ph.D. dissertation (1910)
— Edge, W. L., "Some Implications of the Geometry of
the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)
Stanley Fish reviewed a new book, Steven Smith's The Disenchantment of Secular Discourse, in yesterday's online NY Times—
…the self-impoverished discourse of secular reason does in fact produce judgments, formulate and defend agendas, and speak in a normative vocabulary. How is this managed? By “smuggling,” Smith answers.
. . . the secular vocabulary within which public discourse is constrained today is insufficient to convey our full set of normative convictions and commitments. We manage to debate normative matters anyway— but only by smuggling in notions that are formally inadmissible, and hence that cannot be openly acknowledged or adverted to.
The notions we must smuggle in, according to Smith, include “notions about a purposive cosmos, or a teleological nature stocked with Aristotelian ‘final causes’ or a providential design,” all banished from secular discourse because they stipulate truth and value in advance rather than waiting for them to be revealed by the outcomes of rational calculation. But if secular discourse needs notions like these to have a direction— to even get started— “we have little choice except to smuggle [them] into the conversations— to introduce them incognito under some sort of secular disguise.”
And how do we do that?
A Jewish Answer
By the Coen brothers in "A Serious Man"–
"When the truth is found to be lies
And all the joy within you dies…."
A Christian answer
Disenchantment author Steven Smith is a a professor at the University of San Diego. This suggests a look at the feast day of San Diego himself… Here are Log24 posts that mention that day, November 12 (which is also Grace Kelly's birthday).
New York Times online front page
Christmas morning:
"Arthur Koestler, Man of Darkness"–
The photo is of Koestler in 1931 on a zeppelin expedition to the North Pole.
"The Act of Creation is, I believe, a more truly creative work than any of Koestler’s novels…. According to him, the creative faculty in whatever form is owing to a circumstance which he calls ‘bisociation.’ And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or ’see,’ for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes.”
– Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964
From Opus Postumum by Immanuel Kant, Eckart Förster, Cambridge U. Press, 1995, p. 260:
"In January 1697, Leibniz accompanied his New Year Congratulations to Rudolf August with the design of a medal with the duke's likeness on one side, and the 'image of Creation' in terms of the binary number system on the other. Concerning the inscription on this side, Leibniz writes: 'I have thought for a while about the Motto dell'impresa and finally have found it good to write this line: omnibus ex nihilo ducendis SUFFICIT UNUM [To make all things from nothing, UNITY SUFFICES], because it clearly indicates what is meant by the symbol, and why it is imago creationis' (G. F. Leibniz, Zwei Briefe über das binäre Zahlensystem und die chinesische Philosophie, ed. Renate Loosen and Franz Vonessen, Chr. Belser Verlag: Stuttgart 1968, p. 21)."
Figure from Rudolf Nolte’s
Gottfried Wilhelms Baron von Leibniz
Mathematischer Beweis der Erschaffung und
Ordnung der Welt in einem Medallion…
(Leipzig: J. C. Langenheim, 1734).
"And so that I won’t come entirely empty-handed this time, I enclose a design of that which I had the pleasure of discussing with you recently. It is in the form of a memorial coin or medallion; and though the design is mediocre and can be improved in accordance with your judgment, the thing is such, that it would be worth showing in silver now and unto future generations, if it were struck at your Highness’s command. Because one of the main points of the Christian Faith, and among those points that have penetrated least into the minds of the worldly-wise and that are difficult to make with the heathen is the creation of all things out of nothing through God’s omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing. And it would be difficult to find a better illustration of this secret in nature or philosophy; hence I have set on the medallion design IMAGO CREATIONIS [in the image of creation]. It is no less remarkable that there appears therefrom, not only that God made everything from nothing, but also that everything that He made was good; as we can see here, with our own eyes, in this image of creation. Because instead of there appearing no particular order or pattern, as in the common representation of numbers, there appears here in contrast a wonderful order and harmony which cannot be improved upon….
Such harmonious order and beauty can be seen in the small table on the medallion up to 16 or 17; since for a larger table, say to 32, there is not enough room. One can further see that the disorder, which one imagines in the work of God, is but apparent; that if one looks at the matter with the proper perspective, there appears symmetry, which encourages one more and more to love and praise the wisdom, goodness, and beauty of the highest good, from which all goodness and beauty has flowed."
See also Parable.
From Peter J. Cameron's
Parallelisms of Complete Designs (pdf)–
"…the Feast of Nicholas Ferrar
is kept on the 4th December."
Cameron's is the usual definition
of the term "non-Euclidean."
I prefer a more logical definition.
The New York Times
on June 17, 2007:
Design Meets Dance,
and Rules Are Broken
Yesterday's evening entry was
on the fictional sins of a fictional
mathematician and also (via a link
to St. Augustine's Day, 2006), on
the geometry of the I Ching* —
The eternal
combined with
the temporal:
The fictional mathematician's
name, noted here (with the Augustine-
I Ching link as a gloss) in yesterday's
evening entry, was Summerfield.
From the above Times article–
"Summerspace," a work by
choreographer Merce Cunningham
and artist Robert Rauschenberg
that offers a competing
vision of summer:
From left, composer John Cage,
choreographer Merce Cunningham,
and artist Robert Rauschenberg
in the 1960's
"When shall we three meet again?"
And then there are
gemlike numbers
set free from words…
Today’s New York lottery:
These numbers also
name parts of a book
cited here Nov. 6, 2007:
|
… The actor is — Wallace Stevens in |
New York Times
banner this morning:
Related material from
July 11, 2008:
|
The HSBC Logo Designer — Henry Steiner
Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors' Club. His Cross-Cultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995). |
|
Charles Taylor,
"Epiphanies of Modernism," Chapter 24 of Sources of the Self (Cambridge U. Press, 1989, p. 477):
"… the object sets up
See also Talking of Michelangelo.
|
Related material suggested by
an ad last night on
ABC's Ugly Betty season finale:
Diamond from last night's
Log24 entry, with
four colored pencils from
Diane Robertson Design:
See also
A Four-Color Theorem.
"By far the most important structure in design theory is the Steiner system
— "Block Designs," 1995, by Andries E. Brouwer
"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24."
— The Miracle Octad Generator (MOG) of R.T. Curtis (webpage)
"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a little-known 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."
The 1931 paper of Carmichael is now available online from the publisher for $10.
Solomon's Cube
continued
"There is a book… called A Fellow of Trinity, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."
— G. H. Hardy, A Mathematician's Apology
"The Solomon Key is the working title of an unreleased novel in progress by American author Dan Brown. The Solomon Key will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were Angels & Demons (2000) and The Da Vinci Code (2003)." — Wikipedia
"One has O+(6) ≅ S8, the symmetric group of order 8! …."
— "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.
"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."
— George M. Conwell, "The 3-space PG(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
| Project MUSE — … and interpretations, “any of the Zingari shoolerim [gypsy schoolchildren] may pick a peck of kindlings yet from the sack of auld hensyne” (FW 112.4-8). … — Patrick McGee, “Reading Authority: |
“The ulterior motive behind this essay [“Reading Authority,” above], the purpose for which I seize this occasion, concerns the question or problem of authority. I stress at the outset my understanding of authority as the constructed repository of value or foundation of a system of values, the final effect of fetishism– in this case, literary fetishism. [Cf. Marx, Das Kapital] Reading– as in the phrase ‘reading authority’– should be grasped as the institutionally determined act of constructing authority….”
“[In Peter Pan] Smee is Captain Hook’s right-hand man… Barrie describes him as ‘Irish’ and ‘a man who stabbed without offence‘….”
Background: In yesterday’s morning entry, James Joyce as Jesuit, with “Dagger Definitions.”
A different Smee appears as an art critic in yesterday’s afternoon entry “Design Theory.”–
“Brock, who has a brisk mind, is a man on a mission. He read mathematical economics and political philosophy at Princeton (he has five degrees in all) and is the founder and president of Strategic Economic Decisions Inc., a think tank specializing in applying the economics of uncertainty to forecasting and risk assessment.
But phooey to all that; Brock has deeper things to think about. He believes he has cracked the secret of beautiful design. He even has equations and graphs to prove it.”
A Jesuit in Portrait of the Artist as a Young Man:
“When may we expect to have something from you on the esthetic question?”
“Our entanglement in the wilderness of Finnegans Wake is exemplified by the neologism ‘Bethicket.’ This word condenses a range of possible meanings and reinforces a diversity of possible syntactic interpretations. Joyce seems to allude to Beckett, creating a portmanteau word that melds ‘Beckett’ with ‘thicket’ (continuing the undergrowth metaphor), ‘thick’ (adding mental density to floral density)…. As a single word ‘Bethicket’ contains the confusion that its context suggests. On the one hand, ‘Bethicket me for a stump of a beech’ has the sound of a proverbial expletive that might mean something like ‘I’ll be damned’ or ‘Well, I’ll be a son of a gun.’….”
At the Oscars, 2009
Related material:Frame Tales and Dickung
Through the
Looking Glass:
A Sort of Eternity
From the new president’s inaugural address:
“… in the words of Scripture, the time has come to set aside childish things.”
The words of Scripture:
“through a glass”—
[di’ esoptrou].
By means of
a mirror [esoptron].
Childish things:
© 2005 The Institute for Figuring
Not-so-childish:
Three planes through
the center of a cube
that split it into
eight subcubes:
Through a glass, darkly:
A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
that is symmetric under one of
these 8 transformations–
a 180-degree rotation:
(Click on image
for further details.)
But then face to face:
A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.
Galois age fifteen,
drawn by a classmate.
These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.
For some generalizations,
see Galois Geometry.
Related material:
The central aim of Western religion–
"Each of us has something to offer the Creator... the bridging of masculine and feminine, life and death. It's redemption.... nothing else matters." -- Martha Cooley in The Archivist (1998) The central aim of Western philosophy– Dualities of Pythagoras as reconstructed by Aristotle: Limited Unlimited Odd Even Male Female Light Dark Straight Curved ... and so on .... “Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy.” — Jamie James in The Music of the Spheres (1993) “In the garden of Adding — The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000) A quotation today at art critic Carol Kino’s website, slightly expanded: “Art inherited from the old religion — Octavio Paz,”Seeing and Using: Art and Craftsmanship,” in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52 From Brian O’Doherty’s 1976 Artforum essays– not on museums, but rather on gallery space: “We have now reached
“Space: what you — James Joyce, Ulysses |
Something Traditional —
“German Chancellor Dr. Angela Merkel is the Charlemagne Prize laureate of 2008…. The prize will be awarded on 1 May, Ascension Day.”
Something Modern —
Previously undescribed in this journal:
| A NOTE BY THE DESIGNER
“The traditional chess set, with its naturalistic images of medieval armies, suggests a game between combatants who enjoy the winning of battles. This chess set, with its articulated images of abstract force, suggests a game between contestants who enjoy the process of thinking.
The primary principle of this design… is that the operating reality or function of each piece– both its value and how it moves– is embodied in a simple self-expressive form….  Design Copyright F. Lanier Graham 1967
— F. Lanier Graham, 1967 |
Related material: this journal on the First of May, 2008, the date of last year’s Charlemagne award.
Robert Stone,
"'That old Jew gave me this here.' Egan looked at the diamond. 'I ain't giving this to you, understand? The old man gave it to me for my boy. It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'
'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal. The Boddhisattva declining nirvana out of compassion. Contemplating the ignorance of you and me, eh? That's a metaphor of our Buddhist friends.' Pablo's eyes glazed over. 'Holy shit,' he said. 'Santa Maria.' He stared at the diamond in his palm with passion." For further details, click on the diamond. |
Today's online Times on
the Saturday, Dec. 27,
death of an artist:
Mr. Wasserman wrote more than 75 scripts for television, the stage and the movies, including screenplays for 'The Vikings' (1958), a seafaring epic with Tony Curtis and Kirk Douglas, and 'A Walk With Love and Death' (1969), a John Huston film set in 14th-century Europe….
He feuded with… John Huston, who gave the lead female role in 'Walk' to his teenage daughter, Anjelica, against Mr. Wasserman's wishes. And he never attended ceremonies to receive the awards he won."
Accepting for Mr. Wasserman:
Mr. Graham's widow,
Anjelica Huston —
"Well…"
The Kohs Block Design
Intelligence Test
Samuel Calmin Kohs, the designer (but not the originator) of the above intelligence test, would likely disapprove of the "Aryan Youth types" mentioned in passing by a film reviewer in today's New York Times. (See below.) The Aryan Youth would also likely disapprove of Dr. Kohs.
1. Wechsler Cubes (intelligence testing cubes derived from the Kohs cubes shown above). See…
Harvard psychiatry and…
The Montessori Method;
The Crimson Passion;
The Lottery Covenant.
2. Wechsler Cubes of a different sort (Log24, May 25, 2008)
3. Manohla Dargis in today's New York Times:
"… 'Momma’s Man' is a touchingly true film, part weepie, part comedy, about the agonies of navigating that slippery slope called adulthood. It was written and directed by Azazel Jacobs, a native New Yorker who has set his modestly scaled movie with a heart the size of the Ritz in the same downtown warren where he was raised. Being a child of the avant-garde as well as an A student, he cast his parents, the filmmaker Ken Jacobs and the artist Flo Jacobs, as the puzzled progenitors of his centerpiece, a wayward son of bohemia….
In American movies, growing up tends to be a job for either Aryan Youth types or the oddballs and outsiders…."
"… I think that the deeper opportunity, the greater opportunity film can offer us is as an exercise of the mind. But an exercise, I hate to use the word, I won't say 'soul,' I won't say 'soul' and I won't say 'spirit,' but that it can really put our deepest psychological existence through stuff. It can be a powerful exercise. It can make us think, but I don't mean think about this and think about that. The very, very process of powerful thinking, in a way that it can afford, is I think very, very valuable. I basically think that the mind is not complete yet, that we are working on creating the mind. Okay. And that the higher function of art for me is its contribution to the making of mind."
— Interview with Ken Jacobs, UC Berkeley, October 1999
5. For Dargis's "Aryan Youth types"–
From a Manohla Dargis
New York Times film review
of April 4, 2007
(Spy Wednesday) —
See also, from August 1, 2008
(anniversary of Hitler's
opening the 1936 Olympics) —
For Sarah Silverman —
and the 9/9/03 entry
Doonesbury,
August 21-22, 2008:
Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein’s birthday):
The play’s title, “Every Good Boy Deserves Favour,” is a mnemonic for the notes of the treble clef EGBDF.
The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.
The “u” in favour is the British way, the Stoppard way, “EGBDF” being “a Play for Actors and Orchestra” by Tom Stoppard (words) and André Previn (music).
And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.
When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard “to write something which had the need of a live full-time orchestra onstage,” the 36-year-old playwright jumped at the chance.
One hitch: Stoppard at the time knew “very little about ‘serious’ music… My qualifications for writing about an orchestra,” he says in his introduction to the 1978 Grove Press edition of “EGBDF,” “amounted to a spell as a triangle player in a kindergarten percussion band.”
Review of the same play as presented at Chautauqua Institution on July 24, 2008:
“Stoppard’s modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience.”
— Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008
“The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul.”
— Dan Fogelberg
“He’s watching us all the time.”
| Finnegans Wake, Book II, Episode 2, pp. 296-297: I’ll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you’d wheeze whyse Salmonson set his seel on a hexengown.1 Hissss!, Arrah, go on! Fin for fun! 1 The chape of Doña Speranza of the Nacion. |
“…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity…. … E. M. Forster famously advised his readers, ‘Only connect.’ ‘Reciprocity’ would be Michael Kruger’s succinct philosophy, with all that the word implies.” — William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994 Last year’s entry on this date:
The picture above is of the complete graph K6 … Six points with an edge connecting every pair of points… Fifteen edges in all. Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester’s synthematic totals as they relate to constructions of the Mathieu group M24. If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. “Reciprocity” in the sense of Lao Tzu. See Reciprocity and Reversal in Lao Tzu. For a sense of “reciprocity” more closely related to Michael Kruger’s alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in Kruger’s novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:
Click on the design for details. Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in A Graphical Representation The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.
Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See |
“Finn MacCool ate the Salmon of Knowledge.”
Wikipedia:
“George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest.”
Related material:
Kindergarten Theology,
Arrangements for
56 Triangles.
For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.
Happy birthday,
Martin Sheen.
|
The HSBC Logo Designer — Henry Steiner
Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors' Club. His Cross-Cultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995). |
Charles Taylor,
"… the object sets up
See also Talking of Michelangelo.
|
Related material
from today —
Escape from a
cartoon graveyard:
| CHANGE FEW CAN BELIEVE IN |
Page 67 —
“… Bill and Violet were married. The wedding was held in the Bowery loft on June 16th, the same day Joyce’s Jewish Ulysses had wandered around Dublin. A few minutes before the exchange of vows, I noted that Violet’s last name, Blom, was only an o away from Bloom, and that meaningless link led me to reflect on Bill’s name, Wechsler, which carries the German root for change, changing, and making change. Blooming and changing, I thought.”
For Hustvedt’s discussion of Wechsler’s art– sculptured cubes, which she calls “tightly orchestrated semantic bombs” (p. 169)– see Log24, May 25, 2008.
(after David Wechsler,
1896-1981, chief
psychologist at Bellevue)
These cubes are used to
make 3×3 patterns for
psychological testing.
Related 3×3 patterns appear
in “nine-patch” quilt blocks
and in the following–
| Don Park at docuverse.com, Jan. 19, 2007: “How to draw an Identicon
A 9-block is a small quilt using only 3 types of patches, out of 16 available, in 9 positions. Using the identicon code, 3 patches are selected: one for center position, one for 4 sides, and one for 4 corners. Positions and Rotations For center position, only a symmetric patch is selected (patch 1, 5, 9, and 16). For corner and side positions, patch is rotated by 90 degree moving clock-wise starting from top-left position and top position respectively.” |
| From a weblog by Scott Sherrill-Mix:
“… Don Park came up with the original idea for representing users with geometric shapes….” Claire | 20-Dec-07 at 9:35 pm | Permalink “This reminds me of a flash demo by Jarred Tarbell |
| Jared Tarbell at levitated.net, May 15, 2002:
“The nine block is a common design pattern among quilters. Its construction methods and primitive building shapes are simple, yet produce millions of interesting variations.  Figure A. Four 9 block patterns,
arbitrarily assembled, show the grid composition of the block. Each block is composed of 9 squares, arranged in a 3 x 3 grid. Each square is composed of one of 16 primitive shapes. Shapes are arranged such that the block is radially symmetric. Color is modified and assigned arbitrarily to each new block.
The basic building blocks of the nine block are limited to 16 unique geometric shapes. Each shape is allowed to rotate in 90 degree increments. Only 4 shapes are allowed in the center position to maintain radial symmetry.
 Figure B. The 16 possible shapes allowed |
Such designs become of mathematical interest when their size is increased slightly, from square arrays of nine blocks to square arrays of sixteen. See Block Designs in Art and Mathematics.
(This entry was suggested by examples of 4×4 Identicons in use at Secret Blogging Seminar.)
(For readers of
the previous entry
who would like to
know more about
purchasing the
Brooklyn Bridge)
“He was a stickler for spelling, insisting that students accurately compose dictated sentences, like this one: ‘Outside a cemetery sat a harassed cobbler and an embarrassed peddler, gnawing on a desiccated potato and gazing on the symmetry of a lady’s ankle with unparalleled ecstasy.'”
| How the ancient crystal skull Synergy came to the Western World…
This skull first came to light when it was acquired about two and a half decades ago by a European businessman and avid hiker, as he traveled around Central and South America. He acquired the skull from a very old native man, in a tiny village in the Andes, near the borders of Peru, Bolivia and Chile. He was just passing through, and had come upon the small settlement while looking for a place to stay for the night. He wandered into the village and was greeted with smiles and an invitation to share a meal. This gentleman, George, speaks several languages, and he usually has at least a few words in common with most of the people he meets in his travels– enough to get by, anyway. Although he didn’t speak the same language as most of the people in this isolated village, there was an instant connection between them, and they managed with the smattering of Spanish and Portuguese that a few of them knew. In need of shelter for the night, George was offered a spot for his sleeping bag, near the fire, in the dwelling of an elderly man. After a peaceful evening in the old man’s company, George gratefully accepted a simple breakfast and got ready to take his leave. As he thanked the man for his generous hospitality, the elder led George to an old chest. Opening the crumbling wooden lid, he took out the crystal skull, touched it reverently, and handed it to George. Awed by an artifact of such obvious antiquity, beauty and value, yet uncertain what he was expected to do with it, George tried to hand it back. But the old man urged it upon him, making it clear that he was to take it with him. Curious about the history of such a thing, George tried to find out what the villagers knew about it. One young fellow explained in halting Spanish that the skull had come into the possession of a much loved Catholic nun, in Peru. She was quite old when she died in the early 1800’s, and she had given it to the old man’s “Grandfather” when he was just a boy. (Note: It’s hard to say if this was really the man’s grandfather, or just the honorary title that many natives use to designate an ancestor or revered relative.) The nun told the boy and his father that the skull was “an inheritance from a lost civilization” and, like the Christian cross, it was a symbol of the transcendence of Soul over death. She said that it carried the message of immortal life and the illumination that we may discover when we lose our fear of death. She gave it to the boy and his father, asking them to safeguard it until the “right” person came to get it– and share its message with the world. It had been brought to that land from “somewhere else” and needed to wait until the right person could help it to continue its journey. “Your heart will know the person,” she said. “What a strange story,” thought George. |
“… ‘Supercholita’ tiene sobre todo una clara vocación divulgadora de la cultura andina. No en vano Valdez recibió su primer premio por explicar mediante este personaje cómo se cocina el ‘chuño,’ una típica patata deshidratada muy consumida en el altiplano boliviano.”
astragalo-, astragal-
(Greek: anklebone, talus ball of ankle joint; dice, die [the Greeks made these from ankle bones])
… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte
— Rubén Darío
Image adapted from
Blue Star Traders
The New York Lottery
mid-day number yesterday–
719– and 7/19.
On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings. Here is a related commentary on a different incarnation of 441. (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)
From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–
“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the n – m elements not in S. Then show that G permutes the m elements S into
m(m – 1) … (m – k + 1)
This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.
For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.
“Well, it changes.”
A related Log24 link from
that same date, November 27:
“Plato hadn’t tried to destroy areté. He had encapsulated it; made a permanent, fixed Idea out of it; had converted it to a rigid, immobile Immortal Truth. He made areté the Good, the highest form, the highest Idea of all. It was subordinate only to Truth itself, in a synthesis of all that had gone before.That was why the Quality that Phaedrus had arrived at in the classroom had seemed so close to Plato’s Good. Plato’s Good was taken from the rhetoricians. Phaedrus searched, but could find no previous cosmologists who had talked about the Good. That was from the Sophists. The difference was that Plato’s Good was a fixed and eternal and unmoving Idea, whereas for the rhetoricians it was not an Idea at all. The Good was not a form of reality. It was reality itself, ever changing, ultimately unknowable in any kind of fixed, rigid way.”
— as well as Cold Mountain —
“So in short order Ada found herself bent backward over the mossy well lip, canted in a pose with little to recommend it in the way of dignity or comfort, back arched, hips forward, legs spraddled for balance. She held a hand mirror above her face, angled to catch the surface of the water below.
Ada had agreed to the well-viewing as a variety of experiment in local custom and as a tonic for her gloom. Her thoughts had been broody and morbid and excessively retrospective for so long that she welcomed the chance to run counter to that flow, to cast forward and think about the future, even though she expected to see nothing but water at the bottom of the well.
She shifted her feet to find better grip on the packed dirt of the yard and then tried to look into the mirror. The white sky above was skimmed over with backlit haze, bright as a pearl or as a silver mirror itself. The dark foliage of oaks all around the edges framed the sky, duplicating the wooden frame of the mirror into which Ada peered, examining its picture of the well depths behind her to see what might lie ahead in her life. The bright round of well water at the end of the black shaft was another mirror. It cast back the shine of sky and was furred around the edges here and there with sprigs of fern growing between stones.
Ada tried to focus her attention on the hand mirror, but the bright sky beyond kept drawing her eye away. She was dazzled by light and shade, by the confusing duplication of reflections and of frames. All coming from too many directions for the mind to take account of. The various images bounced against each other until she felt a desperate vertigo, as if she could at any moment pitch backward and plunge head first down the well shaft and drown there, the sky far above her, her last vision but a bright circle set in the dark, no bigger than a full moon.
Her head spun and she reached with her free hand and held to the stonework of the well. And then just for a moment things steadied, and there indeed seemed to be a picture in the mirror.”
— and Log24 on December 3 —
Wallace Stevens,
opening lines of
The Necessary Angel:
We recognize at once, in this figure, Plato’s pure poetry; and at the same time we recognize what Coleridge called Plato’s dear, gorgeous nonsense. The truth is that we have scarcely read the passage before we have identified ourselves with the charioteer, have, in fact, taken his place and, driving his winged horses, are traversing the whole heaven.”
Stevens, who was educated at Harvard, adds:
“Then suddenly we remember, it may be, that the soul no longer exists and we droop in our flight and at last settle on the solid ground. The figure becomes antiquated and rustic.”
Many who lack a Harvard education to make them droop will prefer to remember Robert Craig Knievel (Oct. 17, 1938 – Nov. 30, 2007) not as antiquated and rustic but as young and soaring.
See also the entries for
last February’s
Academy Awards night:
Hollywood Sermon and
Between Two Worlds.
“Yo es que nací un 8 de marzo,
Día de la Mujer Trabajadora,
y no he hecho más que
trabajar toda mi vida.”
For background on Aldecoa,
see a paper (pdf) by
Sara Brenneis:
“Josefina Aldecoa intertwines
history, collective memory
and individual testimony in her
historical memory trilogy…”
The Triangle Shirtwaist Factory Fire in New York City on March 25, 1911, was the largest industrial disaster in the history of the city of New York, causing the death of 146 garment workers who either died in the fire or jumped to their deaths.
Propaganda, March 1977:
“On March 8, 1908, after the death of 128 women trapped in a fire at the Triangle Shirtwaist Factory in New York City, 15,000 women workers from the garment and textile industry marched echoing the demands of their sisters 50 years earlier…”
Propaganda, March 2006:
“First of all, on March 8th, 1857, a large number of factory workers in the United States took to the streets to demand their economic and political rights. The owners called the police who arrived immediately and opened fire, engaging in blind repression… Later on, in 1908, the same date of March 8th was once again a memorable date of struggle. On this day, capitalist bosses in Chicago set fire to a textile factory where over a thousand women worked. A very large number was terribly burnt. 120 died!”
Propaganda disguised as news, March 2007:
From today’s top story in 24 HoursTM, a commuter daily in Vancouver published by Sun Media Corporation:
Fight still on for equality
By Robyn Stubbs and Carly Krug
“International Women’s Day commemorates a march by female garment workers protesting low wages, 12-hour workdays and bad working conditions in New York City on March 8, 1857.
Then in 1908, after 128 women were trapped and killed in a fire at a New York City garment and textile factory, 15,000 women workers again took their protests to the street.”
Related historical fiction:
A version of the
I Ching’s Hexagram 19:
— Katherine Neville, The Eight
|
“What does this have to do with why we’re here?”
“I saw it in a chess book Mordecai showed me. The most ancient chess service ever discovered was found at the palace of King Minos on Crete– the place where the famous Labyrinth was built, named after this sacred axe. The chess service dates to 2000 B.C. It was made of gold and silver and jewels…. And in the center was carved a labrys.” … “But I thought chess wasn’t even invented until six or seven hundred A.D.,” I added. “They always say it came from Persia or India. How could this Minoan chess service be so old?” “Mordecai’s written a lot himself on the history of chess,” said Lily…. “He thinks that chess set in Crete was designed by the same guy who built the Labyrinth– the sculptor Daedalus….” Now things were beginning to click into place…. “Why was this axe carved on the chessboard?” I asked Lily, knowing the answer in my heart before she spoke. “What did Mordecai say was the connection?”…. “That’s what it’s all about,” she said quietly. “To kill the King.” The sacred axe was used to kill the King. The ritual had been the same since the beginning of time. The game of chess was merely a reenactment. Why hadn’t I recognized it before? |
Perhaps at the center of
Aldecoa’s labyrinth lurk the
capitalist bosses from Chicago
who, some say, set fire
to a textile factory
on this date in 1908.
For a Freudian perspective
on the above passage,
see yesterday’s entry
In the Labyrinth of Time,
with its link to
John Irwin‘s essay
“The False Artaxerxes:
Borges and the
Dream of Chess.”
Symbols
S. H. Cullinane
March 7, 2007
Today, by the way, is the
feast of a chess saint.
Today is the 21st birthday of my note “The Relativity Problem in Finite Geometry.”
Some relevant quotations:
“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
Describing the branch of mathematics known as Galois theory, Weyl says that it
“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”
— Weyl, Symmetry, Princeton University Press, 1952, p. 138
Weyl’s set Sigma is a finite set of complex numbers. Some other sets with “discrete and finite character” are those of 4, 8, 16, or 64 points, arranged in squares and cubes. For illustrations, see Finite Geometry of the Square and Cube. What Weyl calls “the relativity problem” for these sets involves fixing “objectively” a class of equivalent coordinatizations. For what Weyl’s “objectively” means, see the article “Symmetry and Symmetry Breaking,” by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy:
“The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), ‘objectivity means invariance with respect to the group of automorphisms [of space-time].‘[22]
22. The significance of the notion of invariance and its group-theoretic treatment for the issue of objectivity is explored in Born (1953), for example. For more recent discussions see Kosso (2003) and Earman (2002, Sections 6 and 7).
References:
Born, M., 1953, “Physical Reality,” Philosophical Quarterly, 3, 139-149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155-167.
Earman, J., 2002, “Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity,’ PSA 2002, Proceedings of the Biennial Meeting of the Philosophy of Science Association 2002, forthcoming [Abstract/Preprint available online]
Kosso, P., 2003, “Symmetry, objectivity, and design,” in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, Cambridge: Cambridge University Press, pp. 410-421.
Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton University Press.
See also
Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)–
“Minkowski, Mathematicians, and the Mathematical Theory of Relativity,” by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 45-86–
“Developing his ideas before Göttingen mathematicians in April 1909, Klein pointed out that the new theory based on the Lorentz group (which he preferred to call ‘Invariantentheorie’) could have come from pure mathematics (1910: 19). He felt that the new theory was anticipated by the ideas on geometry and groups that he had introduced in 1872, otherwise known as the Erlangen program (see Gray 1989: 229).”
References:
Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford University Press.
Klein, Felix. (1910). “Über die geometrischen Grundlagen der Lorentzgruppe.” Jahresbericht der deutschen Mathematiker-Vereinigung 19: 281-300. [Reprinted: Physikalische Zeitschrift 12 (1911): 17-27].
Related material: A pathetically garbled version of the above concepts was published in 2001 by Harvard University Press. See Invariances: The Structure of the Objective World, by Robert Nozick.
Bead Game
Those who clicked on Rieff’s concept in the previous entry will know about the book that Rieff titled Sacred Order/Social Order: My Life among the Deathworks.
That entry, from Tuesday, July 18, was titled “Sacred Order,” and gave as an example the following figure:
For the use of this same figure to represent a theatrical concept–
“It’s like stringing beads on a necklace. By the time the play ends, you have the whole necklace.”
— see Ursprache Revisited (June 9, 2006).
Of course, the figure also includes a cross– or “deathwork”– of sorts. These incidental social properties of the figure (which is purely mathematical in origin) make it a suitable memorial for a theatre critic who died on the date of the previous entry– July 18– and for whom the American Theatre Wing’s design awards, the Henry Hewes Awards, are named.
“The annual awards honor designers… recognizing not only the traditional design categories of sets, costumes and lighting, but also ‘Notable Effects,’ which encompasses sound, music, video, puppets and other creative elements.” —BroadwayWorld.com
For more on life among the deathworks, see an excellent review of the Rieff book mentioned above.
Point Counter Point
From the Encyclopaedia Britannica, 11th Edition, 1911:
COUNTERPOINT (Lat. contrapunctus, “point counter point,” “note against note”)
“In music, the art happily defined by Sir Frederick Gore Ouseley as that ‘of combining’ melodies….
Double Counterpoint is a combination of melodies so designed that either can be taken above or below the other. When this change of position is effected by merely altering the OCTAVE (from Lat. octavus, eighth, octo, eight) of either or both melodies (with or without transposition of the whole combination to another KEY), the artistic value of the device is simply that of the raising of the lower melody to the surface. The harmonic scheme remains the same, except in so far as some of the chords are not in their fundamental position, while others, not originally fundamental, have become so. But double counterpoint may be in other intervals than the octave; that is to say, while one of the parts remains stationary, the other may be transposed above or below it by some interval other than an octave, thus producing an entirely different set of harmonies.”
and the entry below.
Related material:
A Contrapuntal Theme.
— Dean G. Hoffman, Auburn U.,
July 2001 Rutgers talk
Diagrams from Dieter Betten’s 1983 proof
of the nonexistence of two orthogonal
6×6 Latin squares (i.e., a proof
of Tarry’s 1900 theorem solving
Euler’s 1782 problem of the 36 officers):
Compare with the partitions into
two 8-sets of the 4×4 Latin squares
discussed in my 1978 note (pdf).
1:00:19 EST
The Lion, the Witch
and the Wardrobe
premieres tonight at
the Royal Albert Hall.
Hexagram 19 in the
Cullinane series:
— Katherine Neville, The Eight
|
“What does this have to do with why we’re here?”
“I saw it in a chess book Mordecai showed me. The most ancient chess service ever discovered was found at the palace of King Minos on Crete– the place where the famous Labyrinth was built, named after this sacred axe. The chess service dates to 2000 B.C. It was made of gold and silver and jewels…. And in the center was carved a labrys.” … “But I thought chess wasn’t even invented until six or seven hundred A.D.,” I added. “They always say it came from Persia or India. How could this Minoan chess service be so old?” “Mordecai’s written a lot himself on the history of chess,” said Lily…. “He thinks that chess set in Crete was designed by the same guy who built the Labyrinth– the sculptor Daedalus….” Now things were beginning to click into place…. “Why was this axe carved on the chessboard?” I asked Lily, knowing the answer in my heart before she spoke. “What did Mordecai say was the connection?”…. “That’s what it’s all about,” she said quietly. “To kill the King.” The sacred axe was used to kill the King. The ritual had been the same since the beginning of time. The game of chess was merely a reenactment. Why hadn’t I recognized it before? |
|
“But what does it all mean?” asked Susan when they were somewhat calmer. “It means,” said Aslan, “that though the Witch knew the Deep Magic, there is a magic deeper still which she did not know. Her knowledge goes back only to the dawn of Time. But if she could have looked a little further back, into the stillness and the darkness before Time dawned, she would have read there a different incantation. She would have known that when a willing victim who had committed no treachery was killed in a traitor’s stead, the Table would crack and Death itself would start working backward.” |
On Saturday, November 5, 2005,
author John Fowles died.
 |
|
| From Log24 on the date of Fowles’ death: Coincidence and Design Related material: |
From Log24 on the date of Fowles’ death: Contrapuntal Themes in a Shadowland Related material: |
“Two years after The Collector had brought him international recognition and a year before he published The Magus, John Fowles set out his ideas on life in The Aristos. The chief inspiration behind them was the fifth century BC philosopher Heraclitus. In the world he posited of constant and chaotic flux the supreme good was the Aristos, ‘of a person or thing, the best or most excellent of its kind.'”
101 101
— Ben Macintyre,
The London Times, June 4:
When Rimbaud Meets Rambo
Classics Illustrated —
Click on picture for details.
(For some mathematics that is actually
from 1984, see Block Designs
and the 2005 followup
The Eightfold Cube.)
or, The Eightfold Cube
Every permutation of the plane's points that preserves collinearity is a symmetry of the plane. The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)
The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle. It does not, however, indicate where the other 162 symmetries come from.
Shown below is a new model of this same projective plane, using partitions of cubes to represent points:
The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.
(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations. But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results. This illustrates the difference between affine and projective spaces over the binary field GF(2). In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices. This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices. Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)
To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.
For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.
For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way. For more on the mathematics of cubic models, see Solomon's Cube.
Morrison
Scientific American columnist,
pioneer of the
Search for Extraterrestrial Intelligence (SETI)
and author of
The Ring of Truth.
From The Measure of a Life:
Does religion play a role in attitudes toward ETIs? Philip Morrison gave his considered opinion… “Well, it might, but I think that it’s just one of the permissive routes; it isn’t an essential factor. My parents were Jewish. Their beliefs were conventional but not very deep. They belonged to the Jewish community; they went to services infrequently, on special occasions—funerals and high holidays”….
Although Sagan did not believe in God, he nevertheless said this about SETI’s importance… “It touches deeply into myth, folklore, religion, mythology; and every human culture in some way or another has wondered about that type of question. It’s one of the most basic questions there is.” In fact, in Sagan’s novel/film Contact, described by Keay Davidson as “one of the most religious science-fiction tales ever written”… Ellie discovers that pi—the ratio of the circumference of a circle to its diameter—is numerically encoded in the cosmos and this is proof that a super-intelligence designed the universe…
The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle—another circle, drawn kilometers downstream of the decimal point. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons, subsuming Caretakers and Tunnel builders, there is an intelligence that antedates the universe.
|
|
See also yesterday's entry Mathematical Style.
Extra credit: |
Aluminum, Your Shiny Friend
(Continued)
|
Vital Statistics: |
 for larger image. |
“From the very beginning, the Citicorp Center (today, the Citigroup Center) in New York City was an engineering challenge. When planning for the skyscraper began in the early 1970s, the northwest corner of the proposed building site was occupied by
The church allowed Citicorp to build the skyscraper under one condition: a new church would have to be built on the same corner, with no connection to the Citicorp building and no columns passing through it.
How did the engineers do it? They set the 59-story tower on four massive columns, positioned at the center of each side, rather than at the corners. This design allowed the northwest corner of the building to cantilever 72 feet over the new church.”
Source: PBS, Building BIG.
Citigroup (NYSE:C) is said to be the largest financial services conglomerate in the world.
For more on the close relationship between churches and banks, see the works of T. S. Eliot and a description of the City of London,
For more on Eliot, architecture, and another Harvard man, use links in the previous entry.
…continued…
|
From a review in today’s By Michael Kimmelman
in Los Angeles The roots of this work go back to Duchamp, the abiding spirit of “Beyond Geometry.” When he acquired his porcelain urinal in 1917 from a plumbing equipment manufacturer on lower Fifth Avenue in Manhattan, signed it R. Mutt and submitted the now infamous “Fountain” to the Society of Independent Artists exhibition, he set the stage for nearly every subsequent attempt to blur the difference between art and everyday life. This was the great breakthrough of modernism or the end of culture as we know it, depending on your perspective. Either way, after Duchamp, as the artist Joseph Kosuth has put it, all art became conceptual. Duchamp predicted that even a breath might end up being called a work of art, and he was right. Gilbert and George started calling their performances sculptures in the 70’s. Chris Burden, James Lee Byars and others said that their actions were sculptures. Smithson declared derelict factories and suburbs to be sculptures. Artists even made light, the ultimate intangible, into sculpture. The show includes sculptures by Richard Serra and Barnett Newman. I recall Mr. Serra once talking about how Barnett Newman’s paintings invite you to walk past them, to experience them not in a single glance but over time, physically. He said the paintings, with their vertical stripes, or “zips,” are “about dividing and placing spaces next to one another, not about illusionism.” “They’re great when you have to walk by them and immerse yourself in the divisions of their spaces,” he added. Meaning, they’re like sculptures. Nomenclature is not the point. What matters is the ethos of countercultural disruption, looking at the world and art through the other end of the telescope, which is the heart of “Beyond Geometry” and the appeal of its best works to young artists. Now is the time to put this period of postwar tumult into global perspective. The show here is a useful step in that direction. |
Meanwhile, in Philadelphia,
other art events:
(Click on logo for details.)
The reader may determine whether the Philadelphia nothing is the sort of nothing deemed, by some, sacred in my note of March 9, 2000.
I personally have a very low opinion of Kimmelman and his “ethos of countercultural disruption.” The sort of light sculpture his words evoke is not that of the Pantheon (illustrated in an entry for St. Peter’s Day) but that of the current Philadelphia “Big Nothing” show, which in turn reminds me of that classic 1973 Hollywood art exhibit, The Exorcist:
Parable
"A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: 'what is thrown beside' or 'juxtaposed'), a term used to designate the geometric application we call a 'parabola.'…. The basic parables are extended similes or metaphors."
— http://religion.rutgers.edu/nt/
primer/parable.html
"If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals. Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind."
— Sharon Begley, "The Puzzle of Genius," Newsweek magazine, June 28, 1993, p. 50
"The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem 'Pied Beauty' has to do with 'creation.' "
— Speaking in Parables, Ch. 2, by Sallie McFague
"The Act of Creation is, I believe, a more truly creative work than any of Koestler's novels…. According to him, the creative faculty in whatever form is owing to a circumstance which he calls 'bisociation.' And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or 'see,' for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes."
— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964
For further details, see
Speaking in Parables:
A Study in Metaphor and Theology
by Sallie McFague
Fortress Press, Philadelphia, 1975
Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
"Perhaps every science must start with metaphor and end with algebra; and perhaps without metaphor there would never have been any algebra."
— attributed, in varying forms (1, 2, 3), to Max Black, Models and Metaphors, 1962
For metaphor and algebra combined, see
"Symmetry invariance in a diamond ring," A.M.S. abstract 79T-A37, Notices of the Amer. Math. Soc., February 1979, pages A-193, 194 — the original version of the 4×4 case of the diamond theorem.
Minimalism
"It's become our form of modern classicism."
— Nancy Spector in
the New York Times of April 23, 2004
Part I: Aesthetics
In honor of the current Guggenheim exhibition, "Singular Forms" — A quotation from the Guggenheim's own website:
"Minimalism refers to painting or sculpture
Discuss these seven points
in relation to the following:
Form,
by S. H. Cullinane
Mark Rothko's reference
to geometry as a "swamp"
and his talk of "the idea" in art
Michael Kimmelman's
remarks on ideas in art
ART WARS:
Judgment Day
(2003, 10/07)
Part II: Theology
Today's previous entry, "Skylark," concluded with an invocation of the Lord. Of course, the Lord one expects may not be the Lord that appears.
"… the idea that, in art at least, less is more.
It is an idea surely as old, as enduringly attractive and as ubiquitous as its opposite. In the beginning was the Word: only later came the Bible, not to mention the three-decker Victorian novel. The oracle at Delphi did not say, 'Exhaustive analysis and comprehension of one's own psyche may be prerequisite to an understanding of one's behavior and of the world at large'; it said, 'Know thyself.' Such inherently minimalist genres as oracles (from the Delphic shrine of Apollo to the modern fortune cookie), proverbs, maxims, aphorisms, epigrams, pensees, mottoes, slogans and quips are popular in every human century and culture–especially in oral cultures and subcultures, where mnemonic staying power has high priority–and many specimens of them are self-reflexive or self-demonstrative: minimalism about minimalism. 'Brevity is the soul of wit.' "
Another form of the oracle at Delphi, in minimalist prose that might make Hemingway proud:
"He would think about Bert. Bert was an interesting man. Bert had said something about the way a gambler wants to lose. That did not make sense. Anyway, he did not want to think about it. It was dark now, but the air was still hot. He realized that he was sweating, forced himself to slow down the walking. Some children were playing a game with a ball, in the street, hitting it against the side of a building. He wanted to see Sarah.
When he came in, she was reading a book, a tumbler of dark whiskey beside her on the end table. She did not seem to see him and he sat down before he spoke, looking at her and, at first, hardly seeing her. The room was hot; she had opened the windows, but the air was still. The street noises from outside seemed almost to be in the room with them, as if the shifting of gears were being done in the closet, the children playing in the bathroom. The only light in the room was from the lamp over the couch where she was reading.
He looked at her face. She was very drunk. Her eyes were swollen, pink at the corners. 'What's the book,' he said, trying to make his voice conversational. But it sounded loud in the room, and hard.
She blinked up at him, smiled sleepily, and said nothing.
'What's the book?' His voice had an edge now.
'Oh,' she said. 'It's Kierkegaard. Soren Kierkegaard.' She pushed her legs out straight on the couch, stretching her feet. Her skirt fell back a few inches from her knees. He looked away.
'What's that?' he said.
'Well, I don't exactly know, myself." Her voice was soft and thick.
He turned his face away from her again, not knowing what he was angry with. 'What does that mean, you don't know, yourself?'
She blinked at him. 'It means, Eddie, that I don't exactly know what the book is about. Somebody told me to read it once, and that's what I'm doing. Reading it.'
He looked at her, tried to grin at her — the old, meaningless, automatic grin, the grin that made everbody like him — but he could not. 'That's great,' he said, and it came out with more irritation than he had intended.
She closed the book, tucked it beside her on the couch. She folded her arms around her, hugging herself, smiling at him. 'I guess this isn't your night, Eddie. Why don't we have a drink?'
'No.' He did not like that, did not want her being nice to him, forgiving. Nor did he want a drink.
Her smile, her drunk, amused smile, did not change. 'Then let's talk about something else,' she said. 'What about that case you have? What's in it?' Her voice was not prying, only friendly, 'Pencils?'
'That's it,' he said. 'Pencils.'
She raised her eyebrows slightly. Her voice seemed thick. 'What's in it, Eddie?'
'Figure it out yourself.' He tossed the case on the couch."
— Walter Tevis, The Hustler, 1959,
Chapter 11
See, too, the invocation of Apollo in
A Mass for Lucero, as well as
GENERAL AUDIENCE OF JOHN PAUL II
Wednesday 15 January 2003:
"The invocation of the Lord is relentless…."
and
JOURNAL ENTRY OF S. H. CULLINANE
Wednesday 15 January 2003:
Karl Cullinane —
"I will fear no evil, for I am the
meanest son of a bitch in the valley."
Ennui of the First Idea
The ennui of apartments described by Stevens in "Notes Toward a Supreme Fiction" (see previous entry) did not, of course, refer to the "apartments" of incidence geometry. A more likely connection is with the apartments — the "ever fancier apartments and
"Language was no more than a collection of meaningless conventional signs, and life could absurdly end at any moment. He [Mallarmé] became aware, in Millan’s* words, 'of the extremely fine line
separating absence and presence, being and nothingness, life and death, which
— John Simon, Squaring the Circle
* A Throw of the Dice: The Life of Stéphane Mallarmé, by Gordon Millan
The illustration of the "fine line" is not by Mallarmé but by myself. (See Songs for Shakespeare, March 5, where the line separates being from nothingness, and Ridgepole, March 7, where the line represents the "great primal beginning" of Chinese philosophy (or, equivalently, Stevens's "first idea" or Mallarmé's line "separating absence and presence, being and nothingness, life and death.")
Reply to Lucifer
The New York State Lottery evening number for Saturday, September 6, 2003, was
666.
See last year’s entries for Mary Shelley’s birthday,
These were written partly in response to the New York State Lottery midday number for Monday, August 26, 2002, which was also
666.
In reply to that occurrence, I commented on the website
 |
In reply to last Saturday’s return of the beastly lottery number, I recommend the following links on software guru Bill Joy:
Sept. 9 – Sun Co-founder Joy Steps Down:
“Joy co-founded Sun, originally an acronym for Stanford University Network, with McNealy in 1982. Before that, Joy was the designer of the Berkeley version of the Unix operating system and helped pioneer the concept of open source.
More recently, Joy found himself at the center of controversy after he wrote a Wired magazine article on the challenges posed to mankind by new technologies such as nanotechnology, robotics and genetic engineering.”
and
Joy’s April 2000 Wired article, titled
Why the future doesn’t need us:
Our most powerful 21st-century technologies – robotics, genetic engineering, and nanotech – are threatening to make humans an endangered species.
Joy says
“I can date the onset of my unease to the day I met Ray Kurzweil….”
I, too, can date, at least approximately, an encounter with the philosophy of
transhumanism (a Lucifer Media link)
that Kurzweil embraces… It was sometime in the first half of January, 1989… I know this because January 9, 1989, is the date of The New Yorker’s review of Hans Moravec’s Mind Children: The Future of Robot and Human Intelligence (Harvard University Press).
Brad Leithauser, reviewing Mind Children, says that if Moravec “is correct in supposing that human minds will be transferred into or otherwise fused with machines, it seems likely that traditional religious questions — and traditional religions themselves — will either melt away or suffer wholesale metamorphosis. Debates about Heaven or Hell — to take but one example — would hold little relevance for an immortal creature.”
Au contraire. Immortal creatures– such as, according to Christianity, human beings– are the only creatures for whom such debates hold relevance.
For an example of such a debate, see
The Contrasting Worldviews of
Sigmund Freud and C.S. Lewis,
by Harvard psychiatrist Armand Nicholi.
For more on Nicholi, see my entry of August 19, 2003,
For the temple tablet associated with Nicholi in that entry, see my entry of September 6, 2003 (the NY Lottery “666” date),
To sum up this entry, a phrase of C. S. Lewis seems appropriate:
Reciprocity
From my entry of Sept. 1, 2003:
"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….
… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."
— William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994
Last year's entry on this date:
|
Today's birthday:
"Mathematics is the music of reason."
Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory. |
The picture above is of the complete graph
Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M24.
If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. "Reciprocity" in the sense of Lao Tzu. See
Reciprocity and Reversal in Lao Tzu.
For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in
Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the
Click on the design for details.
Those who prefer a Jewish approach to physics can find the star of David, in the form of
A Graphical Representation
of the Dirac Algebra.
The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.
Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See
O'Hara's Fingerpost
In The New York Times Book Review of next Sunday (August 24, 2003), Book Review editor Charles McGrath writes that author John O'Hara
"… discovered a kind of story… in which a line of dialogue or even a single observed detail indicates that something crucial has changed."
From the Online Etymology Dictionary:
crucial – 1706, from Fr. crucial… from L. crux (gen. crucis) "cross." The meaning "decisive, critical" is extended from a logical term, Instantias Crucis, adopted by Francis Bacon (1620); the notion is of cross fingerboard signposts at forking roads, thus a requirement to choose.
The remainder of this note deals with the "single observed detail" 162.
|
|
Instantias Crucis
Francis Bacon says
"Among Prerogative Instances I will put in the fourteenth place Instances of the Fingerpost, borrowing the term from the fingerposts which are set up where roads part, to indicate the several directions. These I also call Decisive and Judicial, and in some cases, Oracular and Commanding Instances. I explain them thus. When in the investigation of any nature the understanding is so balanced as to be uncertain to which of two or more natures the cause of the nature in question should be assigned on account of the frequent and ordinary concurrence of many natures, instances of the fingerpost show the union of one of the natures with the nature in question to be sure and indissoluble, of the other to be varied and separable; and thus the question is decided, and the former nature is admitted as the cause, while the latter is dismissed and rejected. Such instances afford very great light and are of high authority, the course of interpretation sometimes ending in them and being completed. Sometimes these instances of the fingerpost meet us accidentally among those already noticed, but for the most part they are new, and are expressly and designedly sought for and applied, and discovered only by earnest and active diligence."
Inter praerogativas instantiarum, ponemus loco decimo quarto Instantias Crucis; translato vocabulo a Crucibus, quae erectae in biviis indicant et signant viarum separationes. Has etiam Instantias Decisorias et Judiciales, et in casibus nonnullis Instantias Oraculi et Mandati, appellare consuevimus. Earum ratio talis est. Cum in inquisitione naturae alicujus intellectus ponitur tanquam in aequilibrio, ut incertus sit utri naturarum e duabus, vel quandoque pluribus, causa naturae inquisitae attribui aut assignari debeat, propter complurium naturarum concursum frequentem et ordinarium, instantiae crucis ostendunt consortium unius ex naturis (quoad naturam inquisitam) fidum et indissolubile, alterius autem varium et separabile ; unde terminatur quaestio, et recipitur natura illa prior pro causa, missa altera et repudiata. Itaque hujusmodi instantiae sunt maximae lucis, et quasi magnae authoritatis; ita ut curriculum interpretationis quandoque in illas desinat, et per illas perficiatur. Interdum autem Instantiae Crucis illae occurrunt et inveniuntur inter jampridem notatas; at ut plurimum novae sunt, et de industria atque ex composito quaesitae et applicatae, et diligentia sedula et acri tandem erutae.
— Francis Bacon, Novum Organum, Book Two, "Aphorisms," Section XXXVI
A Cubist Crucifixion
An alternate translation:
"When in a Search of any Nature the Understanding stands suspended, the Instances of the Fingerpost shew the true and inviolable Way in which the Question is to be decided. These Instances afford great Light…"
From a review by Adam White Scoville of Iain Pears's novel titled An Instance of the Fingerpost:
"The picture, viewed as a whole, is a cubist description, where each portrait looks strikingly different; the failings of each character's vision are obvious. However, in a cubist painting the viewer often can envision the subject in reality. Here, even after turning the last page, we still have a fuzzy view of what actually transpired. Perhaps we are meant to see the story as a cubist retelling of the crucifixion, as Pilate, Barabbas, Caiaphas, and Mary Magdalene might have told it. If so, it is sublimely done so that the realization gradually and unexpectedly dawns upon the reader. The title, taken from Sir Francis Bacon, suggests that at certain times, 'understanding stands suspended' and in that moment of clarity (somewhat like Wordsworth's 'spots of time,' I think), the answer will become apparent as if a fingerpost were pointing at the way. The final narrative is also titled An Instance of the Fingerpost, perhaps implying that we are to see truth and clarity in this version. But the biggest mystery of this book is that we have actually have no reason to credit the final narrative more than the previous three and so the story remains an enigma, its truth still uncertain."
For the "162" enigma, see
The Matthias Defense, and
The Still Point and the Wheel.
See also the December 2001 Esquire and
the conclusion of my previous entry.
Venn's Trinity
Today is the birthday of logician John Venn.
From the St. Andrews History of Mathematics site:
"Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 non-overlapping regions, the unions of which give 256 different Boolean combinations of the original sets R, S, T."
Last night's entry, "A Queer Religion," gave a Catholic view of the Trinity. Here are some less interesting but more fruitful thoughts inspired by Venn's diagram of the Trinity (or, indeed, of any three entities):
"To really know a subject you've got to learn a bit of its history…."
— John Baez, August 4, 2002
"We both know what memories can bring;
They bring diamonds and rust."
— Joan Baez, April 1975
For the "diamonds" brought by memories of the 28 combinations described above, consider how the symmetric group S8 is related to the symmetries of the finite projective space PG(3,2). (See Diamond Theory.)
For the "rust," consider the following:
"Lay not up for yourselves treasures upon earth, where moth and rust doth corrupt…."
— Matthew 6:19
The letters R, U, S, T in the Venn diagram above are perhaps relevant here, symbolizing, if you will, the earthly confusion of language, as opposed to the heavenly clarity of mathematics.
As for MOTH, see the article Hometown Zeroes (which brings us yet again to the Viper Room, scene of River Phoenix's death) and the very skillfully designed website MOTHEMATICS.
Powered by WordPress
.
He is an internationally recognized corporate identity consultant. Based in Hong Kong, his work for clients such as HongkongBank, IBM and Unilever is a major influence in Pacific Rim design.
