Underlying Structure of the Design —
Schoolgirl Problem —
Various posts here on the geometry underlying the Mathieu group M_{24}
are now tagged with the phrase “Geometry of Even Subsets.”
For example, a post with this diagram . . .
On a recently deceased professor emeritus of architecture
at Princeton —
"… Maxwell 'established the school as a principal
center of design research, history and theory.' ”
"This is not the Maxwell you're looking for."
From The New York Times on Dec. 11 —
See also some other posts in this journal now tagged "Design Notes Dec. 11."
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 quartertothree post as well as
the webpage "Grids, You Say?" by Norwegian artist Josefine Lyche.
The previous post suggests a review of the saying
"There is such a thing as a 4set."
* Title of a 1959 musical
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
Fanoplane 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
3space over the twoelement 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 —
From April 23, 2013, in
"Classical Geometry in Light of Galois Geometry"—
Click above image for some background from 1986.
Related material on sixset geometry from the classical literature—
Baker, H. F., "Note II: On the Hexagrammum Mysticum of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219236
Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen (1900), Volume 53, Issue 12, pp 161176
Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125160
The configurations recently discussed in
Classical Geometry in Light of Galois Geometry
are not unrelated to the 27 "Solomon's Seal Lines"
extensively studied in the 19th century.
See, in particular—
The following figures supply the connection of Henderson's sixset
to the Galois geometry previously discussed in "Classical Geometry…"—
The previous post displayed part of a page from
a newspaper published the day Olivia NewtonJohn
turned 21 — Friday, September 26, 1969.
A meditation, with apologies to Coleridge:
In Xanadu did NewtonJohn
A stately pleasuresquare 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 socalled education programs likewise degenerated into sundry forms of entertainment and promotion. The critics were coopted, 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, 19852005 , publ. by Ivan R. Dee on Oct. 26, 2006, pp. 146147
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
* 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 fourcolor diamond illustrating design at Microsoft —
For some related mathematics see . . .
The FourColor Diamond’s 2007 Source —
See also Log24 posts from August 2007 now tagged The FourColor 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 . . .
“Mr. Caplan, an essayist, professor, lecturer and consultant on design,
died on June 4 in his apartment on the Upper West Side of Manhattan.
He was 95.” — Penelope Green in The New York Times today.
This journal on that date —
Related cultural icons —
” James, Alec. Alec, James.”
The above Nat Friedman is not to be confused with
the Nat Friedman of “Hyperseeing,” discussed here June 12.
“One game is real and one’s a metaphor.
Untold times this wisdom’s come too late.
Battle of White has raged on endlessly.
Everywhere Black will strive to seal his fate.
Continue a search for thirtythree and three.
Veiled forever is the secret door.”
— Katherine Neville, aka Cat Velis, in The Eight,
Ballantine Books, January 1989, page 140
Related literary remarks —
The Old Man and the Bull
The Old Man and the Topic
A version more explicitly connected to finite geometry —
For the six synthematic totals , see The Joy of Six.
A screenshot from 10:07 PM EDT —
See also this journal on Sunset Boulevard.
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 —
The 15 2subsets of a 6set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)
The 35 3subsets of a 7set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)
The 56 3subsets of an 8set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)
Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.
See also Correspondences in this journal.
* The above 1986 construction of PG(3,2) from a 6set also appeared
in the work of other authors in 1994 and 2002 . . .
Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:
See as well the word correspondences in
“James Joyce and the Hermetic Tradition,” by William York Tindall
(Journal of the History of Ideas , Jan. 1954).
Epigraph from Ch. 4 of Design Theory , Vol. I:
"Es is eine alte Geschichte,
doch bleibt sie immer neu "
—Heine (Lyrisches Intermezzo XXXIX)
This epigraph was quoted here earlier on
the morning of September 1, 2011.
“The key is the cocktail that begins the proceedings.”
– Brian Harley, Mate in Two Moves
“Just as these lines that merge to form a key
Are as chess squares . . . .” — Katherine Neville, The Eight
“The complete projective group of collineations and dualities of the
[projective] 3space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3space there corresponds
a transformation of the [projective] 5space. In the 5space, there are
determined 8 sets of 7 points each, ‘heptads’ ….”
— George M. Conwell, “The 3space PG (3, 2) and Its Group,”
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 6076.
“It must be remarked that these 8 heptads are the key to an elegant proof….”
— Philippe Cara, “RWPRI Geometries for the Alternating Group A_{8},” in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 1621, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 6197.
Exercise: Use the Guitart 7cycles below to relate the 56 triples
in an 8set (such as the eightfold cube) to the 56 triangles in
a wellknown Kleinquartic hyperbolicplane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M_{24}.
Click image below to download a Guitart PowerPoint presentation.
See as well earlier posts also tagged Triangles, Spreads, Mathieu.
(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)
The architecture of the recent post
Geometry of 6 and 8 is in part
a reference to the Klein quadric.
Just as
the finite space PG(3,2) is
the geometry of the 6set, so is
the finite space PG(5,2)
the geometry of the 8set.*
Selah.
* Consider, for the 6set, the 32
(16, modulo complementation)
0, 2, 4, and 6subsets,
and, for the 8set, the 128
(64, modulo complementation)
0, 2, 4, 6, and 8subsets.
Update of 11:02 AM ET the same day:
See also Eightfold Geometry, a note from 2010.
The post “Triangles, Spreads, Mathieu” of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.
Related material — A search in this journal for “56 Triangles.”
There are many approaches to constructing the Mathieu
group M_{24}. The exercise below sketches an approach that
may or may not be new.
Exercise:
It is wellknown that …
There are 56 triangles in an 8set.
There are 56 spreads in PG(3,2).
The alternating group A_{n }is generated by 3cycles.
The alternating group A_{8 }is isomorphic to GL(4,2).
Use the above facts, along with the correspondence
described below, to construct M_{24}.
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 —
Note that in the pictures below of the 15 twosubsets of a sixset,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).
Related narrative — The "Quantum Tesseract Theorem."
A revision of the above diagram showing
the Galoisadditiontable structure —
Related tables from August 10 —
See "Schoolgirl Space Revisited."
arXiv.org > quantph > 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 wellknown Pauli spin or LieClifford 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 threedimensional simplex of 15 points are key objects that feature in this study. Comments:16 pages, 10 figures Subjects:Quantum Physics (quantph) Cite as:arXiv:1905.06914 [quantph] (or arXiv:1905.06914v1 [quantph] 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. A193194, 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 4element 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)).
From this journal on April 23, 2013 —
From this journal in 2003 —
From Wikipedia on Groundhog Day, 2019 —
Some related material in this journal — See a search for k6.gif.
Some related material from Harvard —
Elkies's "15 simple transpositions" clearly correspond to the 15 edges of
the complete graph K_{6} and to the 15 2subsets of a 6set.
For the connection to PG(3,2), see Finite Geometry of the Square and Cube.
The following "manifestation" of the 2subsets of a 6set might serve as
the desired Wikipedia citation —
See also the above 1986 construction of PG(3,2) from a 6set
in the work of other authors in 1994 and 2002 . . .
__________________________________________________________________________
See also the previous post.
I prefer the work of Josefine Lyche on the smallest perfect number/universe.
Context —
Lyche's Lynx760 installations and Vigeland's nearby Norwegian clusterfuck.
"The purpose of mathematics cannot be derived from an activity
inferior to it but from a higher sphere of human activity, namely,
religion."
— Igor Shafarevitch, 1973 remark published as above in 1982.
"Perhaps."
— Steven H. Cullinane, February 13, 2019
From Log24 on Good Friday, April 18, 2003 — . . . What, indeed, is truth? I doubt that the best answer can be learned from either the Communist sympathizers of MIT or the “Red Mass” leftists of Georgetown. For a better starting point than either of these institutions, see my note of April 6, 2001, Wag the Dogma. See, too, In Principio Erat Verbum , which notes that “numbers go to heaven who know no more of God on earth than, as it were, of sun in forest gloom.” Since today is the anniversary of the death of MIT mathematics professor GianCarlo Rota, an example of “sun in forest gloom” seems the best answer to Pilate’s question on this holy day. See
“Examples are the stained glass windows Motto of Plato’s Academy † The Exorcist, 1973 
Detail from an image linked to in the above footnote —
"And the darkness comprehended it not."
Id est :
A Good Friday, 2003, article by
a student of Shafarevitch —
"… there are 25 planes in W . . . . Of course,
replacing {a,b,c} by the complementary set
does not change the plane. . . ."
Of course.
See. however, SixSet Geometry in this journal.
This journal ten years ago today —
Surprise Package
From a talk by a Melbourne mathematician on March 9, 2018 —
The source — Talk II below —
Search Resultspdf of talk I (March 8, 2018)www.math.sci.hiroshimau.ac.jp/branched/…/AitchisonHiroshima2018Talk12.pdf Iain Aitchison. Hiroshima University March 2018 … Immediate: Talk given last year at Hiroshima (originally Caltech 2010). pdf of talk II (March 9, 2018) (with model for M24)www.math.sci.hiroshimau.ac.jp/branched/files/…/AitchisonHiroshima22018.pdf Iain Aitchison. Hiroshima University March 2018. (IRA: Hiroshima 032018). Highly symmetric objects II. Abstractwww.math.sci.hiroshimau.ac.jp/branched/files/2018/abstract/Aitchison.txt Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces , related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some … 
Related material —
The 56 triangles of the eightfold cube . . .
Image from Christmas Day 2005.
From the former date above —
Saturday, September 17, 2016 
From the latter date above —
Tuesday, October 18, 2016
Parametrization

From March 2018 —
Click image to enlarge —
A portrait from the home page of David Eppstein,
a professor at the University of California, Irvine
“… how can an image with 8 points and 8 lines
possibly represent a space with 7 points and 7 lines???“
— David Eppstein, 21 December 2015
See ” Projective spaces as ‘collapsed vector spaces,’ ”
page 203 in Geometry and Symmetry by Paul B. Yale,
published by HoldenDay in 1968.
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”
Related structures —
Greg Egan’s animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets or Mozart’s late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. 
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube —
“Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eighttriangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eighttriangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”
Related material from 1975 —
More recently …
From the Diamond Theorem Facebook page —
A question three hours ago at that page —
“Is this Time Cube?”
Notes toward an answer —
And from SixSet Geometry in this journal . . .
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 nonverbal reasoning like
the NNAT, Raven's or nonverbal sections of the CogAT or OLSAT. "
For a webpage that is perhaps un ethical and in appropriate,
see Block Designs in Art and Mathematics.
Hume, from posts tagged "fourset" in this journal —
"The mind is a kind of theatre, where several perceptions
successively make their appearance; pass, repass, glide away,
and mingle in an infinite variety of postures and situations.
There is properly no simplicity in it at one time, nor identity
in different, whatever natural propension we may have
to imagine that simplicity and identity."
Paz, from a search for Paz + Identity in this journal —
"At the point of convergence by Octavio Paz, translated by Helen Lane

"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."
— From the Occupy Space gallery in Ireland
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 a Los Angeles Times piece on Epiphany (Jan. 6), 1988 —
“Some 30 paces east of the spooky old Chateau Marmont is
the intersection of Selma and Sunset Boulevard.” . . . .
“Though it is not much of an intersection, the owner of
the liquor store on that corner might resent that you have
slotted his parking lot in the Twilight Zone. . . .
And directly across Sunset from Selma looking south is
where the infamous Garden of Allah used to stand. . . .”
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="20171016T00:01:3804:00" />
Cover design by Jarrod Taylor.
Book published on July 14, 2015.
For this journal on that date, see posts tagged Perspective.
"Fire and Fury is one of the most popular
historical military miniatures wargames . . . .
For the new player it is an easy game
to learn and enjoy."
A post of March 22, 2017, was titled "The Story of Six."
Related material from that date —
"I meant… a larger map." — Number Six in "The Prisoner"
A sketch, adapted tonight from Girl Scouts of Palo Alto —
From the April 14 noon post High Concept —
From the April 14 3 AM post Hudson and Finite Geometry —
From the April 24 evening post The Trials of Device —
Note that Hudson’s 1905 “unfolding” of even and odd puts even on top of
the square array, but my own 2013 unfolding above puts even at its left.
The above fourelement sets of black subsquares of a 4×4 square array
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .
Hudson did not view these 35 tetrads as planes through the origin in a finite
affine 4space (or, equivalently, as lines in the corresponding finite projective
3space).
In order to view them in this way, one can view the tetrads as derived,
via the 15 twoelement subsets of a sixelement set, from the 16 elements
of the binary Galois affine space pictured above at top left.
This space is formed by taking symmetricdifference (Galois binary)
sums of the 15 twoelement subsets, and identifying any resulting four
element (or, summing three disjoint twoelement subsets, sixelement)
subsets with their complements. This process was described in my note
"The 2subsets of a 6set are the points of a PG(3,2)" of May 26, 1986.
The space was later described in the following —
On a psychotherapist who died at 86 on Monday —
"He studied mathematics and statistics at the Courant Institute,
a part of New York University — he would later write … a
mathematical fable, Numberland (1987)."
— The New York Times online this evening
From Publishers Weekly

See also The Prisoner in this journal.
See also Log24 posts related to “Go Set a Structure”
as well as “New Haven” + Grid.
Hollywood, from the Alto Nido Apartments
to Sunset Boulevard —
See also the Jan. 31 post "Sunset Passion."
For a concise historical summary of the interplay between
the geometry of an 8set and that of a 16set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M_{24}, see Section 2 of …
Alan R. Prince
A near projective plane of order 6 (pp. 97105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).
This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell’s heptads .
Update, morning of the following day (7:07 ET) — related material:
See also “56 spreads” in this journal.
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 littleknown 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 5fach transitiven Gruppen von Mathieu,
— William M. Kantor, book review (pdf), 
The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vectorspace nature.
Examples: The labelings of a 4×4 array by a blank space
plus the 15 twosubsets of a sixset (Hudson, 1905) or by a
blank plus the 5 elements and the 10 twosubsets of a fiveset
(derived in 2014 from a 1906 page by Whitehead), or by
a blank plus the 15 line diagrams of the diamond theorem.
Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —
“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
Note, however, that Weyl's definition of "coordinatization"
is not limited to vectorspace coordinates. He describes it
as simply a mapping to a set of reproducible symbols .
(But Weyl does imply that these symbols should, like vectorspace
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the pointspace
being coordinatized.)
The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).
* For the definition of "perfect number," see any introductory
numbertheory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
the projective 3space over the 2element Galois field GF(2),
was coined by math writer Burkard Polster. Cullinane's square
model of PG(3,2) differs from the earlier tetrahedral model
discussed by Polster.
The previous post discussed the parametrization of
the 4×4 array as a vector 4space over the 2element
Galois field GF(2).
The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2subsets of a 6set, as in Hudson's
1905 classic Kummer's Quartic Surface —
Hudson in 1905:
These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2element sets — were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator," what turned out to be 15 of Hudson's
1905 "Göpel tetrads":
A recap by Cullinane in 2013:
Click images for further details.
The New York Times interviews Alan Moore —
“A version of this article appears in print on September 11, 2016,
on page BR9 of the Sunday Book Review ….”
“What genres do you prefer? And which do you avoid?”
“To be honest, having worked in genre for so long, I’m happiest
when I’m outside it altogether, or perhaps more accurately,
when I can conjure multiple genres all at once, in accordance
with my theory (now available, I believe, as a greeting card and
fridge magnet) that human life as we experience it is a
simultaneous multiplicity of genres. I put it much more elegantly
on the magnet.”
See …
At the Still Point … (February 12, 2008)
For Balanchine's Birthday (January 9, 2007)
Go Set a Structure (Various dates)
and …
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 —
(A sequel to the previous post, Perfect Number)
Since antiquity, six has been known as
"the smallest perfect number." The word "perfect"
here means that a number is the sum of its
proper divisors — in the case of six: 1, 2, and 3.
The properties of a sixelement set (a "6set")
divided into three 2sets and divided into two 3sets
are those of what Burkard Polster, using the same
adjective in a different sense, has called
"the smallest perfect universe" — PG(3,2), the projective
3dimensional space over the 2element Galois field.
A Google search for the phrase "smallest perfect universe"
suggests a turnaround in meaning , if not in finance,
that might please Yahoo CEO Marissa Mayer on her birthday —
The semantic turnaround here in the meaning of "perfect"
is accompanied by a model turnaround in the picture of PG(3,2) as
Polster's tetrahedral model is replaced by Cullinane's square model.
Further background from the previous post —
See also Kirkman's Schoolgirl Problem.
"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—
For the birthday of Marissa Mayer, who turns 41 today —
VOGUE Magazine,
AUGUST 16, 2013 12:01 AM
by JACOB WEISBERG —
"As she works to reverse the fortunes of a failing Silicon Valley
giant, Yahoo’s Marissa Mayer has fueled a national debate
about the office life, motherhood, and what it takes to be the
CEO of the moment.
'I really like even numbers, and
I like heavily divisible numbers.
Twelve is my lucky number—
I just love how divisible it is.
I don’t like odd numbers, and
I really don’t like primes.
When I turned 37,
I put on a strong face, but
I was not looking forward to 37.
But 37 turned out to be a pretty amazing year.
Especially considering that
36 is divisible by twelve!'
A few things may strike you while listening to Marissa Mayer
deliver this riff . . . . "
Yes, they may.
A smaller number for Marissa's meditations:
Six has been known since antiquity as the first "perfect" number.
Why it was so called is of little interest to anyone but historians
of number theory (a discipline that is not, as Wikipedia notes,
to be confused with numerology .)
What part geometry , on the other hand, played in Marissa's education,
I do not know.
Here, for what it's worth, is a figure from a review of posts in this journal
on the key role played by the number six in geometry —
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 emptyhanded 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 worldlywise 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.
See a search for "large Desargues configuration" in this journal.
The 6 Jan. 2015 preprint "Danzer's Configuration Revisited,"
by Boben, Gévay, and Pisanski, places this configuration,
which they call the CayleySalmon configuration , in the
interesting context of Pascal's Hexagrammum Mysticum .
They show how the CayleySalmon configuration is, in a sense,
dual to something they call the SteinerPlücker configuration .
This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3space." The sixsets at
the bottom of that note, together with Figures 3 and 4
of Boben et. al. , indicate how this works.
The duality was, as they note, previously described in 1898.
Related material on sixset geometry from the classical literature—
Baker, H. F., "Note II: On the Hexagrammum Mysticum of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219236
Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen (1900), Volume 53, Issue 12, pp 161176
Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125160
Related material on sixset geometry from a more recent source —
Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage
Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation ). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.
References:
Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213214
Cullinane, Steven H., Notes on Groups and Geometry, 19781986
Related material:
The 6set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —
Background reading:
Ron Shaw on finite geometry, Clifford algebras, and Dirac groups
(undated compilation of publications from roughly 19941995)—
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 7set [i.e., the 35 partitions
of an 8set into two 4sets] 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 M_{24},
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 "endofproof 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 3space over the
twoelement Galois field GF(2), or, if you prefer, on the Galois
field Ω = GF(8).
Related fiction: The Eight , by Katherine Neville.
Related nonfiction: 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— 
Continued from yesterday, the date of death for German
billionaire philanthropist Klaus Tschira —
For Tschira in this journal, see Stiftung .
For some Würfel illustrations, see this morning's post
Manifest O. A related webpage —
The title was suggested by
http://benmarcus.com/smallwork/manifesto/.
The "O" of the title stands for the octahedral group.
See the following, from http://finitegeometry.org/sc/map.html —

An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. 
831001  Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. 
831016  Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. 
840915  Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. 
"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 20thcentury 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 unitsquare “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.
A professor at Harvard has written about
“the urge to seize and display something
real beyond artifice.”
He reportedly died on January 3, 2015.
An image from this journal on that date:
Another Gitterkrieg image:
The 24set Ω of R. T. Curtis
Click on the images for related material.
In memory of Rod Taylor, who
reportedly died at 84 on Wednesday,
the seventh day of 2015 —
“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 ‘mashup’ 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 (19232014)
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, 194446). 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 195658), University of Arizona (Chairman 19581967), 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 greatgranddaughter.
— 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 Lfunctions 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 onesemester 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 onesemester 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 selfstudy (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 BorevichShafarevich 'Number Theory'. The latter is a fantastic book which covers a large superset of the material in Cohn's book. BorevichShafarevich is, however, a much more demanding read and it is out of print. For gentle selfstudy (and perhaps as a preparation to later read BorevichShafarevich), Cohn's book is a fine read."
In the Miracle Octad Generator (MOG):
The above details from a onepage note of April 26, 1986, refer to the
Miracle Octad Generator of R. T. Curtis, as it was published in 1976:
From R. T. Curtis (1976). A new combinatorial approach to M_{24},
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 2542. doi:10.1017/S0305004100052075.
The 1986 note assumed that the reader would be able to supply, from the
MOG itself, the missing top row of each heavy brick.
Note that the interchange of the two squares in the top row of each
heavy brick induces the diamondtheorem correlation.
Note also that the 20 pictured 3subsets of a 6set in the 1986 note
occur as paired complements in two pictures, each showing 10 of the
3subsets.
This pair of pictures corresponds to the 20 Rosenhain tetrads among
the 35 lines of PG(3,2), while the picture showing the 2subsets
corresponds to the 15 Göpel tetrads among the 35 lines.
See Rosenhain and Göpel tetrads in PG(3,2). Some further background:
Some background for the part of the 2002 paper by Dolgachev and Keum
quoted here on January 17, 2014 —
Related material in this journal (click image for posts) —
A sequel to the 1974 film
Thunderbolt and Lightfoot :
Contingent and Fluky
Some variations on a thunderbolt theme:
These variations also exemplify the larger
Verbum theme:
A search today for Verbum in this journal yielded
a Georgetown University Chomskyite, Professor
David W. Lightfoot.
"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."
Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.
See Tecumseh Fitch in this journal.
For other material related to the large Verbum cube,
see posts for the 18th birthday of Harry Potter.
That birthday was also the upload date for the following:
See esp. the comments section.
Rivka Galchen, in a piece mentioned here in June 2010—
On Borges: Imagining the Unwritten Book
"Think of it this way: there is a vast unwritten book that the heart reacts to, that it races and skips in response to, that it believes in. But it’s the heart’s belief in that vast unwritten book that brought the book into existence; what appears to be exclusively a response (the heart responding to the book) is, in fact, also a conjuring (the heart inventing the book to which it so desperately wishes to respond)."
Related fictions
Galchen's "The Region of Unlikeness" (New Yorker , March 24, 2008)
Ted Chiang's "Story of Your Life." A film adaptation is to star Amy Adams.
… and nonfiction
"There is such a thing as a 4set." — January 31, 2012
Or: The Nutshell
What about Pascal?
For some background on Pascal's mathematics,
not his wager, see…
Richmond, H. W.,
"On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics ,
Volume 31 (1900), pp. 125160,
dated by Richmond March 30,1899
Richmond, H. W.,
"The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen ,
Volume 53 (1900), Issue 12, pp 161176,
dated by Richmond February 1, 1899
See also Nocciolo in this journal.
Recall as well that six points in space may,
if constrained to lie on a circle, be given
a religious interpretation. Richmond's
six points are secular and more general.
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.
The Kummer 16_{6} configuration is the configuration of sixteen
6sets within a 4×4 square array of points in which each 6set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.
See Configurations and Squares.
The Wikipedia article Kummer surface uses a rather poetic
phrase* to describe the relationship of the 16_{6} to a number
of other mathematical concepts — "geometric incarnation."
Related material from finitegeometry.org —
* Apparently from David Lehavi on March 18, 2007, at Citizendium .
A useful article on finite geometry,
"21 – 6 = 15: A Connection between Two Distinguished Geometries,"
by Albrecht Beutelspacher, American Mathematical Monthly ,
Vol. 93, No. 1, January 1986, pp. 2941, is available for purchase
at JSTOR.
This article is related to the geometry of the sixset.
For some background, see remarks from 1986 at finitegeometry.org.
Found this morning in a search:
A logline is a onesentence 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.
… And the history of geometry —
Desargues, Pascal, Brianchon and Galois
in the light of complete npoints in space.
(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)
Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:
"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space fivepoint."
Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black points and dashed lines indicate the
complete space fivepoint and lines connecting it to the plane section
containing the Desargues configuration.
In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six point to construct a configuration of
15 points and 20 lines in the context not of Desargues ' theorem, but
rather of Brianchon 's theorem and of the Pascal hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6point in space can be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large Desargues configuration. See Classical Geometry in Light of
Galois Geometry.)
For this large Desargues configuration see April 19.
For Henderson's complete six –point, see The SixSet (April 23).
That post ends with figures relating the large Desargues configuration
to the Galois geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator and the large Mathieu group M_{24} —
See also Note on the MOG Correspondence from April 25, 2013.
That correspondence was also discussed in a note 28 years ago, on this date in 1985.
In light of the April 23 post "The SixSet,"
the caption at the bottom of a note of April 26, 1986
seems of interest:
"The R. T. Curtis correspondence between the 35 lines and the
2subsets and 3subsets of a 6set. This underlies M_{24}."
A related note from today:
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 uptodate version of little Colva,
see Primitive Groups and Maximal Subgroups.)
Click image for some background.
Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M_{24},"
Math. Proc. Camb. Phil. Soc., 79 (1976), 2542.)
The 8subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirtyfive 3subsets of a 7set.
Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the hexagrammum mysticum of Pascal.
On Danzer's 35_{4} Configuration:
"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3sets and all 4sets that can be formed
by the elements of a 7element set; each 'point' is represented
by one of the 3sets, and it is incident with those lines
(represented by 4sets) that contain the 3set."
— Branko Grünbaum, "Musings on an Example of Danzer's,"
European Journal of Combinatorics , 29 (2008),
pp. 1910–1918 (online March 11, 2008)
"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."
— Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited," arXiv.org, Jan. 6, 2013
For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see
Classical Geometry in Light of Galois Geometry.
Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).
"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.
The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8set into two 4sets and the 35 lines in the
finite projective space PG(3,2).
Closely related to the Moore correspondence
is a correspondence (or analogy) between the
15 2subsets of a 6set and the 15 points of PG(3,2).
An analogy between the two above analogies
is supplied by the exceptional outer automorphism of S_{6}.
See…
The 2subsets of a 6set are the points of a PG(3,2),
Picturing outer automorphisms of S_{6}, and
A linear complex related to M_{24}.
(Background: Inscapes, Inscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3space.)
* For some context, see Analogies and
"Smallest Perfect Universe" in this journal.
(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…."
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