Log24

Tuesday, October 29, 2024

Hallucinated Geometry

Filed under: General — Tags: — m759 @ 5:51 pm

For fans of the "story theory of truth" —

An example of artificial stupidity:

The phrases "midpoints of opposite faces" and "essentially
creating a smaller cube" are hallucinated bullshit.

The above AI description was created by inanely parroting
verbiage from the Wikipedia article "Diamond cubic" —
which it credits as a source. (See wider view of search.)
That article contains neither the word "theorem" nor the
phrase "unit cube " from the search-request prompt.

AI, like humans, is likely to fall victim to the notorious
"story theory of truth" purveyed by Richard J. Trudeau.
A real  "diamond shape formed within a unit cube" is the
octahedron, one of the five classical Platonic solids.

Fans of the opposing "diamond theory of truth" rejected by
Trudeau may prefer . . .

Inside the Exploded Cube

(Log24, July 1, 2019).

Tuesday, October 15, 2024

Recycling for Suburbanites

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

The sneering reference to a fictional boy band in the recent film
"The Idea of You" as "so seventh-grade" suggests a flashback to
the seventh-grade class where I first encountered platonic solids.

The school where the class was given is apparently no longer
a school, but on the bright side . . .

Friday, September 22, 2023

Figurate Space

Filed under: General — Tags: , , — m759 @ 11:01 am

For the purpose of defining figurate geometry , a figurate space  might be
loosely described as any space consisting of finitely many congruent figures  —
subsets of Euclidean space such as points, line segments, squares, 
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them. 

Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate  space, considered as a collection of figures —  vertices, edges,
faces —
seen in the nineteenth century as acted upon by a group  of symmetries .

More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator 
of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M24 . That group may be viewed as acting
on various subsets of a 24-set for instance, the 759 octads  that are analogous
to the faces  of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.

Counting symmetries with the orbit-stabilizer theorem

Monday, October 3, 2022

The Abstract and the Concrete

Filed under: General — Tags: — m759 @ 9:42 am

Counting symmetries with the orbit-stabilizer theorem

The above art by Steven H. Cullinane is not unrelated to
art by Josefine Lyche. Her work includes sculpted replicas
of the above abstract  Platonic solids, as well as replicas of
my own work related to properties of the 4×6 rectangle above.
Symmetries of both the solids and the rectangle may be
viewed as permutations of  parts — In the Platonic solids,
the parts are permuted by continuous  rotations of space itself.
In the rectangle, the parts are permuted by non-continuous 
transformations, as in the I Ching . . . i.e., by concrete  illustrations
of change.

Friday, February 17, 2017

Kostant Is Dead

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

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

MIT News, story dated Feb. 16, 2017

See also a search for Kostant in this journal.

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

Kostant:

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

Cullinane:

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

Analogously

Platonic solids' symmetry groups   

Wednesday, November 26, 2014

A Tetrahedral Fano-Plane Model

Filed under: General,Geometry — Tags: , — m759 @ 5:30 pm

Update of Nov. 30, 2014 —

It turns out that the following construction appears on
pages 16-17 of A Geometrical Picture Book , by 
Burkard Polster (Springer, 1998).

"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"

—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya

For a similar but more difficult problem involving the
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."

The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois Interplay" 
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.

Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.

Wednesday, December 25, 2013

Rotating the Facets

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

Previous post

“… her mind rotated the facts….”

Related material— hypercube rotation,* in the context
of rotational symmetries of the Platonic solids:

IMAGE- Count rotational symmetries by rotating facets. Illustrated with 'Plato's Dice.'

“I’ve heard of affairs that are strictly Platonic”

Song lyric by Leo Robin

* Footnote added on Dec. 26, 2013 —

 See Arnold Emch, “Triple and Multiple Systems, Their Geometric
Configurations and Groups
,” Trans. Amer. Math. Soc.  31 (1929),
No. 1, 25–42.

 On page 42, Emch describes the above method of rotating a
hypercube’s 8 facets (i.e., three-dimensional cubes) to count
rotational symmetries —

See also Diamond Theory in 1937.

Also on p. 42, Emch mentions work of Carmichael on a
Steiner system with the Mathieu group M11 as automorphism
group, and poses the problem of finding such systems and
groups that are larger. This may have inspired the 1931
discovery by Carmichael of the Steiner system S(5, 8, 24),
which has as automorphisms the Mathieu group M24 .

Friday, November 18, 2011

Hypercube Rotations

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

The hypercube has 192 rotational symmetries.
Its full symmetry group, including reflections,
is of order 384.

See (for instance) Coxeter

http://www.log24.com/log/pix11C/111118-Coxeter415.jpg

Related material—

The rotational symmetry groups of the Platonic solids
(from April 25, 2011)—

Platonic solids' symmetry groups

— and the figure in yesterday evening's post on the hypercube

http://www.log24.com/log/pix11C/11117-HypercubeFromMIQELdotcom.gif

(Animation source: MIQEL.com)

Clearly hypercube rotations of this sort carry any
of the eight 3D subcubes to the central subcube
of a central projection of the hypercube—

http://www.log24.com/log/pix11C/111118-CentralProjection.gif

The 24 rotational symmeties of that subcube induce
24 rigid rotations of the entire hypercube. Hence,
as in the logic of the Platonic symmetry groups
illustrated above, the hypercube has 8 × 24 = 192
rotational symmetries.

Monday, April 25, 2011

Poetry and Physics

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

One approach to the storied philosophers' stone, that of Jim Dodge in Stone Junction , was sketched in yesterday's Easter post. Dodge described a mystical "spherical diamond." The symmetries of the sphere form what is called in mathematics a Lie group . The "spherical" of Dodge therefore suggests a review of the Lie group Ein Garrett Lisi's poetic theory of everything.

A check of the Wikipedia article on Lisi's theory yields…

http://www.log24.com/log/pix11A/110425-WikipediaE8.jpg

       Diamond and E8 at Wikipedia

Related material — Eas "a diamond with thousands of facets"—

http://www.log24.com/log/pix11A/110425-Kostant.jpg

Also from the New Yorker  article

“There’s a dream that underlying the physical universe is some beautiful mathematical structure, and that the job of physics is to discover that,” Smolin told me later. “The dream is in bad shape,” he added. “And it’s a dream that most of us are like recovering alcoholics from.” Lisi’s talk, he said, “was like being offered a drink.”

A simpler theory of everything was offered by Plato. See, in the Timaeus , the Platonic solids—

Platonic solids' symmetry groups

Figure from this journal on August 19th, 2008.
See also July 19th, 2008.

It’s all in Plato, all in Plato:
bless me, what do  they
teach them at these schools!”
— C. S. Lewis

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.

Tuesday, August 19, 2008

Tuesday August 19, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 8:30 am
Three Times

"Credences of Summer," VII,

by Wallace Stevens, from
Transport to Summer (1947)

"Three times the concentred
     self takes hold, three times
The thrice concentred self,
     having possessed
The object, grips it
     in savage scrutiny,
Once to make captive,
     once to subjugate
Or yield to subjugation,
     once to proclaim
The meaning of the capture,
     this hard prize,
Fully made, fully apparent,
     fully found."

Stevens does not say what object he is discussing.

One possibility —

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:

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

Another possibility —
 

The 4x4 square

  A more modest object —
the 4×4 square.

Update of Aug. 20-21 —

Symmetries and Facets

Kostant's poetic comparison might be applied also to this object.

The natural rearrangements (symmetries) of the 4×4 array might also be described poetically as "thousands of facets, each facet offering a different view of… internal structure."

More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.

For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.

* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–

Platonic solids' symmetry groups

The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the number of natural rearrangements, or symmetries, of the 4×4 array. Actually, the true analogy is between the 16 unit squares of the 4×4 array, regarded as the 16 points of a finite 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (which has infinitely many symmetries).

If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that

"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."

The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
 

Thursday, March 6, 2008

Thursday March 6, 2008

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

This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA's Icosahedron Society.

Royal Road

"The historical road
from the Platonic solids
to the finite simple groups
is well known."

— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture

Euclid is said to have remarked that "there is no royal road to geometry." The road to the end of the four-color conjecture may, however, be viewed as a royal road from geometry to the wasteland of mathematical recreations.* (See, for instance, Ch. VIII, "Map-Colouring Problems," in Mathematical Recreations and Essays, by W. W. Rouse Ball and H. S. M. Coxeter.) That road ended in 1976 at the AMS-MAA summer meeting in Toronto– home of H. S. M. Coxeter, a.k.a. "the king of geometry."

See also Log24, May 21, 2007.

A different road– from Plato to the finite simple groups– is, as I noted in November 2000, well known. But new roadside attractions continue to appear. One such attraction is the role played by a Platonic solid– the icosahedron– in design theory, coding theory, and the construction of the sporadic simple group M24.

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics, Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

This Steiner system is closely connected to M24 and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of  M24):

"Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix (I  J-N) generate the extended binary Golay code." [Here I is the identity matrix and J is the matrix of all 1's.]

Op. cit., p. 719

Related material:

Finite Geometry of
the Square and Cube

and
Jewel in the Crown

"There is a pleasantly discursive
treatment of Pontius Pilate's
unanswered question
'What is truth?'"
— H. S. M. Coxeter, 1987,
introduction to Trudeau's
"story theory" of truth

Those who prefer stories to truth
may consult the Log24 entries
 of March 1, 2, 3, 4, and 5.

They may also consult
the poet Rubén Darío:

Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.


* For a road out of this wasteland, back to geometry, see The Kaleidoscope Puzzle and Reflection Groups in Finite Geometry.

Wednesday, March 5, 2008

Wednesday March 5, 2008

Filed under: General,Geometry — Tags: — m759 @ 1:09 pm
(Context: March 2-4)

For CENTRAL
Central Intelligence:

"God does not play dice."
— Paraphrase of a remark
by Albert Einstein

Another Nobel Prize winner,
Isaac Bashevis Singer

"a God who speaks in deeds,
not in words, and whose
vocabulary is the Cosmos"

From "The Escapist:
The Reality of Fantasy Games
"–

Platonic solids as Dungeons & Dragons dice
Dungeons & Dragons Dice

 

From today's New York Times:

NY Times obituaries online, March 5, 2008: Gary Gygax, Wm. F. Buckley, Kaddish ad by Hadassah

A Kaddish for Gygax:

 

 

"I was reading Durant's section on Plato, struggling to understand his theory of the ideal Forms that lay in inviolable perfection out beyond the phantasmagoria. (That was the first, and I think the last, time that I encountered that word.)"


Related material:

For more on the word
"phantasmagoria," see
Log24 on Dec. 12, 2004
and on Sept. 23, 2006.

For phantasmagoria in action,
see Dungeons & Dragons
and Singer's (and others')
Jewish fiction.

For non-phantasmagoria,
see (for instance) the Elements
of Euclid, which culminates
in the construction of the
Platonic solids illustrated above.

See also Geometry for Jews.

Wednesday, May 23, 2007

Wednesday May 23, 2007

Filed under: General,Geometry — m759 @ 7:00 am
 
Strong Emergence Illustrated:
 
The Beauty Test
 
"There is no royal road
to geometry"

— Attributed to Euclid

There are, however, various non-royal roads.  One of these is indicated by yesterday's Pennsylvania lottery numbers:

PA Lottery May 22, 2007: Mid-day 515, Evening 062

The mid-day number 515 may be taken as a reference to 5/15. (See the previous entry, "Angel in the Details," and 5/15.)

The evening number 062, in the context of Monday's entry "No Royal Roads" and yesterday's "Jewel in the Crown," may be regarded as naming a non-royal road to geometry: either U. S. 62, a major route from Mexico to Canada (home of the late geometer H.S.M. Coxeter), or a road less traveled– namely, page 62 in Coxeter's classic Introduction to Geometry (2nd ed.):

The image “http://www.log24.com/log/pix07/070523-Coxeter62.jpg” cannot be displayed, because it contains errors.

The illustration (and definition) is
of regular tessellations of the plane.

This topic Coxeter offers as an
illustration of remarks by G. H. Hardy
that he quotes on the preceding page:

The image “http://www.log24.com/log/pix07/070523-Hardy.jpg” cannot be displayed, because it contains errors.

One might argue that such beauty is strongly emergent because of the "harmonious way" the parts fit together: the regularity (or fitting together) of the whole is not reducible to the regularity of the parts.  (Regular triangles, squares, and hexagons fit together, but regular pentagons do not.)

The symmetries of these regular tessellations of the plane are less well suited as illustrations of emergence, since they are tied rather closely to symmetries of the component parts.

But the symmetries of regular tessellations of the sphere— i.e., of the five Platonic solids– do emerge strongly, being apparently independent of symmetries of the component parts.

Another example of strong emergence: a group of 322,560 transformations acting naturally on the 4×4 square grid— a much larger group than the group of 8 symmetries of each component (square) part.

The lottery numbers above also supply an example of strong emergence– one that nicely illustrates how it can be, in the words of Mark Bedau, "uncomfortably like magic."

(Those more comfortable with magic may note the resemblance of the central part of Coxeter's illustration to a magical counterpart– the Ojo de Dios of Mexico's Sierra Madre.)

Tuesday, May 22, 2007

Tuesday May 22, 2007

Filed under: General,Geometry — m759 @ 7:11 am
 
Jewel in the Crown

A fanciful Crown of Geometry

The Crown of Geometry
(according to Logothetti
in a 1980 article)

The crown jewels are the
Platonic solids, with the
icosahedron at the top.

Related material:

"[The applet] Syntheme illustrates ways of partitioning the 12 vertices of an icosahedron into 3 sets of 4, so that each set forms the corners of a rectangle in the Golden Ratio. Each such rectangle is known as a duad. The short sides of a duad are opposite edges of the icosahedron, and there are 30 edges, so there are 15 duads.

Each partition of the vertices into duads is known as a syntheme. There are 15 synthemes; 5 consist of duads that are mutually perpendicular, while the other 10 consist of duads that share a common line of intersection."

— Greg Egan, Syntheme

Duads and synthemes
(discovered by Sylvester)
also appear in this note
from May 26, 1986
(click to enlarge):

 

Duads and Synthemes in finite geometry

The above note shows
duads and synthemes related
to the diamond theorem.

See also John Baez's essay
"Some Thoughts on the Number 6."
That essay was written 15 years
ago today– which happens
to be the birthday of
Sir Laurence Olivier, who,
were he alive today, would
be 100 years old.

Olivier as Dr. Christian Szell

The icosahedron (a source of duads and synthemes)

"Is it safe?"

Tuesday, September 17, 2024

Geometry for Jews:
“Kickin’ Down the Cobblestones…” — Song lyric

Filed under: General — Tags: — m759 @ 2:58 am

Kirkenes at newyorker.com

Some less stressful material . . .

"a medley cobbled together" —

Sunday, June 25, 2023

High Concept: The Dreaming Gemstones

Filed under: General — Tags: , , — m759 @ 12:07 am

Sturgeon versus Plato —

Sturgeon's Dreaming Jewels meet Plato's Righteous Gemstones.

Wednesday, November 26, 2014

Class Act

Filed under: General,Geometry — Tags: , — m759 @ 7:18 am

Update of Nov. 30, 2014 —

For further information on the geometry in
the remarks by Eberhart below, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.

A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:

The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and corner points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of corners, totalling 13 axes (the octahedron simply interchanges the roles of faces and corners); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of corners, totalling 31 axes (the icosahedron again interchanging roles of faces and corners). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former.

[9] I am aware only of a series of in-house publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie  I-X.

— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge, 
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science 
, 1998,
archive.bridgesmathart.org/1998/bridges1998-121.pdf

Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…


… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled.  So 1984 to 2002 I taught math (esp. non-Euclidean geometry) at C.S.U. Northridge.  It’s been a rich life.  I’m grateful. 
 
Steve
 

See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.

Saturday, November 16, 2013

Mathematics and Rhetoric

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

Jim Holt in the current (Dec. 5) New York Review of Books

One form of Eros is the sexual desire aroused by the physical beauty of a particular beloved person. That, according to Diotima, is the lowest form. With philosophical refinement, however, Eros can be made to ascend toward loftier and loftier objects. The penultimate of these—just short of the Platonic idea of beauty itself—is the perfect and timeless beauty discovered by the mathematical sciences. Such beauty evokes in those able to grasp it a desire to reproduce—not biologically, but intellectually, by begetting additional “gloriously beautiful ideas and theories.” For Diotima, and presumably for Plato as well, the fitting response to mathematical beauty is the form of Eros we call love.

Consider (for example) the beauty of the rolling donut

http://www.log24.com/log/pix11C/11117-HypercubeFromMIQELdotcom.gif
            (Animation source: MIQEL.com)

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — Tags: , , — m759 @ 4:30 am

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Thursday, August 11, 2005

Thursday August 11, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 8:16 am

Kaleidoscope, continued

From Clifford Geertz, The Cerebral Savage:

"Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately).  And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought.  Both anecdotal and geometric, it builds coherent structures out of 'the odds and ends left over from psychological or historical process.'

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore….  as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular.   But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort….  Levi-Strauss generalizes this permutational view of thinking to savage thought in general.  It is all a matter of shuffling discrete (and concrete) images–totem animals, sacred colors, wind directions, sun deities, or whatever–so as to produce symbolic structures capable of formulating and communicating objective (which is not to say accurate) analyses of the social and physical worlds.

…. And the point is general.  The relationship between a symbolic structure and its referent, the basis of its meaning,  is fundamentally 'logical,' a coincidence of form– not affective, not historical, not functional.  Savage thought is frozen reason and anthropology is, like music and mathematics, 'one of the few true vocations.'

Or like linguistics."

Edward Sapir on Linguistics, Mathematics, and Music:

"… linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms.  Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music.  But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."

— Edward Sapir, "The Grammarian and his Language,"
  American Mercury 1:149-155,1924

From Robert de Marrais, Canonical Collage-oscopes:

"…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups.  You know, those things the non-professionals call . . . kaleidoscopes! *  (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism' **— then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)

* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry  (A polytope is an n-dimensional analog of a polygon or polyhedron.  Chapter V of this book is entitled 'The Kaleidoscope'….)

** … contemporary with the Johns Hopkins hatchet job that won him American marketshare, Derrida was also being subjected to a series of probing interviews in Paris by the hometown crowd.  He first gained academic notoriety in France for his book-length reading of Husserl's two-dozen-page essay on 'The Origin of Geometry.'  The interviews were collected under the rubric of Positions (Chicago: U. of Chicago Press, 1981…).  On pp. 34-5 he says the following: 'the resistance to logico-mathematical notation has always been the signature of logocentrism and phonologism in the event to which they have dominated metaphysics and the classical semiological and linguistic projects…. A grammatology that would break with this system of presuppositions, then, must in effect liberate the mathematization of language…. The effective progress of mathematical notation thus goes along with the deconstruction of metaphysics, with the profound renewal of mathematics itself, and the concept of science for which mathematics has always been the model.'  Nice campaign speech, Jacques; but as we'll see, you reneged on your promise not just with the kaleidoscope (and we'll investigate, in depth, the many layers of contradiction and cluelessness you put on display in that disingenuous 'playing to the house'); no, we'll see how, at numerous other critical junctures, you instinctively took the wrong fork in the road whenever mathematical issues arose… henceforth, monsieur, as Joe Louis once said, 'You can run, but you just can't hide.'…."

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