Log24

Saturday, October 7, 2017

Byte Space

Filed under: Uncategorized — m759 @ 7:59 PM

The Eightfold Cube

"Before time began,
there was the Cube."

Optimus Prime

Tuesday, April 4, 2017

White Cube

Filed under: Uncategorized — Tags: , — m759 @ 12:21 PM

"Inside the White Cube" —

"We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art."

http://www.log24.com/log/pix09/090205-cube2x2x2.gif

"Space: what you
damn well have to see."

— James Joyce, Ulysses  

Saturday, January 7, 2017

Conceptualist Minimalism

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

"Clearly, there is a spirit of openhandedness in post-conceptual art
uses of the term 'Conceptualism.' We can now endow it with a
capital letter because it has grown in scale from its initial designation
of an avant-garde grouping, or various groups in various places, and
has evolved in two further phases. It became something like a movement,
on par with and evolving at the same time as Minimalism. Thus the sense
it has in a book such as Tony Godfrey’s Conceptual Art.  Beyond that,
it has in recent years spread to become a tendency, a resonance within
art practice that is nearly ubiquitous." — Terry Smith, 2011

See also the eightfold cube

The Eightfold Cube

 

Sunday, October 23, 2016

Quartet

Filed under: Uncategorized — m759 @ 12:48 PM

“The man who lives in contact with what he believes to be a living Church
is a man always expecting to meet Plato and Shakespeare to-morrow
at breakfast.”

— G. K. Chesterton

Or Sunday dinner.

The Eightfold Cube

Platonic
solid

Jack in the Box, Natasha Wescoat, 2004
Natasha Wescoat, 2004

Shakespearean
Fool

Not to mention Euclid and Picasso.

 

The image “http://www.log24.com/theory/images/Pythagoras-I47.gif” cannot be displayed, because it contains errors.


The image “http://www.log24.com/log/pix06A/RobertFooteAnimation.gif” cannot be displayed, because it contains errors.

In the above pictures, Euclid is represented by 
Alexander Bogomolny, Picasso by Robert Foote.

Monday, August 8, 2016

A Point of Identity

Filed under: Uncategorized — m759 @ 6:00 PM

For a  Monkey Grammarian  (Viennese Version)

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

A logo that may be interpreted as one-eighth of a 2x2x2 array
of cubes —

The figure in white above may be viewed as a subcube representing,
when the eight-cube array is coordinatized, the identity (i.e., (0, 0, 0)).

Shown below are a few variations on the figure by VCQ,
the Vienna Center for Quantum Science and Technology —
 

(Click image to enlarge.)

Tuesday, June 14, 2016

Model Kit

Filed under: Uncategorized — m759 @ 12:14 PM

The title refers to the previous post, which quotes a 
remark by a poetry critic in the current New Yorker .

Scholia —

From the post Structure and Sense of June 6, 2016 —

Structure

Sense

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

From the post Design Cube of July 23, 2015 —

Broken Symmetries  in  Diamond Space 

Monday, June 6, 2016

Structure and Sense

Filed under: Uncategorized — Tags: — m759 @ 2:01 PM

"… the war of 70-some years ago
has already become something like the Trojan War
had been for the Homeric bards:
a major event in the mythic past
that gives structure and sense to our present reality."

— Justin E. H. Smith, a professor of philosophy at
     the University of Paris 7–Denis Diderot,
     in the New York Times  column "The Stone"
     (print edition published Sunday, June 5, 2016)

In memory of a British playwright who reportedly
died at 90 this morning —

Structure

Sense

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

Sunday, June 5, 2016

Sunday School: Seven Seals

Filed under: Uncategorized — m759 @ 7:00 AM

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

Click image for some background.

See also Standard Disclaimer.

Tuesday, April 19, 2016

The Folding

Filed under: Uncategorized — m759 @ 2:00 PM

(Continued

A recent post about the eightfold cube  suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :

As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array from Turyn's
1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields the 2x2x2 eightfold cube .

Those who prefer an entertainment  approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space." 

Sunday, April 17, 2016

The Thing and I

Filed under: Uncategorized — m759 @ 2:01 PM

The New York Times  philosophy column yesterday —

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

The eightfold cube —

Definition of Epiphany

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

Three Times:

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

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

— What?

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

— Yes? said Cranly absently.

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

— Yes …

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

— And then?

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

— Let us turn back, said Cranly.

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

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

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

— It has not epiphanised yet, he said.

Monday, April 4, 2016

Cube for Berlin

Filed under: Uncategorized — m759 @ 11:00 AM

Foreword by Sir Michael Atiyah —

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

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

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

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

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

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

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

The Eightfold Cube

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

See also the recent post Cube Bricks 1984

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

Related remark from the literature —

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

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

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

Saturday, June 27, 2015

A Single Finite Structure

Filed under: Uncategorized — m759 @ 12:15 PM

"It is as if one were to condense
all trends of present day mathematics
onto a single finite structure…."

— Gian-Carlo Rota, foreword to
A Source Book in Matroid Theory ,
Joseph P.S. Kung, Birkhäuser, 1986

"There is  such a thing as a matroid."

— Saying adapted from a novel by Madeleine L'Engle

Related remarks from Mathematics Magazine  in 2009 —

See also the eightfold cube —

The Eightfold Cube

 .

Thursday, February 26, 2015

A Simple Group

Filed under: Uncategorized — m759 @ 7:59 PM
The Eightfold Cube

The previous post's
illustration was 
rather complicated.

This is a simpler
algebraic figure.

Thursday, June 5, 2014

Twisty Quaternion Symmetry

Filed under: Uncategorized — m759 @ 9:11 PM

The previous post told how user58512 at math.stackexchange.com
sought in 2013 a geometric representation of Q, the quaternion group.
He ended up displaying an illustration that very possibly was drawn,
without any acknowledgement of its source, from my own work.

On the date that user58512 published that illustration, he further
pursued his March 1, 2013, goal of a “twisty” quaternion model.

On March 12, 2013,  he suggested that the quaternion group might be
the symmetry group of the following twisty-cube coloring:

IMAGE- Twisty-cube coloring illustrated by Jim Belk

Illustration by Jim Belk

Here is part of a reply by Jim Belk from Nov. 11, 2013, elaborating on
that suggestion:

IMAGE- Jim Belk's proposed GAP construction of a 2x2x2 twisty-cube model of the quaternion group 

Belk argues that the colored cube is preserved under the group
of actions he describes. It is, however, also preserved under a
larger group.  (Consider, say, rotation of the entire cube by 180
degrees about the center of any one of its checkered faces.)  The
group Belk describes seems therefore to be a  symmetry group,
not the  symmetry group, of the colored cube.

I do not know if any combination puzzle has a coloring with
precisely  the quaternion group as its symmetry group.

(Updated at 12:15 AM June 6 to point out the larger symmetry group
and delete a comment about an arXiv paper on quaternion group models.)

Saturday, May 11, 2013

Core

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

Promotional description of a new book:

"Like Gödel, Escher, Bach  before it, Surfaces and Essences  will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking."

"Like Gödel, Escher, Bach  before it…."

Or like Metamagical Themas

Rubik core:

Swarthmore Cube Project, 2008

Non- Rubik cores:

Of the odd  nxnxn cube:

Of the even  nxnxn cube:

The image “http://www.log24.com/theory/images/cube2x2x2.gif” cannot be displayed, because it contains errors.

Related material: The Eightfold Cube and

"A core component in the construction
is a 3-dimensional vector space  over F."

—  Page 29 of "A twist in the M24 moonshine story," 
      by Anne Taormina and Katrin Wendland.
      (Submitted to the arXiv on 13 Mar 2013.)

Tuesday, February 19, 2013

Configurations

Filed under: Uncategorized — Tags: — m759 @ 12:24 PM

Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in  a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular  to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.

My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010

For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's 
symmetry planes , contradicting the usual use of of that term.

That argument concerns the interplay  between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois  structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)

Related material:  Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.

* Earlier guides: the diamond theorem (1978), similar theorems for
  2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
  (1985). See also Spaces as Hypercubes (2012).

Wednesday, November 14, 2012

Group Actions

Filed under: Uncategorized — Tags: — m759 @ 4:30 PM

The December 2012 Notices of the American
Mathematical Society  
has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on "Low-dimensional
Topology, Geometry, and Dynamics"—

(Only the top part of the ad is shown; for further details
see an ICERM page.)

(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)

The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—

The logo, apparently a stylized image of the architecture 
of the Providence building housing ICERM, is not unlike
a picture of Froebel's Third Gift—

 

Froebel's third gift, the eightfold cube

© 2005 The Institute for Figuring

Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)

The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.

These cubes are not without relevance to the workshops' topics—
low-dimensional exotic geometric structures, group theory, and dynamics.

See The Eightfold Cube, A Simple Reflection Group of Order 168, and 
The Quaternion Group Acting on an Eightfold Cube.

Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—

.

Tuesday, January 10, 2012

Defining Form

Filed under: Uncategorized — Tags: , — m759 @ 9:00 AM

(Continued from Epiphany and from yesterday.)

Detail from the current American Mathematical Society homepage

http://www.log24.com/log/pix12/120110-AMS_page-Detail.jpg

Further detail, with a comparison to Dürer's magic square—

http://www.log24.com/log/pix12/120110-Donmoyer-Still-Life-Detail.jpg http://www.log24.com/log/pix12/120110-DurerSquare.jpg

The three interpenetrating planes in the foreground of Donmoyer's picture
provide a clue to the structure of the the magic square array behind them.

Group the 16 elements of Donmoyer's array into four 4-sets corresponding to the
four rows of Dürer's square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.

Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—

http://www.log24.com/log/pix12/120110-DiamondPuzzleFigure.jpg

Thus the Donmoyer array also enjoys the structural  symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.

Just as the decomposition theorem's interpenetrating lines  explain the structure
of a 4×4 square , the foreground's interpenetrating planes  explain the structure
of a 2x2x2 cube .

For an application to theology, recall that interpenetration  is a technical term
in that field, and see the following post from last year—

Saturday, June 25, 2011

 

Theology for Antichristmas

— m759 @ 12:00 PM

Hypostasis (philosophy)

"… the formula 'Three Hypostases  in one Ousia '
came to be everywhere accepted as an epitome
of the orthodox doctrine of the Holy Trinity.
This consensus, however, was not achieved
without some confusion…." —Wikipedia

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

Ousia

Click for further details:

http://www.log24.com/log/pix11A/110625-ProjectiveTrinitySm.jpg

 

Sunday, August 28, 2011

The Cosmic Part

Filed under: Uncategorized — Tags: — m759 @ 6:29 PM

Yesterday's midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik's mechanical contrivance as a rather absurd "Cosmic Cube."

A simpler candidate for the "Cube" part of that phrase:

http://www.log24.com/log/pix10/100214-Cube2x2x2.gif

The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

"Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions."

Alexandre V. Borovik in "Coxeter Theory: The Cognitive Aspects"

Borovik has a such a diagram—

http://www.log24.com/log/pix11B/110828-BorovikM.jpg

The planes in Borovik's figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik's remarks, the eightfold cube might serve to illustrate the "Cosmic" part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine's Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

Friday, December 17, 2010

Fare Thee Well

Filed under: Uncategorized — m759 @ 4:01 AM

Excerpt from a post of 8 AM May 26, 2006

A Living Church
continued from March 27, 2006

"The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast."

– G. K. Chesterton

The Eightfold Cube

Platonic Solid

The image “http://www.log24.com/log/pix06A/060526-JackInTheBox.jpg” cannot be displayed, because it contains errors.

Shakespearean Fool
© 2004 Natasha Wescoat

A related scene from the opening of Blake Edwards's "S.O.B." —

http://www.log24.com/log/pix10B/101217-SOBintro.jpg

Click for Julie Andrews in the full video.

Monday, June 21, 2010

Cube Spaces

Filed under: Uncategorized — Tags: — m759 @ 11:30 AM

Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.

Example 1— The 2×2×2 Cube—

also known as the eightfold  cube

2x2x2 cube

Group actions on the eightfold cube, 1984—

http://www.log24.com/log/pix10A/100621-diandwh-detail.GIF

Version by Laszlo Lovasz et al., 2003—

http://www.log24.com/log/pix10A/100621-LovaszCubeSpace.gif

Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
 

Example 2— The 3×3×3 Cube

A note from 1985 describing group actions on a 3×3 plane array—

http://www.log24.com/log/pix10A/100621-VisualizingDetail.gif

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
  3×3 group actions from 1985 described above—

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 
note by Cullinane

Pegg gives no reference to the 1985 work on group actions.
 

Example 3— The 4×4×4 Cube

A note from 27 years ago today—

http://www.log24.com/log/pix10A/100621-Cube830621.gif

As far as I know, this version of the
group-actions theorem has not yet been ripped off.

Tuesday, February 16, 2010

Mysteries of Faith

Filed under: Uncategorized — m759 @ 9:00 AM

From today's NY Times

http://www.log24.com/log/pix10/100216-NYTobits.jpg

Obituaries for mystery authors
Ralph McInerny and Dick Francis

From the date (Jan. 29) of McInerny's death–

"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.'"

Harvard University Press on Persons and Things (Walpurgisnacht, 2008), by Barbara Johnson

From the date (Feb. 14) of Francis's death–

2x2x2 cube

The EIghtfold Cube

The "something missing" in the above figure is an eighth cube, hidden behind the others pictured.

This eighth cube is not, as Johnson would have it, a void and "vanishing point," but is instead the "still point" of T.S. Eliot. (See the epigraph to the chapter on automorphism groups in Parallelisms of Complete Designs, by Peter J. Cameron. See also related material in this journal.) The automorphism group here is of course the order-168 simple group of Felix Christian Klein.

For a connection to horses, see
a March 31, 2004, post
commemorating the birth of Descartes
  and the death of Coxeter–

Putting Descartes Before Dehors

     Binary coordinates for a 4x2 array  Chess knight formed by a Singer 7-cycle

For a more Protestant meditation,
see The Cross of Descartes

Descartes

Descartes's Cross

"I've been the front end of a horse
and the rear end. The front end is better."
— Old vaudeville joke

For further details, click on
the image below–

Quine and Derrida at Notre Dame Philosophical Reviews

Notre Dame Philosophical Reviews

Sunday, February 14, 2010

Sunday School

Filed under: Uncategorized — m759 @ 9:00 AM

"Simplify, simplify." — Henry David Thoreau

"Because of their truly fundamental role in mathematics, even the simplest diagrams concerning finite reflection groups (or finite mirror systems, or root systems– the languages are equivalent) have interpretations of cosmological proportions."

Alexandre Borovik, 2010 (See previous entry.)

Exercise: Discuss Borovik's remark
that "the languages are equivalent"
in light of the web page

http://www.log24.com/log/pix10/100214-Cube2x2x2.gif

A Simple Reflection Group
of Order 168
.

Background:

Theorems 15.1 and 15.2 of Borovik's book (1st ed. Nov. 10, 2009)
Mirrors and Reflections: The Geometry of Finite Reflection Groups

15.1 (p. 114): Every finite reflection group is a Coxeter group.

15.2 (p. 114): Every finite Coxeter group is isomorphic to a finite reflection group.

Consider in this context the above simple reflection group of order 168.

(Recall that "…there is only one simple Coxeter group (up to isomorphism); it has order 2…" —A.M. Cohen.)

Friday, April 10, 2009

Friday April 10, 2009

Filed under: Uncategorized — m759 @ 8:00 AM

Pilate Goes
to Kindergarten

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

— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
 remarks on the “Story Theory
 of truth as opposed to the
Diamond Theory” of truth in
 The Non-Euclidean Revolution

Consider the following question in a paper cited by V. S. Varadarajan:

E. G. Beltrametti, “Can a finite geometry describe physical space-time?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.

Simplifying:

“Can a finite geometry describe physical space?”

Simplifying further:

“Yes. VideThe Eightfold Cube.'”

Froebel's 'Third Gift' to kindergarteners: the 2x2x2 cube

Saturday, February 7, 2009

Saturday February 7, 2009

Filed under: Uncategorized — m759 @ 2:02 PM
Childish Things

(continued from Thursday's
"Through the Looking Glass")

DENNIS OVERBYE

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

A letter the physicist wrote in 1954 to the philosopher Eric Gutkind, in which he described the Bible as 'pretty childish' and scoffed at the notion that the Jews could be a 'chosen people,' sold for $404,000 at an auction in London. That was 25 times the presale estimate."

Einstein did not, at least in the place alleged, call the Bible "childish." Proof:

(Click for larger version.)
 
Proof that Einstein did not call the Bible 'childish'

The image of the letter is
from the Sept./Oct. 2008
Search Magazine
.

By the way, today is
the birthday of G. H. Hardy.

Here is an excerpt from his
thoughts on childish things:

"What 'purely aesthetic' qualities can we distinguish in such theorems as Euclid's or Pythagoras's?…. In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions."

Eightfold (2x2x2) cube

"Space: what you
damn well have to see."

— James Joyce, Ulysses  

Thursday, February 5, 2009

Thursday February 5, 2009

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM

Through the
Looking Glass:

A Sort of Eternity

From the new president's inaugural address:

"… in the words of Scripture, the time has come to set aside childish things."

The words of Scripture:

9 For we know in part, and we prophesy in part.
10 But when that which is perfect is come, then that which is in part shall be done away.
11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
12 For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.

 

First Corinthians 13

"through a glass"

[di’ esoptrou].
By means of
a mirror [esoptron]
.

Childish things:

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

 

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

Not-so-childish:

 

Three planes through
the center of a cube
that split it into
eight subcubes:
Cube subdivided into 8 subcubes by planes through the center
Through a glass, darkly:

A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
 that is symmetric under one of
these 8 transformations–
a 180-degree rotation:

Design Cube 2x2x2 for demonstrating Galois geometry

(Click on image
for further details.)

But then face to face:

A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.

Galois age fifteen, drawn by a classmate.

Galois age fifteen,
drawn by a classmate.

These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.

For some generalizations,
see Galois Geometry.

Related material:

The central aim of Western religion–

 

 

"Each of us has something to offer the Creator...
the bridging of
 masculine and feminine,
 life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)

The central aim of Western philosophy–

 Dualities of Pythagoras
 as reconstructed by Aristotle:
  Limited Unlimited
  Odd Even
  Male Female
  Light Dark
  Straight Curved
  ... and so on ....

"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."

— Jamie James in The Music of the Spheres (1993)

"In the garden of Adding
live Even and Odd…
And the song of love's recision
is the music of the spheres."

— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)

A quotation today at art critic Carol Kino's website, slightly expanded:

"Art inherited from the old religion
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history."

— Octavio Paz,"Seeing and Using: Art and Craftsmanship," in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52 

From Brian O'Doherty's 1976 Artforum essays– not on museums, but rather on gallery space:

"Inside the White Cube"

"We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art."

http://www.log24.com/log/pix09/090205-cube2x2x2.gif

"Space: what you
damn well have to see."

— James Joyce, Ulysses  

Wednesday, January 14, 2009

Wednesday January 14, 2009

Filed under: Uncategorized — m759 @ 2:45 AM
Eight is a Gate

'The Eight,' by Katherine Neville

Customer reviews of Neville's 'The Eight'

From the most highly
rated negative review:

“I never did figure out
what ‘The Eight’ was.”

Various approaches
to this concept
(click images for details):

The Fritz Leiber 'Spider' symbol in a square

A Singer 7-cycle in the Galois field with eight elements

The Eightfold (2x2x2) Cube

The Jewel in Venn's Lotus (photo by Gerry Gantt)

Tom O'Horgan in his loft. O'Horgan died Sunday, Jan. 11, 2009.

Bach, Canon 14, BWV 1087

Friday, December 19, 2008

Friday December 19, 2008

Filed under: Uncategorized — Tags: , , — m759 @ 1:06 PM
Inside the
White Cube

Part I: The White Cube

The Eightfold Cube

Part II: Inside
 
The Paradise of Childhood'-- Froebel's Third Gift

Part III: Outside

Mark Tansey, 'The Key' (1984)

Click to enlarge.

Mark Tansey, The Key (1984)

For remarks on religion
related to the above, see
Log24 on the Garden of Eden
and also Mark C. Taylor,
"What Derrida Really Meant"
(New York Times, Oct. 14, 2004).

For some background on Taylor,
see Wikipedia. Taylor, Chairman
of the Department of Religion
at
Columbia University, has a
1973 doctorate in religion from
Harvard University. His opinion
of Derrida indicates that his
sympathies lie more with
the serpent than with the angel
in the Tansey picture above.

For some remarks by Taylor on
the art of Tansey relevant to the
structure of the white cube
(Part I above), see Taylor's
The Picture in Question:
Mark Tansey and the
Ends of Representation

(U. of Chicago Press, 1999):

From Chapter 3,
"Sutures* of Structures," p. 58:

"What, then, is a frame, and what is frame work?

This question is deceptive in its simplicity. A frame is, of course, 'a basic skeletal structure designed to give shape or support' (American Heritage Dictionary)…. when the frame is in question, it is difficult to determine what is inside and what is outside. Rather than being on one side or the other, the frame is neither inside nor outside. Where, then, Derrida queries, 'does the frame take place….'"

* P. 61:
"… the frame forms the suture of structure. A suture is 'a seamless [sic**] joint or line of articulation,' which, while joining two surfaces, leaves the trace of their separation."

 ** A dictionary says "a seamlike joint or line of articulation," with no mention of "trace," a term from Derrida's jargon.

Wednesday, July 9, 2008

Wednesday July 9, 2008

Filed under: Uncategorized — m759 @ 8:28 AM
God, Time, Epiphany

8:28:32 AM

Anthony Hopkins, from
All Hallows’ Eve
last year
:

“For me time is God,
God is time. It’s an equation,
like an Einstein equation.”

James Joyce, from
June 26 (the day after
Anti-Christmas) this year
:

“… he glanced up at the clock
of the Ballast Office and smiled:
— It has not epiphanised yet,
he said.”

Ezra Pound (from a page
linked to yesterday morning):

“It seems quite natural to me
that an artist should have
just as much pleasure in an
arrangement of planes
or in a pattern of figures,
  as in painting portraits….”

From Epiphany 2008:

An arrangement of planes:

http://www.log24.com/log/pix08/080709-Epiphany.gif

From May 10, 2008:

A pattern of figures:

Seven partitions of the 2x2x2 cube in a book from 1906

See also Richard Wilhelm on
Hexagram 32 of the I Ching:

“Duration is a state whose movement is not worn down by hindrances. It is not a state of rest, for mere standstill is regression. Duration is rather the self-contained and therefore self-renewing movement of an organized, firmly integrated whole, taking place in accordance with immutable laws and beginning anew at every ending. The end is reached by an inward movement, by inhalation, systole, contraction, and this movement turns into a new beginning, in which the movement is directed outward, in exhalation, diastole, expansion.”

'The Middle-English Harrowing of Hell,' by Hulme, 1907, page 64, line 672: 'with this he gaf the gaste'

The Middle-English
    Harrowing of Hell…

    by Hulme, 1907, page 64

Saturday, December 23, 2006

Saturday December 23, 2006

Filed under: Uncategorized — m759 @ 9:00 AM
Black Mark

Bernard Holland in The New York Times on Monday, May 20, 1996:

“Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday….”

Log24 on Monday,
Dec. 18, 2006:

“I did a column in
Scientific American
on minimal art, and
I reproduced one of
Ed Rinehart’s [sic]
black paintings.”

Martin Gardner (pdf)

“… the entire profession
has received a very public
and very bad black mark.”

Joan S. Birman (pdf)

Lottery on Friday,
Dec. 22, 2006:

The image “http://www.log24.com/log/pix06B/061222-PAlottery.jpg” cannot be displayed, because it contains errors.

5/04
, 2005:

Analysis of the structure
of a 2x2x2 cube

The Eightfold Cube

via trinities of
projective points
in a Fano plane.

7/15, 2005:

“Art history was very personal
through the eyes of Ad Reinhardt.”

  — Robert Morris,
Smithsonian Archives
of American Art

Also on 7/15, 2005,
a quotation on Usenet:

“A set having three members is a
single thing wholly constituted by
its members but distinct from them.
After this, the theological doctrine
of the Trinity as ‘three in one’
should be child’s play.”

— Max Black,
Caveats and Critiques:
Philosophical Essays in
Language, Logic, and Art

Sunday, October 8, 2006

Sunday October 8, 2006

Filed under: Uncategorized — m759 @ 12:00 AM
Today’s Birthday:
Matt Damon
 
Enlarge this image

The image “http://www.log24.com/log/pix06A/061008-Departed2.jpg” cannot be displayed, because it contains errors.

“Cubistic”

New York Times review
of Scorsese’s The Departed

Related material:

Log24, May 26, 2006

“The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast.”

— G. K. Chesterton
 

The image “http://www.log24.com/log/pix06A/060526-JackInTheBox.jpg” cannot be displayed, because it contains errors.
Natasha Wescoat, 2004

Shakespearean
Fool

Not to mention Euclid and Picasso

(Log24, Oct. 6, 2006) —

The image “http://www.log24.com/theory/images/Pythagoras-I47.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix06A/RobertFooteAnimation.gif” cannot be displayed, because it contains errors.

(Click on pictures for details. Euclid is represented by Alexander Bogomolny, Picasso by Robert Foote.)

See also works by the late Arthur Loeb of Harvard’s Department of Visual and Environmental Studies.

“I don’t want to be a product of my environment.  I want my environment to be a product of me.” — Frank Costello in The Departed

For more on the Harvard environment,
see today’s online Crimson:

The Harvard Crimson,
Online Edition
Sunday,
Oct. 8, 2006

POMP AND
CIRCUS-STANCE


CRIMSON/ MEGHAN T. PURDY

Friday, Oct. 6:

The Ringling Bros. Barnum & Bailey Circus has come to town, and yesterday the animals were disembarked near MIT and paraded to their temporary home at the Banknorth Garden.

OPINION

At Last, a
Guiding Philosophy

The General Education report is a strong cornerstone, though further scrutiny is required.

After four long years, the Curricular Review has finally found its heart.

The Trouble
With the Germans

The College is a little under-educated these days.

By SAHIL K. MAHTANI
Harvard College– in the best formulation I’ve heard– promulgates a Japanese-style education, where the professoriate pretend to teach, the students pretend to learn, and everyone is happy.

Friday, May 26, 2006

Friday May 26, 2006

Filed under: Uncategorized — m759 @ 8:00 AM

A Living Church
continued from March 27

"The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast."

— G. K. Chesterton

The image “http://www.log24.com/log/pix06A/060526-JackInTheBox.jpg” cannot be displayed, because it contains errors.
Shakespearean
Fool

Related material:


Yesterday's entries

and their link to
The Line

as well as

Galois Geometry

and the remarks
of Oxford professor
Marcus du Sautoy,
who claims that
"the right side of the brain
is responsible for mathematics."

Let us hope that Professor du Sautoy
is more reliable on zeta functions,
his real field of expertise,
than on neurology.

The picture below may help
to clear up his confusion
between left and right.

His confusion about
pseudoscience may not
be so easily remedied.

The image “http://www.log24.com/log/pix06A/060526-BrainLR1.jpg” cannot be displayed, because it contains errors.
flickr.com/photos/jaycross/3975200/

(Any resemblance to the film
"Hannibal" is purely coincidental.)
 

Tuesday, August 2, 2005

Tuesday August 2, 2005

Filed under: Uncategorized — Tags: — m759 @ 7:00 AM
Today's birthday:
Peter O'Toole

"What is it, Major Lawrence,
 that attracts you personally
 to the desert?"

"It's clean."

Visible Mathematics,
continued —

From May 18:

Lindbergh's Eden

"The Garden of Eden is behind us
and there is no road
back to innocence;
we can only go forward."

— Anne Morrow Lindbergh,
Earth Shine, p. xii
 

 
On Beauty
 
"Beauty is the proper conformity
of the parts to one another
and to the whole."

— Werner Heisenberg,
"Die Bedeutung des Schönen
in der exakten Naturwissenschaft,"
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg's Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974

Related material:

The Eightfold Cube

The Eightfold Cube

(in Arabic, ka'b)

and

The image “http://www.log24.com/log/pix05B/050802-Geom.jpg” cannot be displayed, because it contains errors.
 

Wednesday, May 18, 2005

Wednesday May 18, 2005

Filed under: Uncategorized — m759 @ 11:07 PM
On Beauty

“Beauty is the proper conformity
  of the parts to one another
  and to the whole.”
 
  — Werner Heisenberg,
Die Bedeutung des Schönen
  in der exakten Naturwissenschaft,”
  address delivered to the
  Bavarian Academy of Fine Arts,
  Munich, 9 Oct. 1970, reprinted in
  Heisenberg’s Across the Frontiers,
  translated by Peter Heath,
  Harper & Row, 1974
 
  Related material:
 
 The Eightfold Cube
 
 The Eightfold Cube


Friday, May 6, 2005

Friday May 6, 2005

Filed under: Uncategorized — Tags: — m759 @ 7:28 PM

Fugues

"To improvise an eight-part fugue
is really beyond human capability."

— Douglas R. Hofstadter,
Gödel, Escher, Bach

The image “http://www.log24.com/theory/images/cube2x2x2.gif” cannot be displayed, because it contains errors.

Order of a projective
 automorphism group:
168

"There are possibilities of
contrapuntal arrangement
of subject-matter."

— T. S. Eliot, quoted in
Origins of Form in Four Quartets.

The image “http://www.log24.com/theory/images/Grid4x4A.gif” cannot be displayed, because it contains errors.

Order of a projective
 automorphism group:
20,160

Wednesday, May 4, 2005

Wednesday May 4, 2005

Filed under: Uncategorized — 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.

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: Uncategorized — Tags: , — m759 @ 10:13 PM
 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.
We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)




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