Log24

Monday, June 25, 2018

The Trials of Device

Filed under: General — Tags: , , — m759 @ 9:34 am

"A blank underlies the trials of device."

Wallace Stevens

"Designing with just a blank piece of paper is very quiet."

Kate Cullinane

Related material —

An image posted at 12 AM ET December 25, 2014:

The image stands for the
phrase "five by five,"
meaning "loud and clear."

Other posts featuring the above 5×5 square with some added structure:

Saturday, June 23, 2018

Meanwhile …

Filed under: General,Geometry — Tags: , , — m759 @ 9:00 pm

Backstory for fiction fans, from Log24 on June 11 —

Related non -fiction —

See as well the structure discussed in today's previous post.

Thursday, June 14, 2018

Wylie’s Bull

Filed under: General — m759 @ 1:00 pm

"There is a structure theory for bull-free graphs
 
 modulo the structure of triangle-free graphs
  and their complements, which again is not easy.
  (The bull has a triangular face, with horns or
  pendant edges at two of its three vertices.)"

— Peter J. Cameron today

For example —

The bull graph in a book by Clarence R. Wylie, Jr.
(author of the poem "Paradox" (1948)). See no. 6 below —

See also Wikipedia.

Related material —

J. Paul Getty and Minotaur, according to Hollywood —

Monday, June 4, 2018

The Trinity Stone Defined

“Unsheathe your dagger definitions.” — James Joyce, Ulysses

The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

An Approach to Symmetric Generation of the Simple Group of Order 168

Some small Galois spaces (the Cullinane models)

Thursday, May 3, 2018

Multifaceted . . .

Filed under: General — Tags: , — m759 @ 2:56 pm

. . . Con Figuras de Espantar

"He Who Searches  is multifaceted in structure …"

Publisher's description of a Helen Lane translation
of "Como en la Guerra ," by Luisa Valenzuela
Also by Valenzuela —

Related material — An obituary from The Boston Globe  today
on the April 5 death of Borinsky's translator, and . . .

"He Who Searches" may consult also posts tagged Date.

Friday, April 13, 2018

Philosophy 101

Filed under: General — Tags: — m759 @ 2:48 pm

See also Log24 posts now tagged "Is and As."

Tuesday, March 27, 2018

Compare and Contrast

Filed under: General,Geometry — Tags: , , — m759 @ 4:28 pm

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Related material on automorphism groups —

The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture 
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.

Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure  above
than they are an eight-cube mechanical puzzle  like the one below.

Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."

Friday, March 9, 2018

The Cemetery Plot

Filed under: General — m759 @ 7:20 pm

The New York Times  today on a philosopher of
history who reportedly died on Monday, March 5 —

“Perhaps White’s most controversial idea,
and one for which he was so often shunned
by his fellow historians, is that ‘all stories
are fictions,’ ” Robert Doran, a professor at
the University of Rochester … said by email.
"White held that while historical facts are
scientifically verifiable, stories are not.
Stories are made, not found in the historical data;
historical meaning is imposed on historical facts
by means of the choice of plot-type, and this choice
is inevitably ethical and political at bottom.

"This is what White called 'emplotment,' a term
he coined," Dr. Doran continued. "Even the most basic
beginning-middle-end structure of a story represents
an imposition: The historian chooses where to begin,
where to end, and what points are important in the middle.
There is no scientific test for 'historical significance.' "

From this  journal on Monday, White's reported date of death —

Plan 9  continues.

Friday, February 16, 2018

Two Kinds of Symmetry

Filed under: General,Geometry — Tags: — m759 @ 11:29 pm

The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter 
revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below. 

In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald 
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)

The adjective "modular"  might aptly be applied to . . .

The adjective "affine"  might aptly be applied to . . .

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.

Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but 
did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For Macdonald's own  use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms," 
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.

Wednesday, December 27, 2017

For Day 27 of December 2017

Filed under: General,Geometry — Tags: — m759 @ 3:57 am

See the 27-part structure of
the 3x3x3 Galois cube

IMAGE- The 3x3x3 Galois cube
as well as Autism Sunday 2015.

Tuesday, December 26, 2017

Raiders of the Lost Stone

Filed under: General,Geometry — Tags: , , — m759 @ 8:48 pm

(Continued

 

Two Students of Structure

A comment on Sean Kelly's Christmas Morning column on "aliveness"
in the New York Times  philosophy series The Stone  —

Diana Senechal's 1999 doctoral thesis at Yale was titled
"Diabolical Structures in the Poetics of Nikolai Gogol."

Her mother, Marjorie Senechal, has written extensively on symmetry
and served as editor-in-chief of The Mathematical Intelligencer .
From a 2013 memoir by Marjorie Senechal —

"While I was in Holland my enterprising student assistant at Smith had found, in Soviet Physics – Crystallography, an article by N. N. Sheftal' on tetrahedral penetration twins. She gave it to me on my return. It was just what I was looking for. The twins Sheftal' described had evidently begun as (111) contact twins, with the two crystallites rotated 60o with respect to one another. As they grew, he suggested, each crystal overgrew the edges of the other and proceeded to spread across the adjacent facet.  When all was said and done, they looked like they'd grown through each other, but the reality was over-and-around. Brilliant! I thought. Could I apply this to cubes? No, evidently not. Cube facets are all (100) planes. But . . . these crystals might not have been cubes in their earliest stages, when twinning occurred! I wrote a paper on "The mechanism of certain growth twins of the penetration type" and sent it to Martin Buerger, editor of Neues Jarbuch für Mineralogie. This was before the Wrinch symposium; I had never met him. Buerger rejected it by return mail, mostly on the grounds that I hadn't quoted any of Buerger's many papers on twinning. And so I learned about turf wars in twin domains. In fact I hadn't read his papers but I quickly did. I added a reference to one of them, the paper was published, and we became friends.[5]

After reading Professor Sheftal's paper I wrote to him in Moscow; a warm and encouraging correspondence ensued, and we wrote a paper together long distance.[6] Then I heard about the scientific exchanges between the Academies of Science of the USSR and USA. I applied to spend a year at the Shubnikov Institute for Crystallography, where Sheftal' worked. I would, I proposed, study crystal growth with him, and color symmetry with Koptsik. To my delight, I was accepted for an 11-month stay. Of course the children, now 11 and 14, would come too and attend Russian schools and learn Russian; they'd managed in Holland, hadn't they? Diana, my older daughter, was as delighted as I was. We had gone to Holland on a Russian boat, and she had fallen in love with the language. (Today she holds a Ph.D. in Slavic Languages and Literature from Yale.) . . . . 
. . .
 we spent the academic year 1978-79 in Moscow.

Philosophy professors and those whose only interest in mathematics
is as a path to the occult may consult the Log24 posts tagged Tsimtsum.

Wednesday, December 20, 2017

January 2018 AMS Notices

Filed under: General,Geometry — Tags: — m759 @ 3:03 pm

Update of 9:29 PM ET Dec. 20, 2017 —

See in particular, in the above Notices , the article

"Algebraic Structures on Polytopes," by Federico Ardila,
within the 2018 Joint Mathematics Meeting Lecture Sampler.

Related reading:

arXiv:1711.09102v1 [hep-th] 24 Nov 2017,

"Scattering Forms and the Positive Geometry of
Kinematics, Color and the Worldsheet," by
Nima Arkani-Hamed, Yuntao Bai, Song He, Gongwang Yan
(Submitted to the arXiv on 24 Nov. 2017).

Monday, December 18, 2017

Mathematics and Art

Filed under: G-Notes,General,Geometry — m759 @ 5:09 pm

From the American Mathematical Society homepage today —

From concinnitasproject.org

"Concinnitas  is the title of a portfolio of fine art prints. . . .
The portfolio draws its name from a word famously used
by the Renaissance scholar, artist, architect, and philosopher
Leon Battista Alberti (1404-1472) to connote the balance of
number, outline, and position (in essence, number, geometry,
and topology) that he believed characterize a beautiful work of art."

The favicon of the Concinnitas Project —

The structure of the Concinnitas favicon —

This structure is from page 15 of
"Diamond Theory," a 1976 preprint —

 .

Sunday, December 10, 2017

Geometry

Google search result for Plato + Statesman + interlacing + interweaving

See also Symplectic in this journal.

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of  linear  (or line ) complex
in the finite projective space PG(3,2) —

Anticommuting Dirac matrices as spreads of projective lines

Ron Shaw on the 15 lines of the classical generalized quadrangle W(2), a general linear complex in PG(3,2)

Thursday, November 23, 2017

The Matrix

Filed under: General — Tags: , — m759 @ 10:00 pm

David Brooks in The New York Times  today

"We once had a unifying national story, celebrated each Thanksgiving.
It was an Exodus story. Americans are the people who escaped oppression,
crossed a wilderness and are building a promised land. The Puritans brought
this story with them. Each wave of immigrants saw themselves in this story.
The civil rights movement embraced this story.

But we have to admit that many today do not resonate with this story. . . .

Today, we have no common national narrative, no shared way
of interpreting the flow of events. Without a common story,
we don’t know what our national purpose is. We have no
common set of goals or ideals.

We need a new national narrative."

From a post of August 15, 2010

http://www.log24.com/log/pix10B/100815-NeoAndOracle.jpg

For some background, see Java Jive and Today's Theology.

Related readings —

From 1928:

From the previous post:

"Thus, instead of Propp's chronological scheme,
in which the order of succession of events
is a feature of the structure . . .
another scheme should be adopted, which would present
a structural model defined as the group of transformations
of a small number of elements. This scheme would appear
as a matrix . . . ."

Claude Lévi-Strauss, 1960 

Lévi-Strauss vs. Propp

Filed under: General,Geometry — Tags: , , , — m759 @ 12:25 pm
​Claude Lévi-Strauss

From his Structure and Form:
Reflections on a Work by Vladimir Propp
” *

To maintain. as I have done. that the permutability of contents is not arbitrary amounts to saying that, if the analysis is carried to a sufficiently deep level, behind diversity we will discover constancy. And, of course. the avowed constancy of form must not hide from us that functions are also permutable.

The structure of the folktale as it is illustrated by Propp presents a chronological succession of qualitatively distinct functions. each constituting an independent genre. One can wonder whether—as with dramatis personae and their attributes— Propp does not stop too soon, seeking the form too close to the level of empirical observation. Among the thirty-one functions that he distinguishes, several are reducible to the same  function reappearing at different  moments of the narrative but after undergoing one or a number of transformations . I have already suggested that this could be true of the false hero (a transformation of the villain), of assigning a difficult task (a transformation of the test), etc. (see p. 181 above), and that in this case the two parties  constituting the fundamental tale would themselves be transformations of each other.

Nothing prevents pushing this reduction even further and analyzing each separate partie  into a small number of recurrent functions, so that several of Propp’s functions would constitute groups of transformations of one and the same function. We could treat the “violation” as the reverse of the “prohibition” and the latter as a negative transformation of the “injunction.” The “departure” of the hero and his “return” would appear as the negative and positive expressions of the same disjunctive function. The “quest” of the hero (hero pursues someone or something) would become the opposite of “pursuit” (hero is pursued by something or someone), etc.

In Vol. I of Structural Anthropology , p. 209, I have shown that this analysis alone can account for the double aspect of time representation in all mythical systems: the narrative is both “in time” (it consists of a succession of events) and “beyond” (its value is permanent). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself  his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 97-98. p. 108) If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns).

These critical remarks are certainly valid for the method used by Propp and for his conclusions. However. it cannot be stressed enough that Propp envisioned them and in several places formulated with perfect clarity the solutions I have just suggested. Let us take up again from this viewpoint the two essential themes of our discussion: constancy of the content (in spite of its permutability) and permutability of functions (in spite of their constancy).

* Translated from a 1960 work in French.  It appeared in English as Chapter VIII
of Structural Anthropology, Volume 2  (U. of Chicago Press, 1976).  Chapter VIII
was originally published in Cahiers de l’Institut de Science 
Économique Appliquée , 
No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).

See also “Lévi-Strauss” + Formula  in this journal.

Some background related to the previous post

Monday, November 13, 2017

In Nomine Patris

Filed under: General — Tags: — m759 @ 12:28 pm

See also Norbert Wiener  in this  journal  and

Related material for the Church of Synchronology

The Log24 post on the above New York Times  death date.

Monday, November 6, 2017

The Chomsky Koan

Filed under: General — Tags: — m759 @ 12:45 pm

"Colorless green ideas sleep furiously   is a sentence
composed by Noam Chomsky in his 1957 book 
Syntactic Structures  as an example of a sentence 
that is grammatically correct, but semantically nonsensical."

Wikipedia article on the sentence

Buddhist midrash from The New York Times  today

"For example, psychology has lately started to let go of its
once-sharp distinction between 'cognitive' and 'affective' 
parts of the mind; it has started to see that feelings are so
finely intertwined with thoughts as to be part of their very
coloration." 

See also other recent Log24 posts now tagged Coloration.

Sunday, October 29, 2017

File System… Unlocked

Filed under: General,Geometry — Tags: — m759 @ 2:16 pm

Logo from the above webpage

See also the similar structure of  the eightfold cube,  and

Related dialogue from the new film "Unlocked"

1057
01:31:59,926 –> 01:32:01,301
Nice to have you back, Alice.

1058
01:32:04,009 –> 01:32:05,467
Don't be a stranger.

Wednesday, October 18, 2017

Dürer for St. Luke’s Day

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:00 pm

Structure of the Dürer magic square 

16   3   2  13
 5  10  11   8   decreased by 1 is …
 9   6   7  12
 4  15  14   1

15   2   1  12
 4   9  10   7
 8   5   6  11
 3  14  13   0 .

Base 4 —

33  02  01  30
10  21  22  13
20  11  12  23 
03  32  31  00 .

Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays

3 0 0 3     3 2 1 0
1 2 2 1     0 1 2 3
2 1 1 2     0 1 2 3
0 3 3 0     3 2 1 0 .

Base 2 –

1111  0010  0001  1100
0100  1001  1010  0111
1000  0101  0110  1011
0011  1110  1101  0000 .

Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —

1001  1001  1100  1010
0110  1001  0011  0101
1001  0110  0011  0101
0110  0110  1100  1010 .

— Steven H. Cullinane,
  October 18, 2017

See also recent related analyses of
noted 3×3 and 5×5 magic squares.

Tuesday, October 17, 2017

Plan 9 Continues

Filed under: General,Geometry — Tags: , , — m759 @ 9:00 pm

See also Holy Field in this journal.

Some related mathematics —

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Analysis of the Lo Shu structure —

Structure of the 3×3 magic square:

4  9  2
3  5  7    decreased by 1 is
8  1  6

3  8  1
2  4  6
7  0  5

In base 3 —

10  22  01
02  11  20
21  00  12

As orthogonal Latin squares
(a well-known construction) —

1  2  0     0  2  1
0  1  2     2  1  0
2  0  1     1  0  2 .

— Steven H. Cullinane,
October 17, 2017

Wednesday, October 4, 2017

Text and Context

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 11:00 am

Text —

"A field is perhaps the simplest algebraic structure we can invent."

— Hermann Weyl, 1952

Context —

See also yesterday's Personalized Book Search.

Full text of Symmetry  – Internet Archive —

https://archive.org/details/Symmetry_482

A field is perhaps the simplest algebraic 143 structure
we can invent. Its elements are numbers. Characteristic
for its structure are the operations of addition and 

From a Log24 search for Mathematics+Nutshell —

IMAGE- History of Mathematics in a Nutshell

Tuesday, September 12, 2017

Think Different

Filed under: General,Geometry — Tags: , — m759 @ 11:00 pm

The New York Times  online this evening

"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."

James Poniewozik 

Review —

Thursday, September 1, 2011

How It Works

Filed under: Uncategorized — Tags:  — m759 @ 11:00 AM 

"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

. . . .

See also 1984 Bricks in this journal.

Saturday, September 9, 2017

How It Works

Filed under: General,Geometry — Tags: , — m759 @ 8:48 pm

Del Toro and the History of Mathematics ,
Or:  Applied Bullshit Continues

 

For del Toro


 

For the history of mathematics —

Thursday, September 1, 2011

How It Works

Filed under: Uncategorized — Tags:  — m759 @ 11:00 AM 

"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

. . . .

Monday, August 14, 2017

Visual Impact

Filed under: General — m759 @ 2:48 pm

Alan Moore on the 9-panel grid

See also posts now tagged "Go Set."

Thursday, July 27, 2017

Keeping It Simple

Filed under: General,Geometry — m759 @ 9:00 pm

Michiko Kakutani in The New York Times

"The detective story genre concerns the finding of clues
and the search for hidden designs, and its very form
underscores Mr. Pynchon’s obsession with conspiracies
and the existence of systems too complicated to understand."

Review of Pynchon's Bleeding Edge , Sept. 10, 2013

Background:  "Moss on the Wall," this  journal on that date.

A less complicated system —

"Plan 9 deals with the resurrection of the dead."

— Bill Murray in "Ed Wood"
 

For The Church of Plan 9

(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)

Sunday, July 2, 2017

Practically Cubist

Filed under: General — m759 @ 3:45 am

From an Anthony Lane movie review in the April 8, 2013,
issue of The New Yorker

"When the Lord God forbade his worshippers to bow down
before any graven image, [Rosario] Dawson’s face was
exactly the kind of thing He had in mind. No other star can
boast such sculptured features—except Vincent Cassel,
who is pretty damn graven himself. When the two of them
make love, in 'Trance,' one strong bone structure pressed
against another, it’s like a clash of major religions. What if
they had a family? The kids would be practically Cubist."

As for the other film Lane reviewed in that issue, "Blancanieves" —

See Snow White + Cube in this  journal.

See as well a related cartoon graveyard, also from April 8, 2013.

Friday, June 30, 2017

Hurriedly Put Together

Filed under: General,Geometry — Tags: , , — m759 @ 9:00 am

The previous post quoted one theologian on a book
by another theologian, saying its tone "is patronizing
and its arguments are hurriedly put together."

For a more leisurely sort of argument, see a 1995* remark 
by a mathematician, Ronald Shaw, quoted here on the morning
of Tuesday, June 27, in an update at the end of the previous day's
post "Upgrading to Six" —

". . . recall the notions of Eddington (1936) . . . ."

* In "Finite Geometry, Dirac Groups and the
Table of Real Clifford Algebras
," pages 59-99 of
R. Ablamowicz and P. Lounesto (eds.),
Clifford Algebras and Spinor Structures ,
Kluwer Academic Publishers, 1995.

Friday, June 23, 2017

“Information from the Middle of the Night”

Filed under: General — Tags: — m759 @ 2:02 am

The title is from an obituary in tonight's online New York Times.

Information —

See also another art publication cover from 1976 —

Thursday, May 18, 2017

Marquee Moon continues

Filed under: General — Tags: — m759 @ 1:01 pm

Exit stage right, enter stage center, exit stage left —

A search for "Darkness Doubled" in this journal yields a link 
to a post on "endgame art" which leads in turn to a post with
the following quotation —

"It is proposed that the two structures of grid and target
are the symbolic vehicles par excellence . . . ."

— Review of Rudolf Arnheim's The Power of the Center:
A Study of Composition in the Visual Arts
  (U. of Calif. Press, 1982).
Review by David A. Pariser, Studies in Art Education , Vol. 24, No. 3
(1983), pp. 210-213.

"Darkness Doubled" is a phrase from a song titled "Marquee Moon."

Sunday, April 23, 2017

A Day in June

Filed under: General — m759 @ 12:25 pm

Friday, April 14, 2017

Hudson and Finite Geometry

Filed under: General,Geometry — Tags: , — m759 @ 3:00 am

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

The above four-element 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 4-space (or, equivalently, as lines in the corresponding finite projective
3-space).

In order to view them in this way, one can view the tetrads as derived,
via the 15 two-element subsets of a six-element set, from the 16 elements
of the binary Galois affine space pictured above at top left.

This space is formed by taking symmetric-difference (Galois binary)
sums of the 15 two-element subsets, and identifying any resulting four-
element (or, summing three disjoint two-element subsets, six-element)
subsets with their complements.  This process was described in my note
"The 2-subsets of a 6-set are the points of a PG(3,2)" of May 26, 1986.

The space was later described in the following —

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

Monday, April 10, 2017

Heidegger for Passover

Filed under: General,Geometry — m759 @ 12:00 pm

From this  journal on August 7, 2010  (footnotes added today) —

The title of this post, "Rift Designs," is taken from Heidegger.

From a recent New Yorker  review of Absence of Mind  by Marilynne Robinson—

"Robinson is eloquent in her defense of the mind’s prerogatives, but her call for a renewed metaphysics might be better served by rereading Heidegger than by dusting off the Psalms."

Following this advice, we find—

"Propriation1 gathers the rift-design2 of the saying and unfolds it3  in such a way that it becomes the well-joined structure4 of a manifold showing."

— p. 415 of Heidegger's Basic Writings , edited by David Farrell Krell, HarperCollins paperback, 1993

"Das Ereignis versammelt den Aufriß der Sage und entfaltet ihn zum Gefüge des vielfältigen Zeigens." 

— Heidegger, Weg zur Sprache

1. "Mirror-Play of the Fourfold"

2. "Christ descending into the abyss"

3. Barrancas of Cuernavaca

4. Combinatorics, Philosophy, Geometry

Tuesday, April 4, 2017

Plan 9 Continues

Filed under: General,Geometry — Tags: — m759 @ 9:00 pm

"Plan 9 deals with the resurrection of the dead."

— Bill Murray in "Ed Wood"
 

For The Church of Plan 9

(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)

Tuesday, March 28, 2017

Backstory

Filed under: General,Geometry — Tags: , — m759 @ 12:06 am

Click here to enlarge.  Click the image for the source page.

The "this page" reference is to …

Finite Geometry of the Square and Cube.

Also from March 14, 2017 —

Related material

'Children of the Central Structure,' adapted from 'Children of the Damned'

Wednesday, March 15, 2017

Middle March:

Filed under: General — Tags: — m759 @ 2:02 pm

The Key to All Mythologies  in a Cartoon Graveyard

This is a sequel to yesterday's post Review, which
suggested a look at Lévi-Strauss's The Raw and The Cooked  
in Derrida's “Structure, Sign, and Play," and then a look at the

Financial Times  of February 26, 2010

"The metaphor for metamorphosis no keys unlock."

Steven H. Cullinane, November 7, 1986

Friday, March 10, 2017

The Transformers

Filed under: General — Tags: — m759 @ 10:00 pm

"The transformed urban interior is the spatial organisation of
an achiever, one who has crossed the class divide and who uses
space to express his membership of, not aspirations towards, 
an ascendant class in our society: the class of those people who 
earn their living by transformation— as opposed to the mere
reproduction— of symbols, such as writers, designers, and
academics"

The Social Logic of Space ,
     by Bill Hillier and Julienne Hanson,
     Cambridge University Press, 1984

For another perspective on the achievers, see The Deceivers .

Related material —

Exhibit A:

Exhibit B:

Edwin Schlossberg, 'Still Changes Through Structure' text piece

Exhibit C:

Thursday, March 9, 2017

Yale Architectural Figure

Filed under: General — Tags: , , — m759 @ 5:48 pm

Edwin Schlossberg, 'Still Changes Through Structure' text piece

See also Log24 posts related to "Go Set a Structure"
as well as "New Haven" + Grid.

Thursday, February 23, 2017

Midnight Special

Filed under: General — m759 @ 12:00 am

Click to enlarge:

See also, in this  journal, "Go Set a Structure,"
"Interior/Exterior," and "Midnight Special."

 

Friday, February 17, 2017

Kostant Is Dead

Filed under: General,Geometry — Tags: , — m759 @ 1:10 pm

"Bertram Kostant, professor emeritus of mathematics at MIT,
died at the Hebrew Senior Rehabilitation Center in Roslindale,
Massachusetts, on Thursday, Feb. 2, at the age of 88."

MIT News, story dated Feb. 16, 2017

See also a search for Kostant in this journal.

Regarding the discussions of symmetries and "facets" found in
that search —

Kostant:

A word about E(8). In my opinion, and shared by others,
E(8) is the most magnificent ‘object’ in all of mathematics.
It is like a diamond with thousands of facets. Each facet
offering a different view of its unbelievable intricate internal
structure.”

Cullinane:

In the Steiner system S(5, 8, 24) each octad might be
regarded as a "facet," with the order of the system's
automorphism group, the Mathieu group M24 , obtained
by multiplying the number of such facets, 759, by the
order of the octad stabilizer group, 322,560. 

Analogously

Platonic solids' symmetry groups   

Heptads and Heptapods

Filed under: General,Geometry — m759 @ 12:00 am

In the recent science fiction film "Arrival," Amy Adams portrays
a linguist, Louise Banks, who must learn to translate the language of
aliens ("Heptapods") who have just arrived in their spaceships.

The point of this tale seems to have something to do with Banks
learning, along with the aliens' language, their skill of seeing into
the future.

Louise Banks wannabes might enjoy the works of one
Metod Saniga, who thinks that finite geometry might have
something to do with perceptions of time.

See Metod Saniga, “Algebraic Geometry: A Tool for Resolving
the Enigma of Time?”, in R. Buccheri, V. Di Gesù and M. Saniga (eds.), 
Studies on the Structure of Time: From Physics to Psycho(patho)logy,
Kluwer Academic / Plenum Publishers, New York, 2000, pp. 137–166.
Available online at www.ta3.sk/~msaniga/pub/ftp/mathpsych.pdf .

Although I share an interest in finite geometry with Saniga —
see, for instance, his remarks on Conwell heptads in the previous post
and my own remarks in yesterday's post "Schoolgirls and Heptads" —
I do not endorse his temporal speculations.

Wednesday, February 15, 2017

Warp and Woof

Filed under: General,Geometry — m759 @ 3:00 pm

Space —

Space structure —

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Space shuttle —

Related ethnic remarks —

As opposed to Michael  Larsen —

Funny, you don't look  Danish.

Sunday, February 12, 2017

Colorful Tales

Filed under: General — Tags: — m759 @ 1:23 pm

“Perhaps the philosophically most relevant feature of modern science
is the emergence of abstract symbolic structures as the hard core
of objectivity behind— as Eddington puts it— the colorful tale of
the subjective storyteller mind.”

— Hermann Weyl, Philosophy of  Mathematics and
    Natural Science 
, Princeton, 1949, p. 237

Harvard University Press on the late Angus Fletcher, author of
The Topological Imagination  and Colors of the Mind

From the Harvard webpage for Colors of the Mind

Angus Fletcher is one of our finest theorists of the arts,
the heir to I. A. Richards, Erich Auerbach, Northrop Frye.
This… book…  aims to open another field of study:
how thought— the act, the experience of thinking—
is represented in literature.

. . . .

Fletcher’s resources are large, and his step is sure.
The reader samples his piercing vision of Milton’s

Satan, the original Thinker,
leaving the pain of thinking
as his legacy for mankind.

A 1992 review by Vinay Dharwadker of Colors of the Mind —

See also the above word "dianoia" in The Echo in Plato's Cave.
Some context 

This post was suggested by a memorial piece today in
the Los Angeles Review of Books

A Florilegium for Angus Fletcher

By Kenneth Gross, Lindsay Waters, V. N. Alexander,
Paul Auster, Harold Bloom, Stanley Fish, K. J. Knoespel,
Mitchell Meltzer, Victoria Nelson, Joan Richardson,
Dorian Sagan, Susan Stewart, Eric Wilson, Michael Wood

Fletcher reportedly died on November 28, 2016.

"I learned from Fletcher how to apprehend
the daemonic element in poetic imagination."

— Harold Bloom in today's Los Angeles florilegium

For more on Bloom and the daemonic, see a Log24 post,
"Interpenetration," from the date of Fletcher's death.

Some backstory:  Dharwadker in this journal.

Sunday, January 8, 2017

A Theory of Everything

Filed under: General,Geometry — Tags: , — m759 @ 7:11 pm

The title refers to the Chinese book the I Ching ,
the Classic of Changes .

The 64 hexagrams of the I Ching  may be arranged
naturally in a 4x4x4 cube. The natural form of transformations
("changes") of this cube is given by the diamond theorem.

A related post —

The Eightfold Cube, core structure of the I Ching

Saturday, December 24, 2016

Early X Piece

Filed under: General,Geometry — Tags: — m759 @ 11:00 am

In memory of an American artist whose work resembles that of
the Soviet constructivist Karl Ioganson (c. 1890-1929).

The American artist reportedly died on Thursday, Dec. 22, 2016.

"In fact, the (re-)discovery of this novel structural principle was made in 1948-49 by a young American artist whom Koleichuk also mentions, Kenneth Snelson. In the summer of 1948, Snelson had gone to study with Joseph Albers who was then teaching at Black Mountain College. . . . One of the first works he made upon his return home was Early X Piece  which he dates to December 1948 . . . . "

— "In the Laboratory of Constructivism:
      Karl Ioganson's Cold Structures,"
      by Maria GoughOCTOBER  Magazine, MIT,
      Issue 84, Spring 1998, pp. 91-117

The word "constructivism" also refers to a philosophy of mathematics.
See a Log24 post, "Constructivist Witness,"  of 1 AM ET on the above
date of death.

Friday, December 23, 2016

Requiem for a Mathematician

Filed under: General,Geometry — m759 @ 2:10 pm

From a Dec. 21 obituary posted by the
University of Tennessee at Knoxville —

"Wade was ordained as a pastor and served
at Oakwood Baptist Church in Knoxville."

Other information —

In a Log24 post, "Seeing the Finite Structure,"
of August 16, 2008, Wade appeared as a co-author
of the Walsh series book mentioned above —

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

Walsh Series: An Introduction
to Dyadic Harmonic Analysis
,
by F. Schipp et al.,
Taylor & Francis, 1990

From the 2008 post —

The patterns on the faces of the cube on the cover
of Walsh Series above illustrate both the 
Walsh functions of order 3 and the same structure
in a different guise, subspaces of the affine 3-space 
over the binary field. For a note on the relationship
of Walsh functions to finite geometry, see 
Symmetry of Walsh Functions.

Friday, December 9, 2016

Snow Dance

Filed under: General,Geometry — Tags: , — m759 @ 9:00 am

See Ballet Blanc  in this journal.

For a darker perspective, click on the image below.

IMAGE- Detail of large 'Search for the Lost Tesseract' image with Amy Adams, Richard Zanuck, 'snowflake' structure, and white gloves

See also Cartier in The Hexagon of Opposition.

Happy birthday to Kirk Douglas.

Kirk Douglas in 'Diamonds'

Wednesday, December 7, 2016

Emch as a Forerunner of S(5, 8, 24)

Filed under: General,Geometry — m759 @ 1:00 pm

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,
Amer. J. Math. 53 (1931), 217-240.

11.  E. Witt, Die 5-fach transitiven Gruppen von Mathieu,
Abh. Hamburg 12 (1938), 256-264. 

— William M. Kantor, book review (pdf), 
Bulletin of the American Mathematical Society, September 1981

Wednesday, November 30, 2016

In Nuce

Filed under: General,Geometry — Tags: , , — m759 @ 3:48 am
 

Excerpts from James C. Nohrnberg, "The Master of the Myth of Literature: An Interpenetrative Ogdoad for Northrop Frye," Comparative Literature  Vol. 53, No. 1 (Winter, 2001), pp. 58-82

From page 58 —
"… the posthumously revealed Notebooks. A major project of the latter was his 'Ogdoad': two groups of four books each. '[T]he second group of four […] were considered to be Blakean "emanations" or counterparts of the first four,' like 'the "double mirror" structure of The Great Code  and Words with Power : two inter-reflecting parts of four chapters apiece,' Michael Dolzani reports.* "

* P. 22 of Rereading Frye: The Published and Unpublished Works , ed. David Boyd and Imre Salusinszky, Frye Studies [series] (Toronto: University of Toronto Press, 1998). [Abbreviated as RF .]


From page 62 —
"Visionaries like Blake and dramatists like Wagner seem to be working from some larger, mythic blueprint present in nuce  from very early on."

From page 63 —
"Frye's hypothetical books and will-to-totality were obviously fruitful; if the beckoning star was illusory, it nonetheless settled on a real birthplace. The sought-for constructs substituted their scaffolding for a backbone-like confidence in pre-given beliefs; possession of the latter is why Tories like Dr. Johnson and T.S. Eliot could do quite nicely without the constructs. Frye's largely imaginary eightfold roman  may have provided him a personal substitute— or alternative— for both ideology and myth."

From page 69 —
"For Frye the chief element of imaginative or expressive form is the myth, which functions structurally in literature like geometric shapes in painting."

From page 71 —
"The metaphysical skyhook lifting the artist free from unreflective social commitment is often a latent or manifest archetype that his work renews or reworks."

From page 77 —
"Frye's treatises— so little annotated themselves— are the notes writ large; the notes in the Notebooks are treatises writ small. They interpenetrate. Denham quotes 'the masters of the T'ien-tai school of Mahayana Buddhism' as saying '[t]he whole world is contained in a mustard seed' (RF  158, 160), and Frye quotes Keats: 'Every point of thought is the center of an intellectual world' (Study  159; cf. Great Code  167-68 and AC  61). …. [Frye’s] complex books were all generated out of the monadic obiter dicta . His kingdom 'is like a grain of mustard seed, which a man took, and cast into his garden, and it grew' (Luke 13:18-19)."

Wednesday, November 23, 2016

Yogiism

Filed under: General,Geometry — Tags: , — m759 @ 12:31 pm

From the American Mathematical Society (AMS) webpage today —

From the current AMS Notices

Related material from a post of Aug. 6, 2014

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

(Here "five point sets" should be "five-point sets.")

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  structure* now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

* The phrase as used here is a deliberate 
abuse of language .  For the real definition of 
“symplectic structure,” see (for instance) 
“Symplectic Geometry,” by Ana Cannas da Silva
(article written for Handbook of Differential
Geometry 
, Vol 2.) To establish that the above
figure is indeed symplectic , see the post 
Zero System of July 31, 2014.

Friday, October 28, 2016

Diamond-Theorem Application

Filed under: General,Geometry — Tags: — m759 @ 1:06 pm
 

Abstract:

"Protection of digital content from being tapped by intruders is a crucial task in the present generation of Internet world. In this paper, we proposed an implementation of new visual secret sharing scheme for gray level images using diamond theorem correlation. A secret image has broken into 4 × 4 non overlapped blocks and patterns of diamond theorem are applied sequentially to ensure the secure image transmission. Separate diamond patterns are utilized to share the blocks of both odd and even sectors. Finally, the numerical results show that a novel secret shares are generated by using diamond theorem correlations. Histogram representations demonstrate the novelty of the proposed visual secret sharing scheme."

— "New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure," by  V. Harish, N. Rajesh Kumar, and N. R. Raajan.

Published in: 2016 International Conference on Circuit, Power and Computing Technologies (ICCPCT).
Date of Conference: 18-19 March 2016. Publisher: IEEE.
Date Added to IEEE Xplore: 04 August 2016

Excerpts —

Related material — Posts tagged Diamond Theorem Correlation.

Friday, September 30, 2016

Desmic Midrash

Filed under: General — Tags: , , — m759 @ 9:19 am

The author of the review in the previous post, Dara Horn, supplies
below a midrash on "desmic," a term derived from the Greek desmé
( δέσμη: bundle, sheaf , or, in the mathematical sense, pencil —
French faisceau ), which is related to the term desmos , bond …

(The term "desmic," as noted earlier, is relevant to the structure of
Heidegger's Sternwürfel .)

The Horn midrash —

(The "medieval philosopher" here is not the remembered pre-Christian
Ben Sirah (Ecclesiasticus ) but the philosopher being read — Maimonides:  
Guide for the Perplexed , 3:51.)

Here of course "that bond" may be interpreted as corresponding to the
Greek desmos  above, thus also to the desmic  structure of the
stellated octahedron, a sort of three-dimensional Star of David.

See "desmic" in this journal.

Wednesday, September 28, 2016

Star Wars

Filed under: General — Tags: , , , — m759 @ 11:00 pm

See also in this journal "desmic," a term related
to the structure of Heidegger's Sternwürfel .

Tuesday, September 27, 2016

Chomsky and Lévi-Strauss in China

Filed under: General,Geometry — Tags: , , — m759 @ 7:31 am

Or:  Philosophy for Jews

From a New Yorker  weblog post dated Dec. 6, 2012 —

"Happy Birthday, Noam Chomsky" by Gary Marcus—

"… two titans facing off, with Chomsky, as ever,
defining the contest"

"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."

Socrates and the slave boy discussed a rather elementary "truth
about geometry" — A diamond inscribed in a square has area 2
(and side the square root of 2) if the square itself has area 4
(and side 2).

Consider that not-particularly-deep structure from the Meno dialogue
in the light of the following…

The following analysis of the Meno diagram from yesterday's
post "The Embedding" contradicts the Lévi-Strauss dictum on
the impossibility of going beyond a simple binary opposition.
(The Chinese word taiji  denotes the fundamental concept in
Chinese philosophy that such a going-beyond is both useful
and possible.)

The matrix at left below represents the feminine yin  principle
and the diamond at right represents the masculine yang .

      From a post of Sept. 22,
"Binary Opposition Illustrated" —

A symbol of the unity of yin and yang —

Related material:

A much more sophisticated approach to the "deep structure" of the
Meno diagram —

The larger cases —

The diamond theorem

Thursday, September 22, 2016

Binary Opposition Illustrated

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm

Click the above image for remarks on
"deep structure" and binary opposition.

See also the eightfold cube.

Tuesday, September 20, 2016

Savage Logic

Filed under: General,Geometry — m759 @ 9:29 pm

From "The Cerebral Savage," by Clifford Geertz —

(Encounter, Vol. 28 No. 4 (April 1967), pp. 25-32.)

From http://www.diamondspace.net/about.html

The diamond theorem

Saturday, September 17, 2016

Composition

Filed under: General — m759 @ 9:29 am

The late Edward Albee, as quoted today in The Telegraph :

“I tell my students, if you want to know something
about the structure of a play, listen to some Bach
preludes and fugues. I discovered classical music
when I was eight, nine, 10 years old, and I think
I learnt something about the nature of dramatic
structure from the nature of the music I was
listening to. I probably think of myself half the
time as a composer.”

See also  Box  as  Bach's.

Friday, September 16, 2016

A Counting-Pattern

Filed under: General,Geometry — Tags: , — m759 @ 10:48 am

Wittgenstein, 1939

Dolgachev and Keum, 2002

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

For some related material, see posts tagged Priority.

Monday, September 12, 2016

The Kummer Lattice

The previous post quoted Tom Wolfe on Chomsky's use of
the word "array." 

An example of particular interest is the 4×4  array
(whether of dots or of unit squares) —

      .

Some context for the 4×4 array —

The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .

Further background on the Kummer lattice:

Alice Garbagnati and Alessandra Sarti, 
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action." 
To appear in Rocky Mountain J. Math.

The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite  geometry, see the website
Finite Geometry of the Square and Cube.

Some further context

"To our knowledge, the relation of the Golay code
to the Kummer lattice is a new observation."

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 
"

As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface.  The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.

* Update of Sept. 14: "Uncoordinatized," but parametrized  by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.

Wednesday, September 7, 2016

Grammar and Patterns

Filed under: General,Geometry — m759 @ 10:00 pm

"May, / The months [sic ] of understanding" — Wallace Stevens

Saturday, May 21, 2016

Grammar

Filed under: Uncategorized — m759 @ 12:00 PM 

Related material 

The Lindbergh Manifesto and The Leibniz Medal.

 

"If pure mathematics does spring from sub-conscious intuitions— already deep-structured as are grammatical patterns in the transformational-generative theory of language?— if the algebraic operation arises from wholly internalized pattern-weaving, how then can it, at so many points, mesh with, correspond to, the material forms of the world?"

— Steiner, George. Grammars of Creation
(Gifford Lectures, 1990). (Kindle Locations 2494-2496).
Open Road Media. Kindle Edition. 

Good question.

See Bedtime Story (Sept. 1, 2016).

Sunday, September 4, 2016

Piled High and Deep

Filed under: General — m759 @ 5:09 pm

Quoted here at 10 PM Pacific Time on Friday night —

"If I should die before I wake,
All my bone and sinew take
Put me in the compost pile
To decompose me for a while . . . ."

— Poem by Lee Hays

Saturday, September 3, 2016

Phenomenology*

Filed under: General — Tags: — m759 @ 8:30 pm

For the Church of Synchronology, a correction of
a recent New York Times  obituary by Daniel Lewis —

Actor Gene Wilder died early Monday, Aug. 29, not, as 
earlier reported, late Sunday, Aug. 28.

NY Times correction: Gene Wilder died early on Mon. Aug. 29, not on Sun. Aug. 28.

See also the last Log24 post of Sunday night, Aug. 28 (Angles of Vision)
and the first post of Monday morning, Aug. 29, 2016 (Roll Credits).

* For some reading related to the title, see an Evil Genius page
by the late David Lavery mentioning Colin Wilson's novel
The Mind Parasites .  Great entertainment for the tinfoil-hat crowd —

"More and more I feel like the narrator of Colin Wilson's 
The Mind Parasites , a phenomenologist who, along with
a dedicated group of compatriots, struggles clandestinely
to overthrow alien invaders that have secretly
taken captive the 'deep structure' of the human mind." 

Friday, August 12, 2016

Dustbucket Physics

Filed under: General,Geometry — Tags: — m759 @ 3:00 pm

Peter Galison, a Harvard professor, is a defender of
the Vienna Circle and the religion of Scientism.

From Galison's “Structure of Crystal, Bucket of Dust,” in
Circles Disturbed: The Interplay of Mathematics and Narrative ,
edited by Apostolos Doxiadis and Barry Mazur, pp. 52-78 
(Princeton: Princeton U. Press, 2012) 

Galison's final paragraph —

"Perhaps, then, it should not surprise us too much if,
as Wheeler approaches the beginning-end of all things,
there is a bucket of Borelian dust. Out of this filth,
through the proposition machine of quantum mechanics
comes pregeometry; pregeometry makes geometry;
geometry gives rise to matter and the physical laws
and constants of the universe. At once close to and far
from the crystalline story that Bourbaki invoked,
Wheeler’s genesis puts one in mind of Genesis 3:19:
'In the sweat of thy face shalt thou eat bread, till thou
return unto the ground; for out of it wast thou taken:
for dust thou art, and unto dust shalt thou return.' "

For fans of Scientism who prefer more colorful narratives —

Thursday, July 14, 2016

Midnight Special

Filed under: General,Geometry — Tags: , — m759 @ 12:00 am

(Continued)

"Poincaré said that science is no more a collection of facts than a house is a collection of bricks. The facts have to be ordered or structured, they have to fit a theory, a construct (often mathematical) in the human mind.

… Mathematics may be art, but to the general public it is a black art, more akin to magic and mystery. This presents a constant challenge to the mathematical community: to explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that architecture is the best of the arts to compare with mathematics. The analogy between the two subjects is not hard to describe and enables abstract ideas to be exemplified by bricks and mortar, in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

A post  from this  journal later in 2010 —

The above post's date — May 20, 2010 — was
the date of death for mathematician Walter Rudin.

The above post from that date has a link to the
Heinlein story "And He Built a Crooked House."
A not-so-crooked house —

Tuesday, July 5, 2016

For the Children in the Apple Tree (continued)

Filed under: General — Tags: , — m759 @ 6:00 am

(See previous posts now tagged Apple Tree Children.)

See as well the comic book in "Midnight Special" —

(Image previously posted in "Common Core vs. Central Structure")

Sunday, July 3, 2016

Articulation

Filed under: General — Tags: — m759 @ 10:10 am

Notes for a monkey grammarian

"Visual forms— lines, colors, proportions, etc.—
are just as capable of articulation ,
i.e. of complex combination, as words.
But the laws that govern this sort of articulation
are altogether different from the laws of syntax
that govern language. The most radical difference
is that visual forms are not discursive .
They do not present their constituents successively,
but simultaneously, so the relations determining
a visual structure are grasped in one act of vision."

— Susanne K. LangerPhilosophy in a New Key

See also Langer's New Key in this journal.

Related material —

Thursday, June 23, 2016

Raiders of the Lost Code

Filed under: General — Tags: — m759 @ 11:00 am

From a web page

Breaking the Code of the Archetypal Self:
An Introductory Overview of the Research Discoveries
Leading to Neo-Jungian Structural Psychoanalysis

Dr. Moore will introduce his research and discoveries
with regard to the deep structures of the Self.
Tracing the foundations in the tradition of Jung’s
affirmation of the collective unconscious, Moore
will present his “decoding of the Diamond Body,”
a mapping of the deep structures of the Great Code
of the psyche. . . .

From the same web site

Googling "Jung" + "Diamond Body" shows that
Moore's terminology differs from Jung's.
The octahedron that Moore apparently associates
with his "diamond body" was discussed by Jung
in a different context. See selections from Ch. 14
of Jung's Aion
 "The Structure and Dynamics of the Self."

Dr. Moore appears as well in the murder-suicide story 
of last night's 11:18 PM ET post.

For the relevance of Aion  to "deep structures,"
see Jung + Diamond + Structure in this  journal
and, more specifically, "Deep  Structure."

Monday, June 20, 2016

Plan 9 Continues

Filed under: General — Tags: — m759 @ 11:00 am

See …

At the Still Point … (February 12, 2008)

For Balanchine's Birthday (January 9, 2007)

Go Set a Structure (Various dates)

and …

Tuesday, June 14, 2016

Model Kit

Filed under: General,Geometry — Tags: — m759 @ 12:14 pm

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

A set of 7 partitions of the 2x2x2 cube that is invariant under PSL(2, 7) acting on the 'knight' coordinatization

From the post Design Cube of July 23, 2015 —

Broken Symmetries  in  Diamond Space 

Wednesday, May 25, 2016

Kummer and Dirac

From "Projective Geometry and PT-Symmetric Dirac Hamiltonian,"
Y. Jack Ng  and H. van Dam, 
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239

(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)

" Studies of spin-½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. 1 "

1 These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is 
related to that of Kummer’s 166 configuration . . . ."

[4]

O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef

E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135

F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished


A remark of my own on the structure of Kummer’s 166 configuration . . . .

See that structure in this  journal, for instance —

See as well yesterday morning's post.

Thursday, May 19, 2016

Kulturkampf

Filed under: General,Geometry — Tags: — m759 @ 2:27 am

From a check tonight of The New York Review of Books

These NYRB  stories from May 15 and May 13 suggest a
review of images on Ratner's Star  and on the Eye of God.

IMAGE- 'Ratner's Star,' by Don DeLillo (1976)

Above image reposted from Jan. 10, 2014

I. The structures in the Diamond Puzzle

Adam and God (Sistine Chapel), with Jungian Self-Symbol and Ojo de Dios (The Diamond Puzzle)

Click on image for Jungian background.

II: The structure on a recent cover of Semiotica

'Semiotica' cover and article by Solomon Marcus on Levi-Strauss's 'canonic formula' of myth

Above images reposted from May 5, 2016

Related material:  The previous post, Dueling Formulas.

Thursday, May 5, 2016

Solomon’s Seal

Filed under: General,Geometry — Tags: — m759 @ 11:00 pm

Excerpt from a post of November 4, 2009

I. The structures in the Diamond Puzzle

Adam and God (Sistine Chapel), with Jungian Self-Symbol and Ojo de Dios (The Diamond Puzzle)

Click on image for Jungian background.

II: The structure on a recent cover of Semiotica

'Semiotica' cover and article by Solomon Marcus on Levi-Strauss's 'canonic formula' of myth

For some related material, see a search 
for Solomon Marcus in this  journal.

Tuesday, April 26, 2016

Interacting

Filed under: General,Geometry — Tags: — m759 @ 8:31 pm

"… I would drop the keystone into my arch …."

— Charles Sanders Peirce, "On Phenomenology"

" 'But which is the stone that supports the bridge?' Kublai Khan asks."

— Italo Calvino, Invisible Cities, as quoted by B. Elan Dresher.

(B. Elan Dresher. Nordlyd  41.2 (2014): 165-181,
special issue on Features edited by Martin Krämer,
Sandra Ronai and Peter Svenonius. University of Tromsø –
The Arctic University of Norway.
http://septentrio.uit.no/index.php/nordlyd)

Peter Svenonius and Martin Krämer, introduction to the
Nordlyd  double issue on Features —

"Interacting with these questions about the 'geometric' 
relations among features is the algebraic structure
of the features."

For another such interaction, see the previous post.

This  post may be viewed as a commentary on a remark in Wikipedia

"All of these ideas speak to the crux of Plato's Problem…."

See also The Diamond Theorem at Tromsø and Mere Geometry.

Monday, April 25, 2016

Seven Seals

Filed under: General,Geometry — Tags: , — m759 @ 11:00 pm

 An old version of the Wikipedia article "Group theory"
(pictured in the previous post) —

"More poetically "

From Hermann Weyl's 1952 classic Symmetry

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

The seven seals from the previous post, with some context —

These models of projective points are drawn from the underlying
structure described (in the 4×4 case) as part of the proof of the
Cullinane diamond theorem .

Sunday, April 17, 2016

The Thing and I

Filed under: General,Geometry — Tags: , — m759 @ 2:01 pm

The New York Times  philosophy column yesterday —

The Times's philosophy column "The Stone" is named after the legendary
"philosophers' stone." The column's name, and the title of its essay yesterday
"Is that even a thing?" suggest a review of the eightfold cube  as "The object
most closely resembling a 'philosophers' stone' that I know of" (Page 51 of
the current issue of a Norwegian art quarterly, KUNSTforum.as).

The eightfold cube —

Definition of Epiphany

From James Joyce’s Stephen Hero , first published posthumously in 1944. The excerpt below is from a version edited by John J. Slocum and Herbert Cahoon (New York: New Directions Press, 1959).

Three Times:

… By an epiphany he meant a sudden spiritual manifestation, whether in the vulgarity of speech or of gesture or in a memorable phase of the mind itself. He believed that it was for the man of letters to record these epiphanies with extreme care, seeing that they themselves are the most delicate and evanescent of moments. He told Cranly that the clock of the Ballast Office was capable of an epiphany. Cranly questioned the inscrutable dial of the Ballast Office with his no less inscrutable countenance:

— Yes, said Stephen. I will pass it time after time, allude to it, refer to it, catch a glimpse of it. It is only an item in the catalogue of Dublin’s street furniture. Then all at once I see it and I know at once what it is: epiphany.

— What?

— Imagine my glimpses at that clock as the gropings of a spiritual eye which seeks to adjust its vision to an exact focus. The moment the focus is reached the object is epiphanised. It is just in this epiphany that I find the third, the supreme quality of beauty.

— Yes? said Cranly absently.

— No esthetic theory, pursued Stephen relentlessly, is of any value which investigates with the aid of the lantern of tradition. What we symbolise in black the Chinaman may symbolise in yellow: each has his own tradition. Greek beauty laughs at Coptic beauty and the American Indian derides them both. It is almost impossible to reconcile all tradition whereas it is by no means impossible to find the justification of every form of beauty which has ever been adored on the earth by an examination into the mechanism of esthetic apprehension whether it be dressed in red, white, yellow or black. We have no reason for thinking that the Chinaman has a different system of digestion from that which we have though our diets are quite dissimilar. The apprehensive faculty must be scrutinised in action.

— Yes …

— You know what Aquinas says: The three things requisite for beauty are, integrity, a wholeness, symmetry and radiance. Some day I will expand that sentence into a treatise. Consider the performance of your own mind when confronted with any object, hypothetically beautiful. Your mind to apprehend that object divides the entire universe into two parts, the object, and the void which is not the object. To apprehend it you must lift it away from everything else: and then you perceive that it is one integral thing, that is a  thing. You recognise its integrity. Isn’t that so?

— And then?

— That is the first quality of beauty: it is declared in a simple sudden synthesis of the faculty which apprehends. What then? Analysis then. The mind considers the object in whole and in part, in relation to itself and to other objects, examines the balance of its parts, contemplates the form of the object, traverses every cranny of the structure. So the mind receives the impression of the symmetry of the object. The mind recognises that the object is in the strict sense of the word, a thing , a definitely constituted entity. You see?

— Let us turn back, said Cranly.

They had reached the corner of Grafton St and as the footpath was overcrowded they turned back northwards. Cranly had an inclination to watch the antics of a drunkard who had been ejected from a bar in Suffolk St but Stephen took his arm summarily and led him away.

— Now for the third quality. For a long time I couldn’t make out what Aquinas meant. He uses a figurative word (a very unusual thing for him) but I have solved it. Claritas is quidditas . After the analysis which discovers the second quality the mind makes the only logically possible synthesis and discovers the third quality. This is the moment which I call epiphany. First we recognise that the object is one  integral thing, then we recognise that it is an organised composite structure, a thing  in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that  thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany.

Having finished his argument Stephen walked on in silence. He felt Cranly’s hostility and he accused himself of having cheapened the eternal images of beauty. For the first time, too, he felt slightly awkward in his friend’s company and to restore a mood of flippant familiarity he glanced up at the clock of the Ballast Office and smiled:

— It has not epiphanised yet, he said.

Monday, April 4, 2016

Cube for Berlin

Foreword by Sir Michael Atiyah —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

 Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010

Judy Bass, Los Angeles Times , March 12, 1989 —

"Like Rubik's Cube, The Eight  demands to be pondered."

As does a figure from 1984, Cullinane's Cube —

The Eightfold Cube

For natural group actions on the Cullinane cube,
see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."

See also the recent post Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Related remark from the literature —

http://www.log24.com/log/pix11B/110918-Felsner.jpg

Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see "Cube Space, 1984-2003."

(From Anatomy of a Cube, Sept. 18, 2011.)

Friday, April 1, 2016

Wonders of the Invisible World

Filed under: General — m759 @ 7:59 pm

Related four-dimensional figure from 1976 —

See as well "Or Only Die."

Friday, March 4, 2016

Cube Bricks 1984

Filed under: General,Geometry — Tags: , — m759 @ 12:06 pm

An Approach to Symmetric Generation of the Simple Group of Order 168

Related aesthetics —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

Sunday, February 21, 2016

Cards

Filed under: General — Tags: — m759 @ 11:00 am

Reposted from an excellent weblog —

Taking note

A blog on the nature of note-taking.

Thursday, April 2, 2009

Nabokov on Index Cards

I came across a recent post on Nabokov's Index Cards by Michael Leddy, which I found interesting.

Nabokov wrote with Index Cards not so much because they allowed associative progression (or because they were somehow like precursors of hypertext for him), but rather because he had such a clear vision of what he meant to create that he could start anywhere in describing it: "The pattern of the thing precedes the thing. I fill in the gaps of the crossword at any spot I happen to choose. These bits I write on index cards until the novel is done. My schedule is flexible but I am rather particular about my instruments: lined Bristol cards and well-sharpened, not too hard, pencils capped with erasers."

"… Since this entire structure, dimly illumined in one's mind, can be compared to a painting, and since you do not have to work gradually from left to right for its proper perception, I may direct my flashlight at any part or particle of the picture when setting it down in writing. I do not begin my novel at the beginning I do not reach chapter three before I reach chapter four… This is why I like writing my stories and novels on index cards, numbering them later when the whole set is complete. Every card is rewritten many times …"

"find a quiet spot (pace the booming surf and rattling wind) where to write. This I do on scrambled index cards (my text existing already there in invisible lead) which I gradually fill in and sort out, using up in the process more pencil sharpeners than pencils; but I have spoken of this in several earlier questionnaires"

Posted by MK at  

Labels: 

From the date of the above Taking Note  post, a post from this  weblog
seems a suitable sermon for the Church of Synchronology.

Friday, February 19, 2016

A Watchman for Nolan

Filed under: General — Tags: , — m759 @ 8:31 pm

The title refers to the Watchman Rorschach in "Go Set a Structure"
and to Christopher Nolan, director of the 2014 film "Interstellar."

"Watchmen"-like art in next Sunday's NY Times Book Review

'Watchmen'-like art in the Feb. 21, 2016, NY Times Book Review

Saturday, January 30, 2016

Pope’s Geometry

Filed under: General,Geometry — m759 @ 10:21 am

From page 56 of The Science Fiction of Mark Clifton ,
Southern Illinois University Press, 1980 —
 

See also the following image in this journal

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven).

Saturday, January 2, 2016

A Very Strange Enchanted Town

Filed under: General — m759 @ 3:00 am

The New Yorker  on March 2, 1992 —

Related material:  Go Set a Structure.

Sunday, November 15, 2015

The Diamond and the Cube

Filed under: General,Geometry — Tags: — m759 @ 8:00 pm

Anyone who clicked on the Dirac search at the end of
the previous post, "Dirac's Diamond," may wonder why the
"Solomon's Cube" post of 11 AM Sunday, March 1, 2009,
appeared in the Dirac search results, since there is no
apparent mention of Dirac in that Sunday post.

Use the source

<!– See also "a linear transformation of V6… which preserves
the Klein quadric; in this way we arrive at the isomorphism of
Sym(8) withthe full orthogonal group O+(6; 2)." in "The
Classification of Flats in PG(9,2) which are External to the
Grassmannian G1,4,2 Authors: Shaw, Ron;
&#160;Maks, Johannes;&#160;Gordon, Neil; Source: Designs,
Codes and Cryptography, Volume 34, Numbers 2-3, February
2005 , pp. 203-227; Publisher: Springer.&#160; For more details,
see "Finite Geometry, Dirac Groups and the Table of Real
Clifford Algebras," by R. Shaw (U. of Hull), pp. 59-99 in
Clifford Algebras and Spinor Structures, by By Albert
Crumeyrolle, Rafa&#322; Ab&#322;amowicz, Pertti Lounesto,
published by Springer, 1995. –>

Saturday, November 7, 2015

Clarifying Dyson

Filed under: General,Geometry — m759 @ 12:00 pm

The previous post quoted a passage from Turing's Cathedral ,
a 2012 book by George Dyson —

A passage in 'Turing's Cathedral' that recalls the Go chip in 'Wild Palms'

It should be noted that Dyson's remarks on "two species of
bits," space, time, "structure and sequence" and logic gates
are from his own idiosyncratic attempt to create a philosophy
based on the workings of computers.  These concepts are not,
so far as I can tell, part of anyone else's approach to the subject.

For a more standard introduction to how computers work, see
(for instance) a book by an author Dyson admires:

The Pattern on the Stone , by W. Daniel Hillis (Basic Books, 1998).

PREFACE: MAGIC IN THE STONE

I etch a pattern of geometric shapes onto a stone.
To the uninitiated, the shapes look mysterious and
complex, but I know that when arranged correctly
they will give the stone a special power, enabling it
to respond to incantations in a language no human
being has ever spoken. I will ask the stone questions
in this language, and it will answer by showing me a
vision: a world created by my spell, a world imagined
within the pattern on the stone.

A few hundred years ago in my native New England,
an accurate description of my occupation would have
gotten me burned at the stake. Yet my work involves
no witchcraft; I design and program computers. The
stone is a wafer of silicon, and the incantations are
software. The patterns etched on the chip and the
programs that instruct the computer may look
complicated and mysterious, but they are generated
according to a few basic principles that are easily
explained. . . . .

Hillis's title suggests some remarks unrelated to computers —

See Philosopher + Stone in this journal.

Saturday, October 31, 2015

Weaving World…

Filed under: General,Geometry — m759 @ 6:00 pm

Continues.

Addendum —


      See also Symplectic Structure 
      and Stevens's Rock.

Thursday, October 22, 2015

Objective Quality

Filed under: General,Geometry — Tags: — m759 @ 2:26 am

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.

Thursday, October 15, 2015

Contrapuntal Interweaving

Filed under: General,Geometry — Tags: — m759 @ 2:01 am

The title is a phrase from R. D. Laing's book The Politics of Experience .
(Published in the psychedelic year 1967. The later "contrapuntal interweaving"
below is of a less psychedelic nature.)

An illustration of the "interweaving' part of the title —
The "deep structure" of the diamond theorem:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven).

The word "symplectic" from the end of last Sunday's (Oct. 11) sermon
describes the "interwoven" nature of the above illustration.

An illustration of the "contrapuntal" part of the title (click to enlarge):

The diamond-theorem correlation

 

Monday, October 5, 2015

Forms that Rhyme:

Filed under: General,Geometry — m759 @ 12:01 am

The 4×4 Latin-Square Structures 

Click image for background.

Monday, September 28, 2015

Cracker Jack Prize

Filed under: General,Geometry — Tags: — m759 @ 11:00 pm

From a post of July 24, 2011

Mira Sorvino in 'The Last Templar'

A review —

“The story, involving the Knights Templar, the Vatican, sunken treasure,
the fate of Christianity and a decoding device that looks as if it came out of 
a really big box of medieval Cracker Jack, is the latest attempt to combine
Indiana Jones derring-do with ‘Da Vinci Code’ mysticism.”

— The New York Times

A feeble attempt at a purely mathematical "decoding device"
from this journal earlier this month

Image that may or may not be related to the extended binary Golay code and the large Witt design

For some background, see a question by John Baez at Math Overflow
on Aug. 20, 2015.

The nonexistence of a 24-cycle in the large Mathieu group
might discourage anyone hoping for deep new insights from
the above figure.

See Marston Conder's "Symmetric Genus of the Mathieu Groups" —

Saturday, September 19, 2015

Geometry of the 24-Point Circle

Filed under: General,Geometry — Tags: , , — m759 @ 1:13 am

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?

Image that may or may not be related to the extended binary Golay code and the large Witt design

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.

Tuesday, September 8, 2015

Point Omega*

Filed under: General — m759 @ 12:00 pm

Fareed Zakaria in an online Aug. 21
New York Times  book review

" Most intellectuals think ideas matter.
In one of his most famous and oft-­quoted lines,
John Maynard Keynes declared, 'Practical men
who believe themselves to be quite exempt from
any intellectual influence are usually the slaves
of some defunct economist. Madmen in authority,
who hear voices in the air, are distilling their frenzy
from some academic scribbler of a few years back.'

Scott L. Montgomery and Daniel Chirot concur,
arguing that ideas 'do not merely matter; they matter
immensely, as they have been the source for decisions
and actions that have structured the modern world.' 
In The Shape of the New: Four Big Ideas and How
They Made the Modern World 
, Montgomery and
Chirot make the case for the importance of four
­powerful ideas, rooted in the European Enlightenment,
that have created the world as we know it.
'Invading armies can be resisted,' they quote
Victor Hugo. 'Invading ideas cannot be.' "

* Related material: Point Omega , a book
   by Don DeLillo, in this journal.

Tuesday, August 18, 2015

A Wrinkle in Terms

Filed under: General,Geometry — m759 @ 8:23 am

The phrase “the permutation group Sn” refers to a
particular  group of permutations that act on an
-element set N— namely, all  of them. For a given n ,
there are, in general, many  permutation groups that
act on N.  All but one are smaller than S.

In other words, the phrase “the permutation group Sn
does not  imply that “Sn ” is a symbol for a structure
associated with n  called “the  permutation group.”
It is instead a symbol for “the symmetric  group,” the largest
of (in general) many permutation groups that act on N.

This point seems to have escaped John Baez.

For two misuses by Baez of the phrase “permutation group” at the
n-Category Café, see “A Wrinkle in the Mathematical Universe”
and “Re: A Wrinkle…” —

“There is  such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle

Wednesday, August 5, 2015

Inking

Filed under: General — Tags: , , , , — m759 @ 9:00 am

From Doctorow's 'Jolene: A Life'

See also Go Set a Structure and Tombstone.

Sunday, July 19, 2015

Sunday School

Filed under: General — m759 @ 9:00 am

See also last night's post.

The above passage was found in a search for thoughts of Heinz Pagels
on "perfect symmetry" (the title of one of his books).  The "If all" part is,
however, apparently not  by Pagels. That part seems to have been
online only in an NYU file that can no longer be accessed.

For perfect symmetry with  structure, see (for instance) 
Go Set a Structure (July 14, 2015) and Tombstone (July 16, 2015).

Thursday, July 16, 2015

Tombstone

Filed under: General — Tags: , — m759 @ 1:44 pm

The black rectangle at the end of Example 1.4
is known as the "end-of-proof symbol," "Halmos,"
or "tombstone."

Thursday, July 9, 2015

Man and His Symbols

Filed under: General,Geometry — m759 @ 2:24 pm

(Continued)

A post of July 7, Haiku for DeLillo, had a link to posts tagged "Holy Field GF(3)."

As the smallest Galois field based on an odd prime, this structure 
clearly is of fundamental importance.  

The Galois field GF(3)

It is, however, perhaps too  small  to be visually impressive.

A larger, closely related, field, GF(9), may be pictured as a 3×3 array

hence as the traditional Chinese  Holy Field.

Marketing the Holy Field

IMAGE- The Ninefold Square, in China 'The Holy Field'

The above illustration of China's  Holy Field occurred in the context of
Log24 posts on Child Buyers.   For more on child buyers, see an excellent
condemnation today by Diane Ravitch of the U. S. Secretary of Education.

Friday, July 3, 2015

Crunching Entities*

Filed under: General — m759 @ 9:19 pm

A figure I prefer to the "Golden Tablet" of Night at the Museum —

IMAGE- The natural symplectic polarity in PG(3,2), illustrating a symplectic structure

The source — The Log24 post "Zero System" of July 31, 2014.

* For the title, see The New Yorker  of Sept. 22, 2014.

Monday, April 27, 2015

The Beast from Hell’s Kitchen

Filed under: General — m759 @ 9:40 am

A comment on the Scholarly Kitchen  piece from
this morning's previous post —

This suggests

The Beast from Hell's Kitchen

For some thoughts on mapping trees into
linear arrays, see The Forking (March 20, 2015).

See also Pitchfork in this journal.

Thursday, April 23, 2015

Colorful Tale

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm

(A sequel to yesterday's ART WARS and this
morning's De Colores )

“Perhaps the philosophically most relevant feature
of modern science is the emergence of abstract
symbolic structures as the hard core of objectivity
behind– as Eddington puts it– the colorful tale
of the subjective storyteller mind.” — Hermann Weyl
(Philosophy of  Mathematics and Natural Science ,
Princeton, 1949, p. 237)

See also Deathly Hallows.

Saturday, April 4, 2015

Harrowing of Hell (continued)

Filed under: General,Geometry — m759 @ 3:28 pm

Holy Saturday is, according to tradition, the day of 
the harrowing of Hell.

Notes:

The above passage on "Die Figuren der vier Modi
im Magischen Quadrat 
" should be read in the context of
a Log24 post from last year's Devil's Night (the night of
October 30-31).  The post, "Structure," indicates that, using
the transformations of the diamond theorem, the notorious
"magic" square of Albrecht Dürer may be transformed
into normal reading order.  That order is only one of
322,560 natural reading orders for any 4×4 array of
symbols. The above four "modi" describe another.

Thursday, March 26, 2015

The Möbius Hypercube

Filed under: General,Geometry — Tags: , — m759 @ 12:31 am

The incidences of points and planes in the
Möbius 8 configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.* 
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and face-planes of two mutually inscribed tetrahedra —

Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his 
Fig. 24, a schematic representation of the point-plane
incidences, as dotless, and some as hollow dots.  The figure,
"Gallucci's version of Möbius's 84," is shown below.
The hollow dots, representing the 8 points  (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.

Here a plane  (represented by a dotless intersection) contains
the four points  that are represented in the square array as lying
in the same row or same column as the plane. 

The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube. 

In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.

Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in 
Coxeter's labels above may be viewed as naming the positions 
of the 1's in (0,1) vectors (x4, x3, x2, x1) over the two-element
Galois field.  In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .

*  "Self-Dual Configurations and Regular Graphs," 
    Bulletin of the American Mathematical Society,
    Vol. 56 (1950), pp. 413-455

The subscripts' usual 1-2-3-4 order is reversed as a reminder
    that such a vector may be viewed as labeling a binary number 
    from 0  through 15, or alternately as labeling a polynomial in
    the 16-element Galois field GF(24).  See the Log24 post
     Vector Addition in a Finite Field (Jan. 5, 2013).

Tuesday, March 24, 2015

Brouwer on the Galois Tesseract

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 pm

Yesterday's post suggests a review of the following —

Andries Brouwer, preprint, 1982:

"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)

Pages 8-9:

Substructures of S(5, 8, 24)

An octad is a block of S(5, 8, 24).

Theorem 5.1

Let B0 be a fixed octad. The 30 octads disjoint from B0
form a self-complementary 3-(16,8,3) design, namely 

the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F2.

Proof….

… (iv) We have AG(4, 2).

(Proof: invoke your favorite characterization of AG(4, 2) 
or PG(3, 2), say 
Dembowski-Wagner or Veblen & Young. 

An explicit construction of the vector space is also easy….)

Related material:  Posts tagged Priority.

Friday, March 20, 2015

The Forking

Filed under: General — m759 @ 10:02 pm

(Continued)

An article in the new April issue of Notices of the American
Mathmatical Society 
suggests a search for connections
between the Calkin-Wilf tree and the modular group.

The search yields, for instance (in chronological order)

"Cutting sequences for geodesic flow on the modular surface
and continued fractions
," David J. Grahinet, Jeffrey C. Lagaria,
arXiv, 2 April 2001

"Orderings of the rationals and dynamical systems,"
Claudio Bonanno, Stefano Isola, arXiv, 14 May 2008.

"Periods of negative-regular continued fractions. Rational numbers."
Sergey Khrushchev and Michael Tyaglov, slides PDF, 11 Sept. 2012

"The Minkowski ?(x) function, a class of singular measures,
theta-constants, and mean-modular forms
," Giedrius Alkauskas,
arXiv, 20 Sept. 2012

"Forests of complex numbers,"
Melvyn B. Nathanson, arXiv, 1 Dec. 2014

Update of March 21, 2015:

For many more related papers, search by combining the
phrase "modular group" with phrases denoting forking structures
other than Calkin-Wilf, such as "cubic tree," "Stern-Brocot tree,"
and "Farey tree" (or "Farey sequence" or "Farey series" or
"Farey graph" ).

Thursday, February 12, 2015

Dead Reckoning

Filed under: General,Geometry — Tags: , — m759 @ 12:48 pm

Continued from yesterday.

The passage on Claude Chevalley quoted here
yesterday in the post Dead Reckoning was, it turns out,
also quoted by Peter Galison in his essay "Structure of Crystal,
Bucket of Dust" in Circles Disturbed: The Interplay of 
Mathematics and Narrative  
(Princeton University Press, 2012,
ed. by Apostolos Doxiadis and Barry Mazur).

Galison gives a reference to his source:

"From 'Claude Chevalley Described by His Daughter (1988),' 
in Michèle Chouchan, Nicolas Bourbaki: Faits et légendes
(Paris: Éditions du Choix, 1995), 36–40, translated and cited
in Marjorie Senechal, 'The Continuing Silence of Bourbaki:
An Interview with Pierre Cartier, June 18, 1997,' 
Mathematical Intelligencer  1 (1998): 22–28."

Galison's essay compares Chevalley with the physicist
John Archibald Wheeler. His final paragraph —

"Perhaps, then, it should not surprise us too much if,
as Wheeler approaches the beginning-end of all things,
there is a bucket of Borelian dust. Out of this filth,
through the proposition machine of quantum mechanics
comes pregeometry; pregeometry makes geometry;
geometry gives rise to matter and the physical laws
and constants of the universe. At once close to and far
from the crystalline story that Bourbaki invoked,
Wheeler’s genesis puts one in mind of Genesis 3:19:
'In the sweat of thy face shalt thou eat bread, till thou
return unto the ground; for out of it wast thou taken:
for dust thou art, and unto dust shalt thou return.'"

See also posts tagged Wheeler.

Wednesday, February 11, 2015

Dead Reckoning

Filed under: General — Tags: , — m759 @ 5:28 pm

Continued from yesterday evening

IMAGE- Bogart in 'Casablanca' with chessboard

Today's mathematical birthday — 

Claude Chevalley, 11 Feb. 1909 – 28 June 1984.

From MacTutor —

Chevalley's daughter, Catherine Chevalley, wrote about
her father in "Claude Chevalley described by his daughter"
(1988):—

For him it was important to see questions as a whole, to see the necessity of a proof, its global implications. As to rigour, all the members of Bourbaki cared about it: the Bourbaki movement was started essentially because rigour was lacking among French mathematicians, by comparison with the Germans, that is the Hilbertians. Rigour consisted in getting rid of an accretion of superfluous details. Conversely, lack of rigour gave my father an impression of a proof where one was walking in mud, where one had to pick up some sort of filth in order to get ahead. Once that filth was taken away, one could get at the mathematical object, a sort of crystallized body whose essence is its structure. When that structure had been constructed, he would say it was an object which interested him, something to look at, to admire, perhaps to turn around, but certainly not to transform. For him, rigour in mathematics consisted in making a new object which could thereafter remain unchanged.

The way my father worked, it seems that this was what counted most, this production of an object which then became inert— dead, really. It was no longer to be altered or transformed. Not that there was any negative connotation to this. But I must add that my father was probably the only member of Bourbaki who thought of mathematics as a way to put objects to death for aesthetic reasons.

Recent scholarly news suggests a search for Chapel Hill
in this journal. That search leads to Transformative Hermeneutics.
Those who, like Professor Eucalyptus of Wallace Stevens's
New Haven, seek God "in the object itself" may contemplate
yesterday's afternoon post on Eightfold Design in light of the
Transformative post and of yesterday's New Haven remarks and
Chapel Hill events.

Thursday, January 1, 2015

New Year’s Greeting from Franz Kafka

Filed under: General,Geometry — m759 @ 5:01 am

An image that led off the year-end review yesterday in
the weblog of British combinatorialist Peter J. Cameron:

See also this  weblog's post final post of 2014,
with a rectangular array illustrating the six faces
of a die, and Cameron's reference yesterday to
a die-related post

"The things on my blog that seem to be
of continuing value are the expository
series like the one on the symmetric group
(the third post in this series was reblogged
by Gil Kalai last month, which gave it a new
lease of life)…."

A tale from an author of Prague:

The Emperor—so they say—has sent a message, directly from his death bed, to you alone, his pathetic subject, a tiny shadow which has taken refuge at the furthest distance from the imperial sun. He ordered the herald to kneel down beside his bed and whispered the message into his ear. He thought it was so important that he had the herald repeat it back to him. He confirmed the accuracy of the verbal message by nodding his head. And in front of the entire crowd of those who’ve come to witness his death—all the obstructing walls have been broken down and all the great ones of his empire are standing in a circle on the broad and high soaring flights of stairs—in front of all of them he dispatched his herald. The messenger started off at once, a powerful, tireless man. Sticking one arm out and then another, he makes his way through the crowd. If he runs into resistance, he points to his breast where there is a sign of the sun. So he moves forward easily, unlike anyone else. But the crowd is so huge; its dwelling places are infinite. If there were an open field, how he would fly along, and soon you would hear the marvelous pounding of his fist on your door. But instead of that, how futile are all his efforts. He is still forcing his way through the private rooms of the innermost palace. He will never he win his way through. And if he did manage that, nothing would have been achieved. He would have to fight his way down the steps, and, if he managed to do that, nothing would have been achieved. He would have to stride through the courtyards, and after the courtyards the second palace encircling the first, and, then again, stairs and courtyards, and then, once again, a palace, and so on for thousands of years. And if he finally did burst through the outermost door—but that can never, never happen—the royal capital city, the centre of the world, is still there in front of him, piled high and full of sediment. No one pushes his way through here, certainly not with a message from a dead man. But you sit at your window and dream of that message when evening comes.

See also a passage quoted in this  weblog on the original
date of Cameron's Prague image, July 26, 2014 —

"The philosopher Graham Harman is invested in
re-thinking the autonomy of objects and is part 
of a movement called Object-Oriented-Philosophy
(OOP)." — From “The Action of Things,” a 2011
M.A. thesis at the Center for Curatorial Studies,
Bard College, by Manuela Moscoso 

— in the context of a search here for the phrase
     "structure of the object." An image from that search:

Saturday, December 27, 2014

More To Be Done

Filed under: General,Geometry — m759 @ 1:44 am

  Ball and Weiner, 'An Introduction to Finite Geometry,' version of Sept. 5, 2011

The Ball-Weiner date above, 5 September 2011,
suggests a review of this journal on that date —

"Think of a DO NOT ENTER pictogram,
a circle with a diagonal slash, a type of ideogram.
It tells you what to do or not do, but not why.
The why is part of a larger context, a bigger picture."

— Customer review at Amazon.com

This passage was quoted here on August 10, 2009.

Also from that date:

The Sept. 5, 2011, Ball-Weiner paper illustrates the
"doily" view of the mathematical structure W(3,2), also
known as GQ(2,2), the Sp(4,2) generalized quadrangle.
(See Fig. 3.1 on page 33, exercise 13 on page 38, and
the answer to that exercise on page 55, illustrated by 
Fig. 5.1 on page 56.)

For "another view, hidden yet true," of GQ(2,2),
see Inscape and Symplectic Polarity in this journal.

Monday, December 15, 2014

Mythic Metaphysics

Filed under: General — Tags: — m759 @ 9:01 pm

Today’s 8:01 PM post quoted Husserl on
the perception of the cube.

Another approach to perception of the cube,
from Narrative  Metaphysics on St. Lucia’s Day —


      See also Symplectic Structure and Stevens’s Rock.

From today’s 11:29 AM post —

John Burt Foster Jr. in Nabokov’s Art of Memory and
European Modernism
  (Princeton U. Press, 1993, p. 224) —

At the time of The Waste Land , in a comment on
Joyce’s Ulysses  that influenced many later definitions
of modernism in the English-speaking world, Eliot
announced, “instead of narrative method, we may
now use the mythical method.”13

For some illuminating remarks on a mythical  approach
to perception of the cube, see Gareth Knight on Schicksalstag   2012.

Saturday, December 13, 2014

Narrative Metaphysics

Filed under: General,Geometry — Tags: , , , — m759 @ 12:00 pm

From "Guardians of the Galaxy" —

"Then the Universe exploded into existence…"

For those who prefer a more traditional approach :

See also Symplectic Structure and Stevens's Rock.

Tuesday, December 2, 2014

Colorful Tale

Filed under: General — Tags: — m759 @ 9:45 am

Continued.

"Perhaps the philosophically most relevant feature
of modern science is the emergence of abstract
symbolic structures as the hard core of objectivity
behind— as Eddington puts it— the colorful tale
of the subjective storyteller mind."

— Hermann Weyl in Philosophy of Mathematics
     and Natural Science
 , Princeton, 1949, p. 237

Tom Wolfe on art theorists in The Painted Word  (1975) :

"It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde  understood, they
were not only theories but … hot news,
straight from the studios, from the scene."

The Weyl quote is a continuing theme in this journal.
The Wolfe quote appeared here on Nov. 18, 2014,
the reported date of death of Yale graduate student 
Natasha Chichilnisky-Heal.

Directions to her burial (see yesterday evening) include
a mention of "Paul Robson Street" (actually Paul
Robeson Place) near "the historic Princeton Cemetery."

This, together with the remarks by Tom Wolfe posted
here on the reported day of her death, suggests a search
for "red green black" —

The late Chichilnisky-Heal was a student of political economy.

The search colors may be interpreted, if one likes, as referring
to politics (red), economics (green), and Robeson (black).

See also Robeson in this journal.

Monday, December 1, 2014

Change Arises

Filed under: General,Geometry — Tags: — m759 @ 11:00 am

Flashback to St. Andrew's Day, 2013 —

Saturday, November 30, 2013

Waiting for Ogdoad

Filed under: Uncategorized — Tags:  — m759 @ 10:30 AM 

Continued from October 30 (Devil's Night), 2013.

“In a sense, we would see that change
arises from the structure of the object.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the
"Classic of Change ," the I Ching .

If the object is a cube, change arises from the fact
that the object has six  faces…

and is the unit cell for the six -dimensional
hyperspace H over the two-element field —

Spaces as Hypercubes

A different representation of the unit cell of
the hyperspace H (and of the I Ching ) —

Sunday, November 30, 2014

View from the Bottom

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm

Reality's Mirror: Exploring the Mathematics of Symmetry —

"Here is a book that explains in laymen language
what symmetry is all about, from the lowliest snowflake
and flounder to the lofty group structures whose
astonishing applications to the Old One are winning
Nobel prizes. Bunch's book is a marvel of clear, witty
science writing, as delightful to read as it is informative
and up-to-date. The author is to be congratulated on
a job well done." — Martin Gardner

"But, sweet Satan, I beg of you, a less blazing eye!"

— Rimbaud,  A Season in Hell

"… the lowliest snowflake and flounder…." 
      — Martin Gardner

Thomas Mann on the deathly precision of snowflakes

Britannica article, 'Flounder'

Thursday, November 13, 2014

Progressive Matrix

Filed under: General,Geometry — Tags: , — m759 @ 6:00 am

Yesterday's post and recent Hollywood news suggest
a meditation on a Progressive Matrix —

Oct. 12-14, 2005:

'A Poem for Pinter,' conclusion: 'Tick Tick Hash.'

'The Interpreter'-- Sean Penn to Nicole Kidman-- 'My Card.'

Click to enlarge.

"My card."

Structurally related images —

A sample Raven's Progressive Matrices  test item
(such items share the 3×3 structure of the hash symbol above):

IMAGE- Raven's Progressive Matrices item with symbols from Cullinane's box-style I Ching

Structural background —

Monday, November 3, 2014

The Rhetoric of Abstract Concepts

Filed under: General,Geometry — Tags: — m759 @ 12:48 pm

From a post of June 3, 2013:

New Yorker  editor David Remnick at Princeton today
(from a copy of his prepared remarks):

“Finally, speaking of fabric design….”

I prefer Tom and Harold:

Tom Wolfe in The Painted Word 

“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”

Harold Rosenberg in The New Yorker  (click to enlarge)

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

Symplectic :

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

— Steven H. Cullinane,
diamond theorem illustration

Friday, October 31, 2014

For the Late Hans Schneider

Filed under: General — Tags: — m759 @ 3:30 am

See a University of Wisconsin obituary for Schneider,
a leading expert on linear algebra who reportedly died
at 87 on Tuesday, October 28, 2014.

Some background on linear algebra and “magic” squares:
tonight’s 3 AM (ET) post and a search in this
journal for Knight, Death, and the Devil.

Click image to enlarge.

Tuesday, October 28, 2014

Figural Processing

Filed under: General,Geometry — Tags: — m759 @ 4:22 am

Part I:

Six-dimensional hypercube from 'Brain and Perception: Holonomy and Structure in Figural Processing,' by Karl H. Pribram

Part II:

Click images for some context.

Tuesday, October 21, 2014

Art as a Tool

Filed under: General,Geometry — Tags: — m759 @ 12:35 pm

Two news items on art as a tool:

Two Log24 posts related to the 3×3 grid, the underlying structure for China’s
ancient Lo Shu “magic” square:

Finally, leftist art theorist Rosalind Krauss in this journal
on AntiChristmas, 2010:

Which is the tool here, the grid or Krauss?

Monday, October 13, 2014

Sallows on “The Lost Theorem”

Filed under: General,Geometry — Tags: — m759 @ 9:30 pm

Parallelograms and the structure of the 3×3 array —

Click to enlarge:

A different approach to parallelograms and arrays —

Click for original post:

Monday, October 6, 2014

Mysterious Correspondences

Filed under: General,Geometry — m759 @ 9:36 am

(Continued from Beautiful Mathematics, Dec. 14, 2013)

“Seemingly unrelated structures turn out to have
mysterious correspondences.” — Jim Holt, opening
paragraph of 
a book review in the Dec. 5, 2013, issue
of 
The New York Review of Books

One such correspondence:

For bibliographic information and further details, see
the March 9, 2014, update to “Beautiful Mathematics.”

See as well posts from that same March 9 now tagged “Story Creep.”

Sunday, September 21, 2014

Saturday-Morning Concept

Filed under: General,Geometry — m759 @ 7:59 pm

Why Is Our Sci-Fi So Glum About A.I.?,”
by Jayson Greene, NY Times Sunday Magazine  today —

“You come to pity these advanced beings, bumping against
the dunderheaded constraints that their less-advanced
creators have placed on them. Johansson’s Lucy grows so
powerful as her cerebral capacity multiplies that she is able
to manipulate her cellular structure. And yet, when pursued
by an entire planet’s worth of law enforcement, she settles
on a disguise straight out of Saturday-morning cartoons —
really big sunglasses and a hairdo change.”

See also this  journal on Saturday morning for a definition, and
Geometry of the I Ching for examples, of

changeable, instantiable entities, i.e., concrete universals.

Above the entrance to Plato's Academy: AGEOMETRETOS MEDEIS EISITO

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