Log24

Saturday, August 28, 2021

Solomon’s Super*  Cube…

Filed under: General — Tags: , , , , — m759 @ 1:33 pm

Geometry for Jews  continues.

210828-Golomb-2x2x2-Super_Cube.jpg (500×373)

The conclusion of Solomon Golomb's
"Rubik's Cube and Quarks,"
American Scientist , May-June 1982 —

Related geometric meditation —
Archimedes at Hiroshima
in posts tagged Aitchison.

 

* As opposed to Solomon's Cube .

Friday, November 10, 2023

Cube Mine

Filed under: General — Tags: , , — m759 @ 12:44 am

In memory of a former president of Boston University
Other posts now tagged Cube Mine.

Related entertainment —

Wednesday, March 9, 2022

Supercube Space

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

The new URL supercube.space forwards to http://box759.wordpress.com/.

The term supercube  is from a 1982 article by Solomon W. Golomb.

The related new URL supercube.group forwards to a page that
describes how the 2x2x2 (or eightfold, or "super") cube's natural
underlying automorphism group is Klein's simple group of order 168.

For further context, see the new URL supercube.art.

For some background, see the phrase Cube Space in this journal. 

Thursday, June 27, 2019

Group Actions on the 4x4x4 Cube

Filed under: General — Tags: — m759 @ 6:23 am

For affine  group actions, see Ex Fano Appollinis  (June 24)
and Solomon's Cube.

For one approach to Mathieu  group actions on a 24-cube subset
of the 4x4x4 cube, see . . .

For a different sort of Mathieu cube, see Aitchison.

Sunday, May 8, 2016

The Three Solomons

Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick's Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.

A quote from LeWitt indicates the depth of the word "conceptual"
in his approach to "conceptual art."

From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:

 

THE SQUARE AND THE CUBE
by Sol LeWitt

"The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed."

"Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America  55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)"

See also the Cullinane models of some small Galois spaces

Some small Galois spaces (the Cullinane models)

Wednesday, May 4, 2016

Solomon Golomb, 1932-2016

Filed under: General,Geometry — m759 @ 4:00 am

Material related to the previous post, "Symmetry" —

This is the group of "8 rigid motions
generated by reflections in midplanes"
of "Solomon's Cube."

Material from this journal on May 1, the date of Golomb's death —

"Weitere Informationen zu diesem Themenkreis
finden sich unter http://​www.​encyclopediaofma​th.​org/
​index.​php/​Cullinane_​diamond_​theorem
und
http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

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. –>

Tuesday, October 16, 2012

Cube Review

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

Last Wednesday's 11 PM post mentioned the
adjacency-isomorphism relating the 4-dimensional 
hypercube over the 2-element Galois field GF(2) to
the 4×4 array made up of 16 square cells, with
opposite edges of the 4×4 array identified.

A web page illustrates this property with diagrams that
enjoy the Karnaugh property— adjacent vertices, or cells,
differ in exactly one coordinate. A brief paper by two German
authors relates the Karnaugh property to the construction
of a magic square like that of Dürer (see last Wednesday).

In a similar way (search the Web for Karnaugh + cube ),
vertex adjacency in the 6-dimensional hypercube over GF(2) 
is isomorphic to cell adjacency in the 4x4x4 cube, with
opposite faces of the 4x4x4 cube identified.

The above cube may be used to illustrate some properties
of the 64-point Galois 6-space that are more advanced
than those studied by enthusiasts of "magic" squares
and cubes.

See

Those who prefer narrative to mathematics may
consult posts in this journal containing the word "Cuber."

Sunday, August 5, 2012

Cube Partitions

Filed under: General,Geometry — Tags: , , , — m759 @ 7:59 am

The second Logos  figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

Click for further details.

Thursday, July 26, 2012

Solomon’s Seal

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

(Mathematics and Narrative, continued)

Narrative—

The Ring and The Stone from yesterday’s post, and…

“In Medieval Jewish, Christian and Islamic legends,
the Seal of Solomon was a magical signet ring
said to have been possessed by King Solomon….”

— Wikipedia article, Seal of Solomon

Mathematics—

IMAGE- Eric Temple Bell on the mathematics of 'Solomon's Seal' (in his 'Development of Mathematics')

A fact related to the mathematical
“Solomon’s seal” described above by Bell:

IMAGE- J.W.P. Hirschfeld on the mathematics of 'Solomon's Seal', with reference to Edge on the same topic

The reference to Edge is as follows—

[3] Edge, W. L., Quadrics over GF(2) and
their relevance for the cubic surface group
,
Canadian J. Maths. 11 (1959) ….

(This reference relates Hirschfeld’s remarks
quoted above to the 64-point affine space
illustrated below (via the associated
63-point projective  space PG (5, 2)).

As for the narrative’s Stone… 

See Solomon’s Cube.

IMAGE- 'Solomon's Cube'

Wednesday, January 11, 2012

Cuber

“Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing.  And once you have made or acquired a new ‘cube’… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube.  What is the essence of each operator?  One senses a deep invariant lying somehow ‘down underneath’ it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment.  In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….

… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube.  It is the  answer; it simply has the right spirit .”

— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern  (Kindle edition, locations 11557-11572)

See also Many Dimensions in this journal and Solomon’s Cube.

Saturday, October 24, 2009

Chinese Cubes Continued

Filed under: General,Geometry — m759 @ 8:28 am

A search for “Chinese Cube” (based on the the previous entry’s title) reveals the existence of a most interesting character, who…

“… has attempted in his books to produce a Science and Art of Reasoning using the simplest of the Platonic solids, the Cube. [His] model also parallels, in some ways, the Cube of Space constructed from the Sepher Yetzirah’s attributions for the Hebrew letters and their direction. [He] elucidated his theories at great length….”

More…

For related remarks, see the link to Solomon’s Cube from the previous entry.

Then of course there is…

http://www.log24.com/log/pix09A/091024-RayFigure.jpg

Click on figure for details.

Thursday, October 22, 2009

Chinese Cubes

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

From the Bulletin of the American Mathematical Society, Jan. 26, 2005:

What is known about unit cubes
by Chuanming Zong, Peking University

Abstract: Unit cubes, from any point of view, are among the simplest and the most important objects in n-dimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all….

From Log24, now:

What is known about the 4×4×4 cube
by Steven H. Cullinane, unaffiliated

Abstract: The 4×4×4 cube, from one point of view, is among the simplest and the most important objects in n-dimensional binary space. In fact, as one will see from the links below, it is not simple at all.

Solomon's Cube

The Klein Correspondence, Penrose Space-Time, and a Finite Model

Non-Euclidean Blocks

Geometry of the I Ching

Related material:

Monday's entry Just Say NO and a poem by Stevens,

"The Well Dressed Man with a Beard."

Wednesday, November 22, 2023

For E. Lily Yu* — Devs Setting

Filed under: General — Tags: , , — m759 @ 12:49 pm
 

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM

See Cube Symbology.

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Da hats ein Eck 

* Author of Jewel Box: Stories  ( Erewhon Books, Oct. 24, 2023).

Sunday, November 19, 2023

Six Dimensions

Filed under: General — Tags: , — m759 @ 9:59 am

Heinlein:

"Sharpie, we have condensed six dimensions into four,
then we either work by analogy into six, or we have to use math
that apparently nobody but Jake and my cousin Ed understands.
Unless you can think of some way to project six dimensions into three–
you seem to be smart at such projections."

I closed my eyes and thought hard. "Zebbie, I don't think it can be done.
Maybe Escher could have done it."

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

Friday, November 17, 2023

Classicism Continued: An Apotheosis of Modernity

Filed under: General — Tags: — m759 @ 12:34 pm
 

From Chapter 23, "Poetry," by Adam Parkes, in
A Companion to Modernist Literature and Culture,
edited by David Bradshaw and Kevin J. H. Dettmar,
Blackwell Companions to Literature and Culture,
© 2006 by Blackwell Publishing Ltd.

Writing in 1910–11, the English poet and critic T. E. Hulme claimed that the two major traditions in poetry, romanticism and classicism, were as different as a well and a bucket. According to the romantic party, Hulme explained, humankind is “intrinsically good, spoilt by circumstance”; that is, our nature is “a well, a reservoir full of possibilities.” For the classical party, however, human nature is “like a bucket”; it is “intrinsically limited, but disciplined by order and tradition to something fairly decent” (Hulme 1987: 117). But it was not only that romanticism and classicism were as dissimilar as a well and a bucket; their contents were different, too. To draw water from the well of romanticism was, in effect, to pour a “pot of treacle over the dinner table,” while the classical bucket was more likely to be full of little stones – or jewels, perhaps. Romanticism, in Hulme’s view, was the result of displaced religious fervor; it represented the return of religious instincts that the “perverted rhetoric of Rationalism” had suppressed, so that “concepts that are right and proper in their own sphere are spread over, and so mess up, falsify and blur the clear outlines of human experience” (Hulme 1987: 118). Classicism, by contrast, traded in dry goods – dry, hard goods, to be precise.

Hulme left little doubt as to which side he was on. “It is essential to prove,” he argued, “that beauty may be in small, dry things. The great aim is accurate, precise and definite description. . . . I prophesy that a period of dry, hard, classical verse is coming” (Hulme 1987: 131–3). If by “dry, hard, classical verse” Hulme meant poems looking like the fragments of Sappho, he didn’t have to wait long to see his prophecy fulfilled.

The hard sand breaks,
and the grains of it
are clear as wine.

Far off over the leagues of it,
the wind,

228

playing on the wide shore,
piles little ridges,
and the great waves
break over it.

So wrote Hilda Doolittle in “Hermes of the Ways,” the first poem that she signed “H. D., Imagiste” at the behest of her fellow American expatriate Ezra Pound. From Pound’s perspective, the Imagist movement that he co-founded in 1912 with H. D. and the English poet Richard Aldington was finished well before the First World War began in August 1914; throughout this war-torn decade, however, Imagism continued to spawn the poetry of “small, dry things” whose coming Hulme had predicted a few years before.

Indeed, modernist poets weren’t content merely to break down the extended heroic narratives – the “spilt religion,” as Hulme put it – of their treacly nineteenthcentury predecessors; they insisted on breaking down small things into ever-smaller particles and subparticles. This logic of disintegration is clearly at work in poems like “Hermes of the Ways,” where each line is metrically unique, creating a sense of perpetual freshness – an apotheosis of modernity, as it were.

REFERENCE

Hulme, T. E. (1987). Speculations: Essays on Humanism and the Philosophy of Art, ed. Herbert Read. London and New York: Routledge and Kegan Paul. First published 1924.

Compare and contrast:

Jeremy Gray,
Plato's Ghost: The Modernist Transformation of Mathematics,
Princeton University Press, first edition Sept. 22, 2008

"Here, modernism is defined as an autonomous body of ideas,
having little or no outward reference, placing considerable emphasis
on formal aspects of the work and maintaining a complicated—
indeed, anxious— rather than a naïve relationship with the
day-to-day world, which is the de facto view of a coherent group
of people, such as a professional or discipline-based group
that has a high sense of the seriousness and value of what it is
trying to achieve. This brisk definition…."

(Quoted at the webpage Solomon's Cube.)

Friday, October 13, 2023

Hungarian Puzzle

Filed under: General — Tags: — m759 @ 8:25 pm

See "Cube Space" + Lovasz.

This search was suggested by . . .

The conclusion of Solomon Golomb's
"Rubik's Cube and Quarks,"
American Scientist , May-June 1982 —

Artbox.group

Filed under: General — Tags: — m759 @ 2:03 pm

This new URL will forward to http://m759.net/wordpress/?s=Solomon+Cube.

Wednesday, May 10, 2023

Saving the Appearances

Filed under: General — Tags: , — m759 @ 2:18 pm

Douglas Hofstadter —

“… I realized that to me,
Gödel and Escher and Bach
were only shadows
cast in different directions by
some central solid essence.
I tried to reconstruct
the central object, and
came up with this book.”

Goedel Escher Bach cover

Related images —

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

Thursday, April 20, 2023

Alphabet Meets Gestalt . . .

Filed under: General — Tags: , , — m759 @ 2:25 pm

Continued from April 18 .

"Working with words to create art
and working with your hands to create art
seem like two separate activities to me."

Cover artist, The New Yorker , on April 17

See also Alphabet Blocks in this  journal
as well as Escher's Verbum.

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

Monday, January 16, 2023

Ready When You Are, C. B.

Filed under: General — m759 @ 12:23 pm

Monday, November 28, 2022

Groups, Spaces, and Ripoffs

Filed under: General — Tags: — m759 @ 8:21 pm

"Rubik's Cube, and the simpler [2x2x2] Super Cube, represent
one form of mathematical and physical reality."

— Solomon W. Golomb, "Rubik's Cube and Quarks:
Twists on the eight corner cells of Rubik's Cube
provide a model for many aspects of quark behavior
,"
American Scientist , Vol. 70, No. 3 (May-June 1982), pp. 257-259 

From the last (Nov. 14, 2022) of the Log24 posts now tagged Groups and Spaces

From the first (June 21, 2010) of the Log24 posts now tagged Groups and Spaces

Thursday, November 10, 2022

For Students of the Forked Tongue

Filed under: General — Tags: , , , — m759 @ 11:42 am

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

The above 1975 book by Robert Greer Cohn, Modes of Art, is
Volume I of a planned three-volume work.

The passage below is from a review of Cohn's Vol. II, Ways of Art — 

Franklin, Ursula (1987) "Book Review: A Critical Work II.
Ways of Art: Literature. Music, Painting in France 
,"
Grand Valley Review : Vol. 3: Iss. 1, Article 19. Available at: http://scholarworks.gvsu.edu/gvr/vol3/iss1/19 .

. . . .

Those not familiar with the author's epistemology should begin with Appendix A of Ways of Art , a schematic demonstration of his tetrapolar-polypolar-dialectic, especially as it concerns the development of the French novel within the European tradition. But this dialectic, which has antecedents in Kierkegaard, Mallarme and Joyce, underlies all art, because: "this dimensional pulsation, or tetrapolar (and polypolar) higher vibrancy is, in short, the stuff of life: life is vibrant in this more complex way as well as in the more bipolar sense" (7). Cohn shows that "far out enough" the male or linear and the female or circular, the male vertical and the female horizontal dimensions "tend to merge as in relativity theory" (19). Ways of Art  shows us the way through a historical becoming of art in its complex dialectic in which the metonymic (horizontal) axis constantly interrelates with the metaphoric (vertical). "Life is the mother, art the father" (vii); hence Cohn's quarrel with most contemporary Feminism, which is pronounced throughout the volume. Firmly grounded in its author's tetra-polypolar epistemology, this beautiful book becomes, however, at no point dryly abstract; it is the mature work of a true humanist who stands in clear and open opposition to the dehumanizing trend of "the quasi-scientific reductionism and abstract gimmickry of a great deal of current academic literary study, bellwethered by the structuralists, post-structuralists, and deconstructionists" (vi). Abundant footnotes constitute a substantial part of Ways of Art , on occasion developing insights almost into essays demonstrating crucial points along the general flow of the tradition from "Obscure Beginnings;' the opening chapter, to our "Contemporaries;' the last.

Cohn reminds us that "In the Beginning was the Word;' for the Judaeo-Christian tradition at least, which his study fervently embraces; thus, for example, in Appendix 0 on "The Dance of the Sexes;' he censures "those who live by slogans, camps, and peer-opinion, the countless little bastard cults which characterize an era which has massively veered away from our free and beautiful Greco-Judaeo-Christian tradition" (332). Cohn traces man's way and that of his myths and rituals culminating in his art from that beginning along the lines of Freud, Neumann and Cassirer, and many others, always demonstrating the underlying polypolar dialectical rhythm. Thus in "From Barbarism to Young Culture;' we follow the Celts to Druidic ritual, Hebrew beginnings to the Psalms, Dionysian ritual to Greek tragedy, and thence to the beginnings of French dramatic literature originating in the Quem quaeritis sequence of the medieval Mass. Along the way arises artistic symbolism, for Cohn synonymous with "effective poetry;' to finally "ripen in France as never before" (99). Table I (134) graphs this development from the twelfth to the late nineteenth and early twentieth centuries. The author traces the rise of the artistic vocation from its antecedents in the double function of bard and priest, with the figure of Ronsard at the crossroads of that dying institution and the nascent concept of personal glory. "The Enlightenment Vocation" is exemplified in Montaigne, who humanizes the French cultural elite and points the way to French classicism and, farther down the road, after the moral collapse with the outgoing reign of Louis XIV, toward the Age of Reason. Clearly the most significant figure of the French Enlightenment for all of Western civilization is Rousseau, and Cohn beautifully shows us why this is so. Subsequently, "the nineteenth-century stage of the writer's journey will lead, starting from the crossroads of Rousseau, primarily in these two directions: the imperialistic and visionary prose of Balzac, the equally ambitious poetry of Mallarme", brothers under the skin" (199). And these two paths will then be reconciled in Proust's monumental A la recherche du temps perdu .

. . . .

Monday, October 31, 2022

Folklore vs. Mathematics

Filed under: General — Tags: , , , — m759 @ 5:59 pm


Folklore —
 

Earlier in that same journal . . .

The 1955 Levi-Strauss 'canonic formula' in its original context of permutation groups


Mathematics —
 

Webpage demonstrating symmetries of 'Solomon's Cube'

Saturday, June 25, 2022

Gödel, Escher, Bach

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

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

Friday, May 6, 2022

Interality and the Bead Game

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

WIkipedia on the URL suffix ".io" —

"In computer science, "IO" or "I/O" is commonly used
as an abbreviation for input/output, which makes the
.io domain desirable for services that want to be
associated with technology. .io domains are often used
for open source projects, application programming
interfaces ("APIs"), startup companiesbrowser games,
and other online services."

An association with the Bead Game from a post of April 7, 2018

IMAGE- 'Solomon's Cube'

Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

The recent use by a startup company of the URL "interality.io" suggests
a fourth  reading for the 1955 list of Lévi-Strauss — in and out
i.e., inner and outer group automorphisms —  from a 2011 post
on the birthday of T. S. Eliot :

A transformation:

Inner and outer group automorphisms

Click on the picture for details.

Interality and the I Ching

Filed under: General — Tags: , , — m759 @ 12:57 am

See "Flusser and the I Ching," by Peter Zhang.

Zhang has written extensively on the concept of "interality,"
a term coined by his colleague Geling Shang.

For interality as the mathematics underlying the natural
automorphism group of the I Ching, see my own work.

Saturday, March 26, 2022

Box Geometry: Space, Group, Art  (Work in Progress)

Filed under: General — Tags: — m759 @ 2:06 am

Many structures of finite geometry can be modeled by
rectangular or cubical arrays ("boxes") —
of subsquares or subcubes (also "boxes").

Here is a draft for a table of related material, arranged
as internet URL labels.

Finite Geometry Notes — Summary Chart
 

Name Tag .Space .Group .Art
Box4

2×2 square representing the four-point finite affine geometry AG(2,2).

(Box4.space)

S4 = AGL(2,2)

(Box4.group)

 

(Box4.art)

Box6 3×2 (3-row, 2-column) rectangular array
representing the elements of an arbitrary 6-set.
S6  
Box8 2x2x2 cube or  4×2 (4-row, 2-column) array. S8 or Aor  AGL(3,2) of order 1344, or  GL(3,2) of order 168  
Box9 The 3×3 square. AGL(2,3) or  GL(2,3)  
Box12 The 12 edges of a cube, or  a 4×3  array for picturing the actions of the Mathieu group M12. Symmetries of the cube or  elements of the group M12  
Box13 The 13 symmetry axes of the cube. Symmetries of the cube.  
Box15 The 15 points of PG(3,2), the projective geometry
of 3 dimensions over the 2-element Galois field.
Collineations of PG(3,2)  
Box16 The 16 points of AG(4,2), the affine geometry
of 4 dimensions over the 2-element Galois field.

AGL(4,2), the affine group of 
322,560 permutations of the parts
of a 4×4 array (a Galois tesseract)

 
Box20 The configuration representing Desargues's theorem.    
Box21 The 21 points and 21 lines of PG(2,4).    
Box24 The 24 points of the Steiner system S(5, 8, 24).    
Box25 A 5×5 array representing PG(2,5).    
Box27 The 3-dimensional Galois affine space over the
3-element Galois field GF(3).
   
Box28 The 28 bitangents of a plane quartic curve.    
Box32 Pair of 4×4 arrays representing orthogonal 
Latin squares.
Used to represent
elements of AGL(4,2)
 
Box35 A 5-row-by-7-column array representing the 35
lines in the finite projective space PG(3,2)
PGL(3,2), order 20,160  
Box36 Eurler's 36-officer problem.    
Box45 The 45 Pascal points of the Pascal configuration.    
Box48 The 48 elements of the group  AGL(2,3). AGL(2,3).  
Box56

The 56 three-sets within an 8-set or
56 triangles in a model of Klein's quartic surface or
the 56 spreads in PG(3,2).

   
Box60 The Klein configuration.    
Box64 Solomon's cube.    

— Steven H. Cullinane, March 26-27, 2022

Friday, December 31, 2021

Aesthetics in Academia

Filed under: General — Tags: — m759 @ 9:33 am

Related art — The non-Rubik 3x3x3 cube —

The above structure illustrates the affine space of three dimensions
over the three-element finite (i.e., Galois) field, GF(3). Enthusiasts
of Judith Brown's nihilistic philosophy may note the "radiance" of the
13 axes of symmetry within the "central, structuring" subcube.

I prefer the radiance  (in the sense of Aquinas) of the central, structuring 
eightfold cube at the center of the affine space of six dimensions over
the two-element field GF(2).

Thursday, March 5, 2020

Pythagorean Letter Meets Box of Chocolates

Filed under: General — Tags: , , , , — m759 @ 10:30 am

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM

See Cube Symbology.

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Da hats ein Eck 

Monday, November 11, 2019

Time and Chance

Filed under: General — Tags: , , , — m759 @ 2:49 pm

http://www.log24.com/log/pix10B/101202-DreidelAndStone.jpg

The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel 
Many Dimensions  published in 1993 by Wm. B. Eerdmans.

Compare and constrast —

Goedel Escher Bach cover

Cover of a book by Douglas Hofstadter

IMAGE- 'Solomon's Cube'

An Invariance of Symmetry

Monday, September 9, 2019

ART WARS at Harvard: The Wertham Professorship

Filed under: General — Tags: , — m759 @ 8:38 pm

See as well an obituary for Mrs. Wertham from 1987.

Related art —

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM 

See Cube Symbology.

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Da hats ein Eck 

For further details, search the Web for "Wertham Professor" + Eck.

Sunday, July 15, 2018

Jewish Oases

Filed under: General,Geometry — Tags: , — m759 @ 10:06 pm

"… Lincoln Plaza Cinemas, the Juilliard String Quartet,
and the Strand Book Store remained  oases
for cultural and intellectual stimulation."

John S. Friedman in The Forward , Jan. 21, 2018

Read more: 

https://forward.com/culture/392483/
how-fred-bass-dan-talbot-robert-mann
-shaped-new-york-culture/

From  the Oasis  in Steven Spielberg's "Ready Player One" (2018) —

I prefer, from a Log24 search for Flux Capacitor

Symbologist Robert Langdon views a corner of Solomon's Cube

From "Raiders of the Lost Images" —

"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."

Sunday, July 8, 2018

Sixers*

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

Eric Temple Bell, 'The Development of Mathematics'

See also Solomon's  cube.

* Title suggested by a 2011 dystopian novel.

Thursday, June 7, 2018

For Dan Brown

Filed under: General,Geometry — Tags: , , , — m759 @ 1:09 pm

See also Eightfold Trinity in this  journal.

Symbologist Robert Langdon views a corner of Solomon's Cube

Saturday, May 19, 2018

Flux Capacitor

Filed under: General,Geometry — Tags: , , , — m759 @ 4:13 pm

For Tom Hanks and Dan Brown —

Symbologist Robert Langdon views a corner of Solomon's Cube

From "Raiders of the Lost Images" —

"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."

See as well a Google search for flux philosophy
https://www.google.com/search?q=flux+philosophy.

Saturday, April 7, 2018

Sides

The FBI holding cube in "The Blacklist" —

" 'The Front' is not the whole story . . . ."

— Vincent Canby, New York Times  film review, 1976,
     as quoted in Wikipedia.

See also Solomon's Cube in this  journal.

IMAGE- 'Solomon's Cube'

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

Monday, March 12, 2018

Stein

Filed under: General,Geometry — m759 @ 9:55 pm

Stein reportedly died at 100 last Friday (March 9).

Related material —

Textiles by Stein arranged on the six faces of a cube —

Ethel Stein, "Circus & Slapstick," 1996

See also a less amusing approach to
patterns on the faces of a cube.

Tuesday, February 27, 2018

Raiders of the Lost Images

Filed under: General,Geometry — Tags: — m759 @ 11:28 am

On the recent film "Justice League" —

From DC Extended Universe Wiki, "Mother Box" —

"However, during World War I, the British rediscovered
mankind's lost Mother Box. They conducted numerous studies
but were unable to date it due to its age. The Box was then
shelved in an archive, up until the night Superman died,
where it was then sent to Doctor Silas Stone, who
recognized it as a perpetual energy matrix. . . ." [Link added.]

The cube shape of the lost Mother Box, also known as the
Change Engine, is shared by the Stone in a novel by Charles Williams,
Many Dimensions . See the Solomon's Cube webpage.

See too the matrix of Claude Lévi-Strauss in posts tagged
Verwandlungslehre .

Some literary background:

Who speaks in primordial images speaks to us
as with a thousand trumpets, he grips and overpowers,
and at the same time he elevates that which he treats
out of the individual and transitory into the sphere of
the eternal. 
— C. G. JUNG

"In the conscious use of primordial images—
the archetypes of thought—
one modern novelist stands out as adept and
grand master: Charles Williams.
In The Place of the Lion  he incarnates Plato’s
celestial archetypes with hair-raising plausibility.
In Many Dimensions  he brings a flock of ordinary
mortals face to face with the stone bearing
the Tetragrammaton, the Divine Name, the sign of Four.
Whether we understand every line of a Williams novel
or not, we feel something deep inside us quicken
as Williams tells the tale.

Here, in The Greater Trumps , he has turned to
one of the prime mysteries of earth . . . ."

— William Lindsay Gresham, Preface (1950) to
Charles Williams's The Greater Trumps  (1932)

For fans of what the recent series Westworld  called "bulk apperception" —

Tuesday, January 16, 2018

The Pentagram Papers

Filed under: General,Geometry — Tags: , , — m759 @ 9:16 am

Other intersection-points-counting material —

The Finkelstein Talisman:

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points

See also Hanks + Cube in this journal —

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube.

Thursday, December 21, 2017

For Winter Solstice 2017

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 10:30 am

A review —

Some context —

Webpage demonstrating symmetries of 'Solomon's Cube'

Tuesday, October 24, 2017

Visual Insight

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

The most recent post in the "Visual Insight" blog of the
American Mathematical Society was by John Baez on Jan. 1, 2017


A visually  related concept — See Solomon's Cube in this  journal.
Chronologically  related — Posts now tagged New Year's Day 2017.
Solomon's cube is the 4x4x4 case of the diamond theorem — 

Sunday, July 30, 2017

Sermon: MS R I

Filed under: General,Geometry — Tags: — m759 @ 9:57 am

From Solomon's Cube

"Here MSRI, an acronym for Mathematical Sciences Research Institute,
is pronounced 'Misery.' See Stephen King [and] K.C. Cole . . . ."

From a manuscript by Mikhail Gromov cited yesterday in MSRI Program —

Quotes from a founder of geometric group theory

Monday, July 24, 2017

Penguin Classics Deluxe Edition

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

The above title was suggested by a film trailer quoted here Saturday

" Jeremy Irons' dry Alfred Pennyworth:
'One misses the days when one's biggest concerns
were exploding wind-up penguins.' "

"Penguin Classics Deluxe Edition" describes, among other books,
an edition of the I Ching  published on December 1, 2015.

Excerpt from this journal on that date

Tuesday, December 1, 2015

Verhexung

Filed under: Uncategorized — m759 @ 9:00 PM 

(Continued)

"The positional meaning of a symbol derives from
its relationship to other symbols in a totality, a Gestalt,
whose elements acquire their significance from the
system as a whole."

— Victor Turner, The Forest of Symbols , Ithaca, NY,
Cornell University Press, 1967, p. 51, quoted by
Beth Barrie in "Victor Turner."

(Turner pioneered the use of the term "symbology,"
a term later applied by Dan Brown to a fictional
scholarly pursuit at Harvard.)

. . . .

Related material —

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

The I Ching's underlying group has 1,290,157,424,640 permutations.

Thursday, June 22, 2017

Face Henge

Filed under: General,Geometry — Tags: , , — m759 @ 3:07 pm

With a hat tip to Vinnie Mancuso

Wednesday, June 21, 2017

Concept and Realization

Filed under: General,Geometry — Tags: , , — m759 @ 1:12 pm

Remark on conceptual art quoted in the previous post

"…he’s giving the concept but not the realization."

A concept See a note from this date in 1983:

IMAGE- 'Solomon's Cube'

A realization  

Webpage demonstrating symmetries of 'Solomon's Cube'

Not the best possible realization, but enough for proof of concept .

Tuesday, May 2, 2017

Image Albums

Filed under: General,Geometry — Tags: , , , , — m759 @ 1:05 pm

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Saturday, April 1, 2017

Beyond All Recognition

Filed under: General,Geometry — m759 @ 10:45 am

Prequel —

Cube symmetry subgroup of order 8 from 'Geometry and Symmetry,' Paul B. Yale, 1968, p.21

Note that Yale's die design and use of the phrase "rigid motions"
differ from those in the webpage "Solomon's Cube."

Wednesday, March 29, 2017

Art Space Illustrated

Another view of the previous post's art space  —

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

More generally, see Solomon's Cube in Log24.

See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above 
cube arrangement from "Affine groups on small binary spaces" (1984).

IMAGE- Current math.stackexchange.com logo and a 1984 figure from 'Notes on Groups and Geometry, 1978-1986'

Friday, March 10, 2017

Transformers

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

Or:  Y  for Yale  continued

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

See also Transformers in this journal and Y for Yale.

Thursday, March 2, 2017

Stories

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

"We tell ourselves stories in order to live." — Joan Didion

The New York Times Magazine  online today —

"As a former believer and now a nonbeliever, Carrère,
seeking answers, sets out, in The Kingdom , to tell
the story of the storytellers. He is trying to understand
what it takes to be able to tell a story, any story.
And what he finds, once again, is that you have to find
your role in it."

Wyatt Mason in The New York Times Magazine ,
     online March 2, 2017 

Like Tom Hanks?

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Click image for related posts.

Sunday, February 26, 2017

Poetic Order

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

Transformations acting on Solomon's Cube
furnish a model of poetic order.

Some backstory for Hollywood —

Hollywood analogue to Solomon's Cube in 'Transformers'

Saturday, February 18, 2017

Solid Symmetry (continued)

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

See Hanks + Cube in this journal For instance

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM 

See Cube Symbology.

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Da hats ein Eck 

Verbum

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

The Log24 version  (Nov. 9, 2005, and later posts) —

VERBUM
SAT
SAPIENTI

 

Escher's 'Verbum'

Escher's Verbum


Solomon's Cube

Solomon's Cube
 

I Ching hexagrams as parts of 4x4x4 cube

Geometry of the I Ching

The Warner Brothers version

The Paramount version

See also related material in the previous post, Transformers.

Thursday, January 12, 2017

The Cherished Gift

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

From "Solomon's Cube" —

Related material —

"Is this a dagger I see before me?

"No." (A line suggested by Polanski's 2010 "The Ghost Writer")

Monday, October 3, 2016

Ein Eck

Filed under: General,Geometry — Tags: , — m759 @ 5:05 pm
 

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM 

See Cube Symbology.

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Da hats ein Eck 

Friday, August 5, 2016

Sleight of Post

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

From an earlier Log24 post —

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM 

See Cube Symbology.

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Da hats ein Eck 

From a post of the next day, July 12, 2014 —

"So there are several different genres and tones
jostling for prominence within Lexicon :
a conspiracy thriller, an almost abstract debate
about what language can do, and an ironic
questioning of some of the things it’s currently used for."

Graham Sleight in The Washington Post 
     a year earlier, on July 15, 2013

For the Church of Synchronology, from Log24 on the next day — 

From a post titled Circles on the date of Marc Simont's death —

See as well Verhexung  in this journal.

Thursday, July 14, 2016

Symmetries and Correspondences

Filed under: General,Geometry — Tags: , , — m759 @ 7:30 pm

The title is that of a large-scale British research project
in mathematics. On a more modest scale

"Hanks + Cube" in this journal —

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Block That Metaphor

Sunday, June 26, 2016

Common Core versus Central Structure

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

Rubik's Cube Core Assembly — Swarthmore Cube Project, 2008 —

"Children of the Common Core" —

There is also a central structure within Solomon's  Cube

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

For a more elaborate entertainment along these lines, see the recent film

"Midnight Special" —

Tuesday, June 21, 2016

The Central Structure

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

“The central poem is the poem of the whole,
The poem of the composition of the whole”

— Wallace Stevens, “A Primitive like an Orb”

The symmetries of the central four squares in any pattern
from the 4×4 version of the diamond theorem  extend to
symmetries of the entire pattern.  This is true also of the
central eight cubes in the 4×4×4  Solomon’s cube .

Tuesday, May 3, 2016

Symmetry

A note related to the diamond theorem and to the site
Finite Geometry of the Square and Cube —

The last link in the previous post leads to a post of last October whose
final link leads, in turn, to a 2009 post titled Summa Mythologica .

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above web page* —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

Update of Sept. 5, 2016 — See also a related remark
by Lévi-Strauss in 1955:  "…three different readings
become possible: left to right, top to bottom, front
to back."

* For the underlying mathematics, see a June 21, 1983, research note.

Wednesday, April 20, 2016

Symmetric Generation of a Simple Group

The reference in the previous post to the work of Guitart and
The Road to Universal Logic  suggests a fiction involving
the symmetric generation of the simple group of order 168.

See The Diamond Archetype and a fictional account of the road to Hell 

'PyrE' in Bester's 'The Stars My Destination'

The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.

IMAGE- 'The Stars My Destination' (with cover slightly changed)

For related symmetric generation of a much larger group, see Solomon’s Cube.

Wednesday, December 2, 2015

Symbology

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

Symbologist Robert Langdon views a corner of Solomon's Cube

Click image to search Log24
for Solomon + Stone.

Saturday, April 25, 2015

Ghosts and Shadows

Filed under: General,Geometry — Tags: , , — m759 @ 5:31 pm

For Poetry Month

From the home page of Alexandre Borovik:

Book in progress: Shadows of the Truth

This book (to be published soon) can be viewed
as a sequel to Mathematics under the Microscope ,
but with focus shifted on mathematics as it was
experienced by children (well, by children who
became mathematicians). The cover is designed
by Edmund Harriss.

See also Harriss's weblog post of Dec. 27, 2008, on the death
of Harold Pinter: "The Search for the Truth Can Never Stop."

This suggests a review of my own post of Dec. 3, 2012,
"The Revisiting." A figure from that post:

The image “http://www.log24.com/theory/images/GEB.jpg” cannot be displayed, because it contains errors.

Wednesday, September 17, 2014

Raiders of the Lost Articulation

Tom Hanks as Indiana Langdon in Raiders of the Lost Articulation :

An unarticulated (but colored) cube:

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

A 2x2x2 articulated cube:

IMAGE- Eightfold cube with detail of triskelion structure

A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:

Image-- Solomon's Cube

Solomon’s Cube

Saturday, July 12, 2014

Mars Package

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

For Ursula K. Le Guin

“For me it is a sign that we have fundamentally different
conceptions of the work of the intelligence services.”

— Germany’s Chancellor Angela Merkel in
theguardian.com, Saturday, 12 July 2014, 14.32 EDT

Another sort of service, thanks to Dan Brown and Tom Hanks:

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM 

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Sequel

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

A sequel to the 1974 film
Thunderbolt and Lightfoot :

Contingent and Fluky

Some variations on a thunderbolt  theme:

Design Cube 2x2x2 for demonstrating Galois geometry

These variations also exemplify the larger
Verbum  theme:

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

A search today for Verbum  in this journal yielded
a Georgetown 
University Chomskyite, Professor
David W. Lightfoot.

"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."

Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.

See Tecumseh Fitch  in this journal.

For other material related to the large Verbum  cube,
see posts for the 18th birthday of Harry Potter.

That birthday was also the upload date for the following:

See esp. the comments section.

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

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

See Cube Symbology.

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Da hats ein Eck 

Friday, May 9, 2014

Models of Everything

Filed under: General,Geometry — Tags: , — m759 @ 11:16 am

“The About page contains detailed descriptions of the project….”

The Illustris project on constructing a model of the universe

For the mathematics of a simpler traditional Chinese model
of everything, see

Thursday, March 27, 2014

Diamond Space

Filed under: General,Geometry — Tags: , , , — m759 @ 2:28 pm

(Continued)

Definition:  A diamond space  — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.

The reason for the name:

IMAGE - The Diamond Theorem, including the 4x4x4 'Solomon's Cube' case

Click to enlarge.

Saturday, March 1, 2014

HaShem

Filed under: General,Geometry — m759 @ 10:30 pm

IMAGE- Josefine Lyche changes her Facebook cover photo to a form of the Tetragrammaton.

From New World Encyclopedia —

See also Tetragrammaton in this journal.

For further context, see Solomon's Cube and Oct. 16, 2013.

Saturday, April 21, 2012

Finding a Form

Filed under: General,Geometry — Tags: , , , — m759 @ 11:30 pm


In "Contact," Dr. Arroway  is shown the key to the Primer

In this journal, fictional symbologist Robert Langdon is shown a cube

Symbologist Robert Langdon views a corner of Solomon's Cube

"Confusion is nothing new." — Song lyric

Wednesday, April 18, 2012

Adam in Eden

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

…. and John Golding, an authority on Cubism who "courted abstraction"—

"Adam in Eden was the father of Descartes." — Wallace Stevens

Fictional symbologist Robert Langdon and a cube

Symbologist Robert Langdon views a corner of Solomon's Cube

From a Log24 post, "Eightfold Cube Revisited,"
on the date of Golding's death—

Dynkin diagram D4 for triality

A related quotation—

"… quaternions provide a useful paradigm
  for studying the phenomenon of 'triality.'"

  — David A. Richter's webpage Zometool Triality

See also quaternions in another Log24 post
from the date of Golding's death— Easter Act.

Monday, April 16, 2012

Carroll Thanks the Academy

Filed under: General,Geometry — Tags: , , , — m759 @ 11:30 am

Gary Gutting, "Arguing About Language," in "The Stone,"
The New York Times  philosophy column, yesterday—

There's a sense in which we speak language
and a sense in which, in Mallarmé's famous phrase,
“language itself speaks.”

Famous? A Google Book Search for

"language itself speaks" Mallarmé

yields 2 results, neither helpful.

But a Google Book Search for

"language itself speaks" Heidegger

yields "about 312 results."

A related search yields the following

Paul Valéry, encountering Un Coup de Dés  in Mallarmé’s worksheets in 1897, described the text as tracing the pattern of thought itself:

It seemed to me that I was looking at the form and pattern of a thought, placed for the first time in finite space. Here space itself truly spoke, dreamed, and gave birth to temporal forms….

… there in the same void with them, like some new form of matter arranged in systems or masses or trailing lines, coexisted the Word! (Leonardo  309*)

* The page number is apparently a reference to The Collected Works of Paul Valéry: Leonardo, Poe, Mallarmé , translated by Malcolm Cowley and James R. Lawler, Princeton University Press, 1972. (As a temporal  form, "309" might be interpreted as a reference to 3/09, March 9, the date of a webpage on the Void.)

For example—

Symbologist Robert Langdon views a corner of Solomon's Cube

Background:
Deconstructing Alice
and Symbology.

Thursday, April 12, 2012

Mythopoetic*

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

"Is Space Digital?" 

Cover storyScientific American  magazine, February 2012

"The idea that space may be digital
  is a fringe idea of a fringe idea
  of a speculative subfield of a subfield."

— Physicist Sabine Hossenfelder
     at her weblog on Feb. 5, 2012

"A quantization of space/time
 is a holy grail for many theorists…."

— Peter Woit in a comment at his physics weblog today

See also 

* See yesterday's Steiner's Systems.

Thursday, April 5, 2012

Meanwhile, back in 1950…

Filed under: General,Geometry — Tags: — m759 @ 10:30 am

See also Solomon's Cube.

Tuesday, February 14, 2012

The Ninth Configuration

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

The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title, 
that of a novel by the author of The Exorcist .

The Ninth Configuration 

The ninth* in a list of configurations—

"There is a (2d-1)d  configuration
  known as the Cox configuration."

MathWorld article on "Configuration"

For further details on the Cox 326 configuration's Levi graph,
a model of the 64 vertices of the six-dimensional hypercube γ6  ,
see Coxeter, "Self-Dual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc.  Vol. 56, pages 413-455, 1950.
This contains a discussion of Kummer's 166 as it 
relates to  γ6  , another form of the 4×4×4 Galois cube.

See also Solomon's Cube.

* Or tenth, if the fleeting reference to 113 configurations is counted as the seventh—
  and then the ninth  would be a 153 and some related material would be Inscapes.

Tuesday, January 17, 2012

Augenmusik

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

In memory of Bach interpreter
Gustav Leonhardt

http://www.log24.com/log/pix12/120117-SolomonsCube.jpg

Augenmusik

Wednesday, January 19, 2011

Intermediate Cubism

Filed under: General,Geometry — Tags: , — m759 @ 2:22 pm

The following is a new illustration for Cubist Geometries

IMAGE- A Galois cube: model of the 27-point affine 3-space

(For elementary cubism, see Pilate Goes to Kindergarten and The Eightfold Cube.
 For advanced, see Solomon's Cube and Geometry of the I Ching .)

Cézanne's Greetings.

Tuesday, December 7, 2010

The Tiffany Puzzle

Filed under: General,Geometry — Tags: — m759 @ 5:00 am

Suggested by Dan Brown's remarks in today's Science Times  special section on puzzles—

http://www.log24.com/log/pix10B/101202-DreidelAndStoneSm.jpg

For a fanciful linkage of the dreidel 's concept of chance
to The Stone 's concept of invariant law, note that the
New York Lottery evening number on Dec. 1 (the
beginning of Hanukkah) was 840. See also the number
840 in the final post (July 20, 2002) of a search for
Solomon's Cube.

http://www.log24.com/log/pix10B/101207-FifthAve5AM.jpg

Thursday, December 2, 2010

Caesarian

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

The Dreidel Is Cast

The Nietzschean phrase "ruling and Caesarian spirits" occurred in yesterday morning's post "Novel Ending."

That post was followed yesterday morning by a post marking, instead, a beginning— that of Hanukkah 2010. That Jewish holiday, whose name means "dedication," commemorates the (re)dedication of the Temple in Jerusalem in 165 BC.

The holiday is celebrated with, among other things, the Jewish version of a die—  the dreidel . Note the similarity of the dreidel  to an illustration of The Stone*  on the cover of the 2001 Eerdmans edition of  Charles Williams's 1931 novel Many Dimensions

http://www.log24.com/log/pix10B/101202-DreidelAndStone.jpg

For mathematics related to the dreidel , see Ivars Peterson's column on this date fourteen years ago.
For mathematics related (if only poetically) to The Stone , see "Solomon's Cube" in this journal.

Here is the opening of Many Dimensions

http://www.log24.com/log/pix10B/101202-WilliamsChOne.jpg

For a fanciful linkage of the dreidel 's concept of chance to The Stone 's concept of invariant law, note that the New York Lottery yesterday evening (the beginning of Hanukkah) was 840. See also the number 840 in the final post (July 20, 2002) of the "Solomon's Cube" search.

Some further holiday meditations on a beginning—

Today, on the first full day of Hanukkah, we may or may not choose to mark another beginning— that of George Frederick James Temple, who was born in London on this date in 1901. Temple, a mathematician, was President of the London Mathematical Society in 1951-1953. From his MacTutor biography

"In 1981 (at the age of 80) he published a book on the history of mathematics. This book 100 years of mathematics (1981) took him ten years to write and deals with, in his own words:-

those branches of mathematics in which I had been personally involved.

He declared that it was his last mathematics book, and entered the Benedictine Order as a monk. He was ordained in 1983 and entered Quarr Abbey on the Isle of Wight. However he could not stop doing mathematics and when he died he left a manuscript on the foundations of mathematics. He claims:-

The purpose of this investigation is to carry out the primary part of Hilbert's programme, i.e. to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced."

For a brief review of Temple's last work, see the note by Martin Hyland in "Fundamental Mathematical Theories," by George Temple, Philosophical Transactions of the Royal Society, A, Vol. 354, No. 1714 (Aug. 15, 1996), pp. 1941-1967.

The following remarks by Hyland are of more general interest—

"… one might crudely distinguish between philosophical and mathematical motivation. In the first case one tries to convince with a telling conceptual story; in the second one relies more on the elegance of some emergent mathematical structure. If there is a tradition in logic it favours the former, but I have a sneaking affection for the latter. Of course the distinction is not so clear cut. Elegant mathematics will of itself tell a tale, and one with the merit of simplicity. This may carry philosophical weight. But that cannot be guaranteed: in the end one cannot escape the need to form a judgement of significance."

— J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.

Here Hyland appears to be discussing semantic ("philosophical," or conceptual) and syntactic ("mathematical," or structural) approaches to proof theory. Some other remarks along these lines, from the late Gian-Carlo Rota

http://www.log24.com/log/pix10B/101202-RotaChXII-sm.jpg

    (Click to enlarge.)

See also "Galois Connections" at alpheccar.org and "The Galois Connection Between Syntax and Semantics" at logicmatters.net.

* Williams's novel says the letters of The Stone  are those of the Tetragrammaton— i.e., Yod, He, Vau, He  (cf. p. 26 of the 2001 Eerdmans edition). But the letters on the 2001 edition's cover Stone  include the three-pronged letter Shin , also found on the dreidel .  What esoteric religious meaning is implied by this, I do not know.

Wednesday, September 1, 2010

September Morn

Filed under: General,Geometry — Tags: , — m759 @ 7:59 am

For Alyssa Milano —

http://www.log24.com/log/pix10B/100901-MilanoFork.jpg

The Forking

(Click here for cheesy Neil Diamond background music.)

For some related philosophical remarks, see Deconstructing Alice

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

and the new Pythagorean thriller The Thousand.

Wednesday, June 16, 2010

Brightness at Noon

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

David Levine's portrait of Arthur Koestler (see Dec. 30, 2009) —

Image-- Arthur Koestler by David Levine, NY Review of Books, Dec. 17, 1964, review of 'The Act of Creation'

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

Image-- The 64 I Ching hexagrams in the 4 layers of the Cullinane cube

Geometry of the I Ching

See also this morning's post as well as
Monday's post quoting George David Birkhoff

"If I were a Leibnizian mystic… I would say that…
God thinks multi-dimensionally — that is,
uses multi-dimensional symbols beyond our grasp."

Geometry of Language

Filed under: General,Geometry — Tags: , — m759 @ 10:31 am

(Continued from April 23, 2009, and February 13, 2010.)

Paul Valéry as quoted in yesterday’s post:

“The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (Cahiers, 15:170 [2: 315])

The geometric example discussed here yesterday as a Self symbol may seem too small to be really impressive. Here is a larger example from the Chinese, rather than European, tradition. It may be regarded as a way of representing the Galois field GF(64). (“Field” is a rather ambiguous term; here it does not, of course, mean what it did in the Valéry quotation.)

From Geometry of the I Ching

Image-- The 64 hexagrams of the I Ching

The above 64 hexagrams may also be regarded as
the finite affine space AG(6,2)— a larger version
of the finite affine space AG(4,2) in yesterday’s post.
That smaller space has a group of 322,560 symmetries.
The larger hexagram  space has a group of
1,290,157,424,640 affine symmetries.

From a paper on GL(6,2), the symmetry group
of the corresponding projective  space PG(5,2),*
which has 1/64 as many symmetries—

(Click to enlarge.)

Image-- Classes of the Group GL(6,)

For some narrative in the European  tradition
related to this geometry, see Solomon’s Cube.

* Update of July 29, 2011: The “PG(5,2)” above is a correction from an earlier error.

Tuesday, May 4, 2010

Mathematics and Narrative, continued

Filed under: General,Geometry — Tags: , , — m759 @ 8:28 pm

Romancing the
Non-Euclidean Hyperspace

Backstory
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937

The cast of 1937's 'King Solomon's Mines' goes back to the future

For the 1937 grid, see Diamond Theory, 1937.

The grid is, as Mere Geometry points out, a non-Euclidean hyperspace.

For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.

Sunday, March 21, 2010

Galois Field of Dreams

Filed under: General,Geometry — Tags: , — m759 @ 10:01 am

It is well known that the seven (22 + 2 +1) points of the projective plane of order 2 correspond to 2-point subspaces (lines) of the linear 3-space over the two-element field Galois field GF(2), and may be therefore be visualized as 2-cube subsets of the 2×2×2 cube.

Similarly, recent posts* have noted that the thirteen (32 + 3 + 1) points of the projective plane of order 3 may be seen as 3-cube subsets in the 3×3×3 cube.

The twenty-one (42 + 4 +1) points of the (unique) projective plane of order 4 may also be visualized as subsets of a cube– in this case, the 4×4×4 cube. This visualization is somewhat more complicated than the 3×3×3 case, since the 4×4×4 cube has no central subcube, and each projective-plane point corresponds to four, not three, subcubes.

These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)

A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field  GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."

Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).

The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…

Number and Time, by Marie-Louise von Franz

See also Geometry of the I Ching and a search in this journal for "Galois + Ching."

* February 27 and March 13

** G20160 in Mitchell 1910,  LF(3,22) in Edge 1965

— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
   of the Finite Projective Plane PG(2,22),"
   Princeton Ph.D. dissertation (1910)

— Edge, W. L., "Some Implications of the Geometry of
   the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)

Friday, March 19, 2010

Garden of Forking Paths

Filed under: General,Geometry — Tags: , — m759 @ 10:18 am

For Alyssa

 

 An Old Magic Symbol

http://www.log24.com/log/pix10/100319-Palermo.gif

… and for Dan Brown —

Symbology
Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Saturday, March 6, 2010

Deconstructing Alice

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

Alyssa is  Wonderland

Manohla Dargis in The New York Times  yesterday

“Of course the character of Carroll’s original Alice is evident in each outrageous creation she dreams up in ‘Wonderland’ and in the sequel, ‘Through the Looking-Glass,’ which means that she’s a straight man to her own imagination. (She is  Wonderland.)”

Alyssa Milano as a child, with fork

From Inside the White Cube

“The sacramental nature of the space becomes clear, and so does one of the great projective laws of modernism: as modernism gets older, context becomes content. In a peculiar reversal, the object introduced into the gallery ‘frames’ the gallery and its laws.”

From Yogi Berra–

“When you come to a fork in the road, take it.”

Related material:  For Baron Samedi and…

Symbology
Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube
Jacques Derrida on the Looking-Glass garden, 'The Time before First,' and Solomon's seal

Wednesday, March 3, 2010

Plato’s Ghost

Filed under: General,Geometry — Tags: , — m759 @ 11:07 am

Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics, Princeton, 2008–

"Here, modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated— indeed, anxious— rather than a naïve relationship with the day-to-day world, which is the de facto view of a coherent group of people, such as a professional or discipline-based group that has a high sense of the seriousness and value of what it is trying to achieve. This brisk definition…."

Brisk? Consider Caesar's "The die is cast," Gray in "Solomon's Cube," and yesterday's post

Group of 8 cube-face permutations generated by reflections in midplanes parallel to faces

This is the group of "8 rigid motions
generated by reflections in midplanes"
of Solomon's Cube.

Related material:

"… the action of G168 in its alternative guise as SL(3; Z/2Z) is also now apparent. This version of G168 was presented by Weber in [1896, p. 539],* where he attributed it to Kronecker."

— Jeremy Gray, "From the History of a Simple Group," in The Eightfold Way, MSRI Publications, 1998

Here MSRI, an acronym for Mathematical Sciences Research Institute, is pronounced "Misery." See Stephen King, K.C. Cole, and Heinrich Weber.

*H. Weber, Lehrbuch der Algebra, Vieweg, Braunschweig, 1896. Reprinted by Chelsea, New York, 1961.

Thursday, February 18, 2010

Theories: An Outline

Filed under: General,Geometry — Tags: , , , , — m759 @ 10:31 am

Truth, Geometry, Algebra

The following notes are related to A Simple Reflection Group of Order 168.

1. According to H.S.M. Coxeter and Richard J. Trudeau

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”

— Coxeter, 1987, introduction to Trudeau’s The Non-Euclidean Revolution

1.1 Trudeau’s Diamond Theory of Truth

1.2 Trudeau’s Story Theory of Truth

2. According to Alexandre Borovik and Steven H. Cullinane

2.1 Coxeter Theory according to Borovik

2.1.1 The Geometry–

Mirror Systems in Coxeter Theory

2.1.2 The Algebra–

Coxeter Languages in Coxeter Theory

2.2 Diamond Theory according to Cullinane

2.2.1 The Geometry–

Examples: Eightfold Cube and Solomon’s Cube

2.2.2 The Algebra–

Examples: Cullinane and (rather indirectly related) Gerhard Grams

Summary of the story thus far:

Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side.  Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)

Tuesday, January 26, 2010

Symbology

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

From this journal:

Friday December 5, 2008

m759 @ 1:06 PM
 
Mirror-Play of
the Fourfold

For an excellent commentary
 on this concept of Heidegger,

View selected pages
from the book

Dionysus Reborn:

Play and the Aesthetic Dimension
in Modern Philosophical and
Scientific Discourse

(Mihai I. Spariosu,
Cornell U. Press, 1989)

Related material:
the logo for a
web page

Logo for 'Elements of Finite Geometry'

– and Theme and Variations.

Transition to the
Garden of Forking Paths–

(See For Baron Samedi)–

The Found Symbol
Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

and Dissemination, by Jacques Derrida,
translated by Barbara Johnson,
London, Athlone Press, 1981–

Pages 354-355
On the mirror-play of the fourfold

Pages 356-357
Shaking up a whole culture

Pages 358-359
Cornerstone and crossroads

Pages 360-361
A deep impression embedded in stone

Pages 362-363
A certain Y, a certain V

Pages 364-365
The world is Zeus's play

Page 366
It was necessary to begin again

 

Saturday, January 23, 2010

For Baron Samedi

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

Yesterday's Times —

NY Times banner with Eve and apple

Today's Times —

NY Times ad for Goldstein's '36 Arguments'-- 'Deconstruct the Arguments'

   Annals of Deconstruction —

Click on image for background.

New Yorker cover on Haiti featuring Baron Samedi

Related material
   for Baron Samedi

The Found Symbol
Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube
Jacques Derrida on the Looking-Glass garden, 'The Time before First,' and Solomon's seal

Saturday, November 14, 2009

Mathematics and Narrative, continued:

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

A graphic novel reviewed in the current Washington Post  features Alfred North Whitehead and Bertrand Russell–

Whitehead and Russell, 'Logicomix' page 181

Related material:

Whitehead on Fano’s finite projective three-space:

“This is proved by the consideration of a three dimensional geometry in which there are only fifteen points.”

The Axioms of Projective Geometry , Cambridge University Press, 1906

A related affine six-space:

Grey cube, 4x4x4

Further reading:

See Solomon’s Cube and the link at the end of today’s previous entry, then compare and contrast the above portraits of Whitehead and Russell with Charles Williams’s portraits of Sir Giles Tumulty and Lord Arglay in the novel Many Dimensions .

It was a dark and stormy night….

Thursday, September 17, 2009

Thursday September 17, 2009

Filed under: General,Geometry — Tags: — m759 @ 8:00 pm
Jennifer's Body

The following remark this evening by Ann Hornaday of The Washington Post serves as an instant review of today's previous cinematic Log24 offering starring the late Patrick Swayze:

"Watch it, forget it, move on."

A perhaps more enduring tribute:

Patrick Swayze in 'King Solomon's Mines'

 

Related material:

Solomon's Cube,
Solomon and Sheba,
and
Raiders of the Lost Stone.

"Ready when you are, C.B."

 

Thursday September 17, 2009

Filed under: General,Geometry — Tags: , — m759 @ 11:07 am
Symbologist Robert Langdon and a corner of Solomon's Cube

Patrick Swayze and Jennifer Grey in  'Dirty Dancing'

“Nobody puts Baby in a corner.”

Wednesday, September 16, 2009

Wednesday September 16, 2009

Filed under: General,Geometry — Tags: , — m759 @ 11:07 am
The Found Symbol
Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube
Jacques Derrida on the Looking-Glass garden, 'The Time before First,' and Solomon's seal

Saturday, September 5, 2009

Saturday September 5, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 10:31 pm
For the
Burning Man

'The Stars My Destination,' current edition (with cover slightly changed)

(Cover slightly changed.)

 
Background —

 
SAT
 
Part I:

Sophists (August 20th)

Part II:

VERBUM
SAT
SAPIENTI

Escher's 'Verbum'

Escher's Verbum


Solomon's Cube



Part III:

From August 25th

Equilateral triangle on a cube, each side's length equal to the square root of two

"Boo, boo, boo,
  square root of two.
"

Sunday, March 1, 2009

Sunday March 1, 2009

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

Solomon's Cube
continued

"There is a book… called A Fellow of Trinity, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."

— G. H. Hardy, A Mathematician's Apology

"The Solomon Key is the working title of an unreleased novel in progress by American author Dan Brown. The Solomon Key will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were Angels & Demons (2000) and The Da Vinci Code (2003)." — Wikipedia

"One has O+(6) ≅ S8, the symmetric group of order 8! …."

 — "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.

"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."

— George M. Conwell, "The 3-space PG(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
 

Tuesday, February 17, 2009

Tuesday February 17, 2009

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

Diamond-Faceted:
Transformations
of the Rock

A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:

For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:

The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...

The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,

Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.

The rock is the habitation of the whole,
Its strength and measure, that which is near,
     point A
In a perspective that begins again

At B: the origin of the mango's rind.

                    (Collected Poems, 528)

A mathematical version of
this poetic concept appears
in a rather cryptic note
from 1981 written with
Stevens's poem in mind:

http://www.log24.com/log/pix09/090217-SolidSymmetry.jpg

For some explanation of the
groups of 8 and 24
motions referred to in the note,
see an earlier note from 1981.

For the Perlis "diamond facets,"
see the Diamond 16 Puzzle.

For a much larger group
of motions, see
Solomon's Cube.

As for "the mind itself"
and "possibilities for
human thought," see
Geometry of the I Ching.

Saturday, December 6, 2008

Saturday December 6, 2008

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

"While feasts of Saint Nicholas are not observed nationally, cities with strong German influences like Milwaukee, Cincinnati, and St. Louis celebrate St. Nick's Day on a scale similar to the German custom." —Wikipedia

A footprint from Germany:

Germany
Python-urllib
/504856559/item.html 12/6/2008
1:21 PM

The link in the above footprint leads
to an entry of July 5, 2006.

The access method:

The urllib Module

"The Python urllib module implements a fairly high-level abstraction for making any web object with a URL act like a Python file: i.e., you open it, and get back an object…."


For more pictures and discussion
of the object fetched by Python,
see AntiChristmas 2007.

For a larger and more sophisticated
relative of that object,
 see Solomon's Cube and
the related three presents
from the German link's target:

Spellbound: A trinity of Christmas presents

1. Many Dimensions
2. Boggle
3. My Space

Sunday, May 18, 2008

Sunday May 18, 2008

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

From the Grave

DENNIS OVERBYE

in yesterday's New York Times:

"From the grave, Albert Einstein
poured gasoline on the culture wars
between science and religion this week…."

An announcement of a
colloquium at Princeton:

Cartoon of Coxedter exhuming Geometry

Above: a cartoon,
"Coxeter exhuming Geometry,"
with the latter's tombstone inscribed

"GEOMETRY

  600 B.C. —
1900 A.D.
R.I.P."

Page from 'The Paradise of Childhood,' 1906 edition

The above is from
The Paradise of Childhood,
a work first published in 1869.

"I need a photo-opportunity,
I want a shot at redemption.
Don't want to end up a cartoon
In a cartoon graveyard."

— Paul Simon

Einstein on TIME cover as 'Man of the Century'

Albert Einstein,
1879-1955:

"It is quite clear to me that the religious paradise of youth, which was thus lost, was a first attempt to free myself from the chains of the 'merely-personal,' from an existence which is dominated by wishes, hopes and primitive feelings.  Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking.  The contemplation of this world beckoned like a liberation…."

Autobiographical Notes, 1949

Related material:

A commentary on Tom Wolfe's
"Sorry, but Your Soul Just Died"–

"The Neural Buddhists," by David Brooks,
 in the May 13 New York Times:

"The mind seems to have
the ability to transcend itself
and merge with a larger
presence that feels more real."

A New Yorker commentary on
a new translation of the Psalms:

"Suddenly, in a world without
Heaven, Hell, the soul, and
eternal salvation or redemption,
the theological stakes seem
more local and temporal:
'So teach us to number our days.'"

and a May 13 Log24 commentary
on Thomas Wolfe's
"Only the Dead Know Brooklyn"–

"… all good things — trout as well as
eternal salvation — come by grace
and grace comes by art
and art does not come easy."

A River Runs Through It

"Art isn't easy."
— Stephen Sondheim,
quoted in
Solomon's Cube.

For further religious remarks,
consult Indiana Jones and the
Kingdom of the Crystal Skull
and The Librarian:
Return to King Solomon's Mines.

Saturday, May 10, 2008

Saturday May 10, 2008

MoMA Goes to
Kindergarten

"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."

— "Was Modernism Born
     in Toddler Toolboxes?"
     by Trip Gabriel, New York Times,
     April 10, 1997
 

RELATED MATERIAL

Figure 1 —
Concept from 1819:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

Froebel's Third Gift

Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.

(Footnote 3)

Figure 3 —
The Third Gift, 1906:

Seven partitions of the eightfold cube in 'Paradise of Childhood,' 1906

Figure 4 —
Solomon's Cube,
1981 and 1983:

Solomon's Cube - A 1981 design by Steven H. Cullinane

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

For some mathematical background, see

Footnotes:
 
1. Image said to be after Holden and Morrison, Crystals and Crystal Growing, 1982
2. Curtis Schuh, "The Library: Biobibliography of Mineralogy," article on Mohs
3. Bart Kahr, "Crystal Engineering in Kindergarten" (pdf), Crystal Growth & Design, Vol. 4 No. 1, 2004, 3-9

Monday, July 23, 2007

Monday July 23, 2007

Daniel Radcliffe
is 18 today.
Daniel Radcliffe as Harry Potter

Greetings.

“The greatest sorcerer (writes Novalis memorably)
would be the one who bewitched himself to the point of
taking his own phantasmagorias for autonomous apparitions.
Would not this be true of us?”

Jorge Luis Borges, “Avatars of the Tortoise”

El mayor hechicero (escribe memorablemente Novalis)
sería el que se hechizara hasta el punto de
tomar sus propias fantasmagorías por apariciones autónomas.
¿No sería este nuestro caso?”

Jorge Luis Borges, “Los Avatares de la Tortuga

Autonomous Apparition

At Midsummer Noon:

“In Many Dimensions (1931)
Williams sets before his reader the
mysterious Stone of King Solomon,
an image he probably drew from
a brief description in Waite’s
The Holy Kabbalah (1929) of
a supernatural cubic stone
on which was inscribed
‘the Divine Name.’”
The image “http://www.log24.com/log/pix07/070624-Waite.gif” cannot be displayed, because it contains errors.
Related material:
It is not enough to cover the rock with leaves.
We must be cured of it by a cure of the ground
Or a cure of ourselves, that is equal to a cure 

Of the ground, a cure beyond forgetfulness.
And yet the leaves, if they broke into bud,
If they broke into bloom, if they bore fruit,

And if we ate the incipient colorings
Of their fresh culls might be a cure of the ground.

– Wallace Stevens, “The Rock”

See also
as well as
Hofstadter on
his magnum opus:
“… I realized that to me,
Gödel and Escher and Bach
were only shadows
cast in different directions by
some central solid essence.
I tried to reconstruct
the central object, and
came up with this book.”
Goedel Escher Bach coverHofstadter’s cover.

Here are three patterns,
“shadows” of a sort,
derived from a different
“central object”:
Faces of Solomon's Cube, related to Escher's 'Verbum'

Click on image for details.

Sunday, June 24, 2007

Sunday June 24, 2007

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm
Raiders of
the Lost Stone

(Continued from June 23)

Scott McLaren on
Charles Williams:
 
"In Many Dimensions (1931)
Williams sets before his reader the
mysterious Stone of King Solomon,
an image he probably drew
from a brief description in Waite's
The Holy Kabbalah (1929)
of a supernatural cubic stone
on which was inscribed
'the Divine Name.'"

The image “http://www.log24.com/log/pix07/070624-Waite.gif” cannot be displayed, because it contains errors.

Related material:

The image “http://www.log24.com/log/pix07/070624-Cube.gif” cannot be displayed, because it contains errors.

Solomon's Cube,

Geometry of the 4x4x4 Cube,

The Klein Correspondence,
Penrose Space-Time,
and a Finite Model

Tuesday, April 3, 2007

Tuesday April 3, 2007

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

Our Judeo-Christian
Heritage –
 
Lottery
Hermeneutics

Part I: Judeo

The Lottery 12/9/06 Mid-day Evening
New York 036

See

The Quest
for the 36

331

See 3/31

“square crystal” and “the symbolism could not have been more perfect.”

Pennsylvania 602

See 6/02

Walter Benjamin
on
“Adamic language.”

111

See 1/11

“Related material:
Jung’s Imago and Solomon’s Cube.”

 

Part II: Christian

The Lottery 4/3/07 Mid-day Evening
New York 115

See 1/15

Inscape

017

See

The image “Primitive roots modulo 17

Pennsylvania 604

See
6/04

Death Valley and the Fisher King

714

See
7/14

Happy Birthday, Esther Dyson

Part III:
Imago Dei

Jung's Four-Diamonds Figure


Click on picture
for details.

 

Related material:

It is perhaps relevant to
this Holy Week that the
date 6/04 (2006) above
refers to both the Christian
holy day of Pentecost and
to the day of the
facetious baccalaureate
of the Class of 2006 in
the University Chapel
at Princeton.

For further context for the
Log24 remarks of that same
date, see June 1-15, 2006.

Sunday, December 10, 2006

Sunday December 10, 2006

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

Time and Chance
on the 90th Birthday
of Kirk Douglas,
star of
The Garden of Allah

The Lottery 12/9/06 Mid-day Evening
New York 036

See

The Quest
for the 36

331

See 3/31

“square crystal” and “the symbolism could not have been more perfect.”

Pennsylvania 602

See 6/02

Walter Benjamin
on
“Adamic language.”

111

See 1/11

“Related material:
Jung’s Imago and Solomon’s Cube.”

See also

The image “http://www.log24.com/log/pix05B/051209-Douglas1.jpg” cannot be displayed, because it contains errors.

Diamonds

The image “http://www.log24.com/theory/images/JungDiamonds.gif” cannot be displayed, because it contains errors.

Friday, November 3, 2006

Friday November 3, 2006

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

First to Illuminate

From the History of a Simple Group” (pdf), by Jeremy Gray:

“The American mathematician A. B. Coble [1908; 1913]* seems to have been the first to illuminate the 27 lines and 28 bitangents with the elementary theory of geometries over finite fields.

The combinatorial aspects of all this are pleasant, but the mathematics is certainly not easy.”

* [Coble 1908] A. Coble, “A configuration in finite geometry isomorphic with that of the 27 lines on a cubic  surface,” Johns Hopkins University Circular 7:80-88 (1908), 736-744.

   [Coble 1913] A. Coble, “An application of finite geometry to the characteristic theory of the odd and even theta functions,” Trans. Amer. Math. Soc. 14 (1913), 241-276.

Related material:

Geometry of the 4x4x4 Cube,

Christmas 2005.

Thursday, October 5, 2006

Thursday October 5, 2006

Filed under: General,Geometry — Tags: , — m759 @ 9:11 am
In Touch with God

(Title of an interview with
the late Paul Halmos, mathematician)

Since Halmos died on Yom Kippur, his thoughts on God may be of interest to some.

From a 1990 interview:

“What’s the best part of being a mathematician? I’m not a religious man, but it’s almost like being in touch with God when you’re thinking about mathematics. God is keeping secrets from us, and it’s fun to try to learn some of the secrets.”

I personally prefer Annie Dillard on God:

“… if Holy the Firm is matter at its dullest, Aristotle’s materia prima, absolute zero, and since Holy the Firm is in touch with the Absolute at base, then the circle is unbroken.  And it is…. Holy the Firm is in short the philosopher’s stone.”

Some other versions of
the philosopher’s stone:

The image �http://www.log24.com/log/pix06/060101-SixOfOne.jpg� cannot be displayed, because it contains errors.

And, more simply,
April 28, 2004:

This last has the virtue of
being connected with Halmos
via his remarks during the
“In Touch with God” interview:

“…at the root of all deep mathematics there is a combinatorial insight… the really original, really deep insights are always combinatorial….”
 
“Combinatorics, the finite case, is where the genuine, deep insight is.”

See also the remark of Halmos that serves as an epigraph to Theme and Variations.

Finally, it should be noted that
the 4×9 black rectangle

The image “http://www.log24.com/log/pix06A/061004-Halmos100x225.jpg” cannot be displayed, because it contains errors.

has also served
at least one interpreter
as a philosopher’s stone,
and is also the original
“Halmos tombstone.”

(See previous entry.)

Wednesday, September 20, 2006

Wednesday September 20, 2006

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

Public Space

"… the Danish cartoons crisis last March showed 'two world views colliding in public space with no common point of reference.'"

George Carey, Archbishop of Canterbury from 1991 to 2002, quoted in today's London Times.

Related material:

Geometry and Christianity
   (Google search yielding
    "about 1,540,000" results)

Geometry and Islam
   (Google search yielding
    "about 1,580,000" results)

MySpace.com/affine

The image “http://www.log24.com/log/pix06A/060828-Cube.jpg” cannot be displayed, because it contains errors.

A Public Space

 

The image “http://www.log24.com/log/pix06A/060920-Motto.jpg” cannot be displayed, because it contains errors.

— Motto of 
Plato's Academy

Background from
Log24 on Feb. 15, 2006:

Hellmut Wilhelm on the Tao
 
If we replace the Chinese word "I" (change, transformation) with the word "permutation," the relevance of Western mathematics (which some might call "the Logos") to the I Ching ("Changes Classic") beomes apparent.

For the relevance of Plato to
Islam, see David Wade's
Pattern in Islamic Art
and a Google search on
Plato and Islam
("about 1,680,000" results).

"We should let ourselves be guided by what is common to all. Yet although the Logos is common to all, most men live as if each had a private intelligence of his own."

Heraclitus of Ephesus, about 500 B.C.

Wednesday, July 5, 2006

Wednesday July 5, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 12:25 pm

And now, from
the author of Sphere

CUBE

He beomes aware of something else… some other presence.
“Anybody here?” he says.
I am here.
He almost jumps, it is so loud. Or it seems loud. Then he wonders if he has heard anything at all.
“Did you speak?”
No.
How are we communicating? he wonders.
The way everything communicates with everything else.
Which way is that?
Why do you ask if you already know the answer?

Sphere, by Michael Crichton, Harvard ’64

“… when I went to Princeton things were completely different. This chapel, for instance– I remember when it was just a clearing, cordoned off with sharp sticks.  Prayer was compulsory back then, and you couldn’t just fake it by moving your lips; you had to know the words, and really mean them.  I’m dating myself, but this was before Jesus Christ.”

Baccalaureate address at Princeton, Pentecost 2006, reprinted in The New Yorker, edited by David Remnick, Princeton ’81

Related figures:

The image “http://www.log24.com/log/pix06A/060617-Spellbound.jpg” cannot be displayed, because it contains errors.

For further details,
see Solomon’s Cube
and myspace.com/affine.

The image “http://www.log24.com/log/pix06A/060705-Cube.jpg” cannot be displayed, because it contains errors.

For further details,
see Jews on Buddhism
and
Adventures in Group Theory.

“In this way we are offered
a formidable lesson
for every Christian community.”

Pope Benedict XVI
on Pentecost,
June 4, 2006,
St. Peter’s Square
.

Saturday, June 17, 2006

Saturday June 17, 2006

Filed under: General,Geometry — Tags: , — m759 @ 7:59 am
In memory of
Barbara Epstein:
 

Spellbound

“Breaking the spell of religion is a
 game that many people can play.”
— Freeman Dyson in the current
   New York Review of Books

Part I:
The Game

The image “http://www.log24.com/log/pix06A/060617-Boggle.jpg” cannot be displayed, because it contains errors.

Part II:
Many People

The image “http://www.log24.com/log/pix06A/060617-Spellbound.jpg” cannot be displayed, because it contains errors.

For further details,
see Solomon’s Cube
and myspace.com/affine.

“The rock cannot be broken.
It is the truth.”
— Wallace Stevens     

Wednesday, May 10, 2006

Wednesday May 10, 2006

Filed under: General,Geometry — Tags: , — m759 @ 4:29 pm
My Space

The image “http://www.log24.com/theory/images/MySpace.jpg” cannot be displayed, because it contains errors.

“… we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections.”
I closed my eyes and thought hard. “Zebbie, I don’t think it can be done. Maybe Escher could have done it.”

— Robert A. Heinlein,
The Number of the Beast

The image “http://www.log24.com/theory/images/Screenshot-64cube.gif” cannot be displayed, because it contains errors.

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the 2-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

“I laugh because I dare not cry.
This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”

— Robert A. Heinlein,
The Number of the Beast

And so…

Compare and contrast:

Solomon’s Cube, the five
Log24 entries ending on 3/14,
and the
American Mathematical Society
on Mathematical Imagery.

Related material:

A more extensive excerpt from
The Number of the Beast, and

Story Theory and
the Number of the Beast.

Thursday, March 9, 2006

Thursday March 9, 2006

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

Finitegeometry.org Update

(Revised May 21, 2006)

Finitegeometry.org now has permutable JavaScript views of the 2x2x2 and 4x4x4 design cubes.  Solomon’s Cube presented a claim that the 4x4x4 design cube retains symmetry under a group of about 1.3 trillion transformations.  The JavaScript version at finitegeometry.org/sc/64/view/ lets the reader visually verify this claim.  The reader should first try the Diamond 16 Puzzle.  The simpler 2x2x2 design cube, with its 1,344 transformations, was described in Diamonds and Whirls; the permutable JavaScript version is at finitegeometry.org/sc/8/view/.

Wednesday, February 15, 2006

Wednesday February 15, 2006

Filed under: General,Geometry — Tags: , — m759 @ 11:07 am
Anthony Hopkins
Writes Screenplay
About God, Life & Death

These topics may be illuminated
by a study of the Chinese classics.

The image “http://www.log24.com/theory/images/GF64-63cycleA495.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/images/IChing/WilhelmHellmut.gif” cannot be displayed, because it contains errors.

If we replace the Chinese word "I"
(change, transformation) with the
word "permutation," the relevance
of Western mathematics (which
some might call "the Logos") to
the I Ching ("Changes Classic")
beomes apparent.

Related material:

Hitler's Still Point,
Jung's Imago,
Solomon's Cube,
Geometry of the I Ching,
and Globe Award.

Yesterday's Valentine
may also have some relevance.

Wednesday, January 11, 2006

Wednesday January 11, 2006

Filed under: General,Geometry — Tags: , — m759 @ 10:30 pm

Time in the Rock

"a world of selves trying to remember the self
before the idea of self is lost–

Walk with me world, upon my right hand walk,
speak to me Babel, that I may strive to assemble
of all these syllables a single word
before the purpose of speech is gone."

— Conrad Aiken, "Prelude" (1932),
    later part of "Time in the Rock,
    or Preludes to Definition, XIX" (1936),
    in Selected Poems, Oxford U. Press
    paperback, 2003, page 156

"The rock is the habitation of the whole,
Its strength and measure, that which is near, point A
In a perspective that begins again

At B: the origin of the mango's rind.
It is the rock where tranquil must adduce
Its tranquil self, the main of things, the mind,

The starting point of the human and the end,
That in which space itself is contained, the gate
To the enclosure, day, the things illumined

By day, night and that which night illumines,
Night and its midnight-minting fragrances,
Night's hymn of the rock, as in a vivid sleep."

— Wallace Stevens in The Rock (1954)

"Poetry is an illumination of a surface,
  the movement of a self in the rock."
— Wallace Stevens, introduction to
    The Necessary Angel, 1951
 

Related material:
Jung's Imago and Solomon's Cube.

 

The following may help illuminate the previous entry:

"I want, as a man of the imagination, to write poetry with all the power of a monster equal in strength to that of the monster about whom I write.  I want man's imagination to be completely adequate in the face of reality."

— Wallace Stevens, 1953 (Letters 790)

The "monster" of the previous entry is of course not Reese Witherspoon, but rather Vox Populi itself.

Wednesday, January 4, 2006

Wednesday January 4, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 4:04 am
Dragon School

In memory of Humphrey Carpenter, author of The Inklings, who attended The Dragon School.  Carpenter died a year ago today.

From Log24 on Nov. 16, 2005:

 

Images

 

Adam Gopnik on C. S. Lewis in the New Yorker:

"Lewis began with a number of haunted images…."

"The best of the books are the ones… where the allegory is at a minimum and the images just flow."

"'Everything began with images,' Lewis wrote…."

The image “http://www.log24.com/log/pix05B/051116-Time.jpg” cannot be displayed, because it contains errors.

 

From Paul Preuss,
Broken Symmetries
(see previous entry):

"Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once.  It was a thin nectar, but he was convinced it was the nectar of the gods…."


From
Verbum Sat Sapienti?

The image “http://www.log24.com/theory/images/EscherVerbum2.jpg” cannot be displayed, because it contains errors.

Escher's Verbum

The image “http://www.log24.com/theory/images/DTinvar246.jpg” cannot be displayed, because it contains errors.

Solomon's Cube


The image “http://www.log24.com/theory/images/HexagramsTable.jpg” cannot be displayed, because it contains errors.

Geometry of the I Ching

 

Wednesday, November 9, 2005

Wednesday November 9, 2005

Filed under: General,Geometry — Tags: , — m759 @ 3:09 pm
In honor of the 120th anniversary
of the birth of Hermann Weyl:

Saturday, November 5, 2005

Saturday November 5, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 4:24 pm

Contrapuntal Themes
in a Shadowland

 
(See previous entry.)

Douglas Hofstadter on his magnum opus:

"… I realized that to me, Gödel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book."

The image “http://www.log24.com/theory/images/GEBcover.jpg” cannot be displayed, because it contains errors.
Hofstadter's cover

Here are three patterns,
"shadows" of a sort,
derived from a different
"central object":

The image “http://www.log24.com/theory/images/GEB.jpg” cannot be displayed, because it contains errors.

For details, see
Solomon's Cube.

Related material:
The reference to a
"permutation fugue"
(pdf) in an article on
Gödel, Escher, Bach.

Wednesday, May 4, 2005

Wednesday May 4, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 1:00 pm
The Fano Plane
Revisualized:

 

 The Eightfold Cube

or, The Eightfold Cube

Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):
 
The image “http://www.log24.com/theory/images/Fano.gif” cannot be displayed, because it contains errors.
 

Every permutation of the plane's points that preserves collinearity is a symmetry of the  plane.  The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group  PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)

The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle.  It does not, however, indicate where the other 162 symmetries come from.  

Shown below is a new model of this same projective plane, using partitions of cubes to represent points:

 

Fano plane with cubes as points
 
The cubes' partitioning planes are added in binary (1+1=0) fashion.  Three partitioned cubes are collinear if and only if their partitioning planes' binary sum equals zero.

 

The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.

 

Fano plane group - generating permutations

For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.

 

(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations.  But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results.  This illustrates the difference between affine and projective spaces over the binary field GF(2).  In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices.  This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices.  Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)

To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.

 

For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.

For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way.  For more on the mathematics of cubic models, see Solomon's Cube.

 

For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.

Sunday, February 20, 2005

Sunday February 20, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 2:20 pm

Relativity Blues

Today, February 20, is the 19th anniversary of my note The Relativity Problem in Finite Geometry.  Here is some related material.

In 1931, the Christian writer Charles Williams grappled with the theology of time, space, free will, and the many-worlds interpretation of quantum mechanics (anticipating by many years the discussion of this topic by physicists beginning in the 1950's).

(Some pure mathematics — untainted by physics or theology — that is nevertheless related, if only by poetic analogy, to Williams's 1931 novel, Many Dimensions, is discussed in the above-mentioned note and in a generalization, Solomon's Cube.)

On the back cover of Williams's 1931 novel, the current publisher, William B. Eerdmans Publishing Company of Grand Rapids, Michigan, makes the following statement:

"Replete with rich religious imagery, Many Dimensions explores the relation between predestination and free will as it depicts different human responses to redemptive transcendence."

One possible response to such statements was recently provided in some detail by a Princeton philosophy professor.  See On Bullshit, by Harry G. Frankfurt, Princeton University Press, 2005.

A more thoughtful response would take into account the following:

1. The arguments presented in favor of philosopher John Calvin, who discussed predestination, in The Death of Adam: Essays on Modern Thought, by Marilynne Robinson

2. The physics underlying Einstein's remarks on free will, God, and dice
 
3. The physics underlying Rebecca Goldstein's novel Properties of Light and Paul Preuss's novels  Secret Passages and Broken Symmetries

4. The physics underlying the recent so-called "free will theorem" of John Conway and Simon Kochen of Princeton University

5. The recent novel Gilead, by Marilynne Robinson, which deals not with philosophy, but with lives influenced by philosophy — indirectly, by the philosophy of the aforementioned John Calvin.

From a review of Gilead by Jane Vandenburgh:  

"In The Death of Adam, Robinson shows Jean Cauvin to be the foremost prophet of humanism whose Protestant teachings against the hierarchies of the Roman church set in motion the intellectual movements that promoted widespread literacy among the middle and lower classes, led to both the American and French revolutions, and not only freed African slaves in the United States but brought about suffrage for women. It's odd then that through our culture's reverse historicism, the term 'Calvinism' has come to mean 'moralistic repression.'"

For more on what the Calvinist publishing firm Eerdmans calls "redemptive transcendence," see various July 2003 Log24.net entries.  If these entries include a fair amount of what Princeton philosophers call bullshit, let the Princeton philosophers meditate on the summary of Harvard philosophy quoted here on November 5 of last year, as well as the remarks of November 5, 2003,  and those of November 5, 2002.

From Many Dimensions (Eerdmans paperback, 1963, page 53):

"Lord Arglay had a suspicion that the Stone would be purely logical.  Yes, he thought, but what, in that sense, were the rules of its pure logic?"

A recent answer:

Modal Theology

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

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