Log24

Friday, April 4, 2014

Dream of the Expanded Field

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

(Continued)

From today’s news:

“His daughter, the poet Jorie Graham, confirmed the death.”

From an artist on Oct. 3, 2013:

“‘This is St. Francis country,’ she says of Umbria.”

Monday, October 14, 2013

Dream of the Expanded Field

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

(Continued)

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman

Further context: Galois I Ching

Sunday, November 7, 2010

Dream of the Expanded Field

Filed under: General — m759 @ 12:00 pm

Continued from July 26, 2010

http://www.log24.com/log/pix10B/101107-JonHanSm.jpg

Wednesday, September 2, 2015

Expanding the Field

Filed under: General — m759 @ 7:59 pm

(Continued)

See The Nordic Journal of Aesthetics ,
Vol. 23, No. 42 (2012), pp. 14-31, 
"Art in an Expanded Field: Wittgenstein and Aesthetics,"
by Noël Carroll.

Abstract:

"This article reviews the various ways in which the later writings
of Ludwig Wittgenstein have been employed to address the question
'What is Art?' These include the family resemblance model, the
cluster concept model and the form of life model. The article defends
a version of the form of life approach. Also, addressed the charge that
it would have been more profitable had aestheticians explored what
Wittgenstein actually said about art instead of trying to extrapolate from
his writings an approach to what Nigel Warburton calls the art question."

Wednesday, June 30, 2010

Field Dream

Filed under: General,Geometry — Tags: , , , , — m759 @ 10:23 am

In memory of Wu Guanzhong, Chinese artist who died in Beijing on Friday

Image-- The Dream of the Expanded Field

"Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game.  Elder Brother laughed.  'Go ahead and try,' he exclaimed.  'You'll see how it turns out.  Anyone can create a pretty little bamboo garden in the world.  But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'"

— Hermann Hesse, The Glass Bead Game, translated by Richard and Clara Winston

"The Chinese painter Wu Tao-tzu was famous because he could paint nature in a unique realistic way that was able to deceive all who viewed the picture. At the end of his life he painted his last work and invited all his friends and admirers to its presentation. They saw a wonderful landscape with a romantic path, starting in the foreground between flowers and moving through meadows to high mountains in the background, where it disappeared in an evening fog. He explained that this picture summed up all his life’s work and at the end of his short talk he jumped into the painting and onto the path, walked to the background and disappeared forever."

Jürgen Teichmann. Teichmann notes that "the German poet Hermann Hesse tells a variation of this anecdote, according to his own personal view, as found in his 'Kurzgefasster Lebenslauf,' 1925."

Saturday, September 3, 2022

1984 Revisited

Filed under: General — m759 @ 2:46 pm

Cube Bricks 1984 —

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

Related material

Note the three quadruplets of parallel edges  in the 1984 figure above.

Further Reading

The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —

Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.

Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —

      And then, more simply, there is the Galois tesseract

For parts of my own  world in June 2010, see this journal for that month.

The above Galois tesseract appears there as follows:

Image-- The Dream of the Expanded Field

See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post

Monday, August 1, 2022

Review

Filed under: General — Tags: , — m759 @ 5:12 am

From Log24 posts tagged Art Space —

From a paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader,
and Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Friday, March 18, 2022

The Dog Far Hence, or Ekphrasis for Anubis

Filed under: General — Tags: — m759 @ 1:26 am

IMAGE- Concepts of Space

The above image suggests a review of Sigaud in this journal and of . . .

Related material from the Web —

"Anubis, easily recognizable as an anthropomorphized jackal or dog,
was the Egyptian god of the afterlife and mummification. He helped
judge souls after their death and guided lost souls into the afterlife.
So, was he evil? No, and in fact just the opposite. In ancient Egyptian
mythology the ultimate evil was chaos. Nearly all of Egyptian mythology
was focused around maintaining the cycles of cosmic order that kept
chaos at bay. Few things were as significant in this goal as the rituals
maintaining the cycle of life, death, and afterlife. Therefore, Anubis was
not evil but rather one of the most important gods who kept evil out of Egypt."

— Christopher Muscato at Study.com

Sunday, February 20, 2022

Ceremonial Space

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

From posts now tagged iching.space (also a URL) —

IMAGE- Concepts of Space

 

Tuesday, November 5, 2019

Non-Woo

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

A followup to earlier posts on Trudeau vs. Euclid —

Geometry from July 6, 2014:

IMAGE- Concepts of Space

Monday, March 11, 2019

Ant-Man Meets Doctor Strange

Filed under: General — m759 @ 1:22 pm

IMAGE- Concepts of Space

The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .

"Think outside the tesseract.

Sunday, July 1, 2018

The Perpetual Motion of T. S. Eliot

Filed under: General — Tags: — m759 @ 10:28 am

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman

Friday, June 29, 2018

For St. Stanley

Filed under: General,Geometry — m759 @ 1:26 pm

The phrase "Blue Dream" in the previous post
suggests a Web search for Traumnovelle .
That search yields an interesting weblog post
from 2014 commemorating the 1999 dies natalis 
(birth into heaven) of St. Stanley Kubrick.

Related material from March 7, 2014,
in this  journal

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

That  2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.

Wednesday, March 29, 2017

Art Space, Continued

Filed under: General — Tags: , , — m759 @ 4:35 am

"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."

—Staff writer Kamila Czachorowski, Harvard Crimson  today

From Log24 posts tagged Art Space

From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Monday, October 26, 2015

Expanding the Spielraum

Filed under: General — Tags: — m759 @ 4:00 pm

(Continued)

Halloween meditation  on  the Tummelplatz  at Innsbruck

"Die Ritter und Knappen des nahegelegenen Ambras
pflegten hier ihre Rosse zu tummeln, woher sich auch
der Name Tummelplatz  schreibt."

"The knights and squires of nearby Ambras used to let their
steeds romp here, whence came the name Tummelplatz ." 

— Quelle: Ludwig von Hörmann, "Der Tummelplatz  bei Amras,"
in: Der Alpenfreund , 1. Band, Gera 1870, S. 72 – 73.

See as well Sigmund Freud, Erinnern, Wiederholen und Durcharbeiten
(1914) —

"Wir eröffnen ihm die Übertragung als den Tummel­platz ,
auf dem ihm gestattet wird, sich in fast völliger Freiheit
zu entfalten, und auferlegt ist, uns alles vorzuführen,
was sich an pathogenen Trieben im Seelenleben des
Analysierten verborgen hat."

"We admit it into the transference as a playground
in which it is allowed to expand in almost complete freedom
and in which it is expected to display to us everything in the
way of pathogenic instincts that is hidden in the patient's mind."

This passage has been discussed by later psychotherapists,
notably Russell Meares.  Dr. Meares, working from a translation
that has "playground" for Freud's Tummelplatz , uses Spielraum  
in place of Freud's own word.

For related material in this  journal, see Expanding the Spielraum.
An illustration from that search —

IMAGE- Concepts of Space

Tuesday, February 3, 2015

Expanding the Spielraum

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

A short poem by several authors:

"The role of
the 16 singular points
on the Kummer surface
is now played by
the 64 singular points
on the Kummer threefold."

— From Remark 2.4 on page 9 of
"The Universal Kummer Threefold,"
by Qingchun Ren, Steven V Sam,
Gus Schrader, and Bernd Sturmfels,
http://arxiv.org/abs/1208.1229v3,
August 6, 2012 — June 12, 2013.

See also "Expanded Field" in this journal.

IMAGE- Concepts of Space

Illustration from "Sunday School," July 20, 2014.

Other Log24 background:  Kummer, Spielraum, Art Space.

Sunday, July 20, 2014

Sunday School

Filed under: General,Geometry — Tags: — m759 @ 9:29 am

Paradigms of Geometry:
Continuous and Discrete

The discovery of the incommensurability of a square’s
side with its diagonal contrasted a well-known discrete 
length (the side) with a new continuous  length (the diagonal).
The figures below illustrate a shift in the other direction.
The essential structure of the continuous  configuration at
left is embodied in the discrete  unit cells of the square at right.

IMAGE- Concepts of Space: The Large Desargues Configuration, the Related 4x4 Square, and the 4x4x4 Cube

See Desargues via Galois (August 6, 2013).

Sunday, July 6, 2014

Sticks and Stones

Filed under: General,Geometry — Tags: , — m759 @ 6:29 am

The title is from this morning's previous post.

From a theater review in that post—

… "all flying edges and angles, a perpetually moving and hungry soul"

… "a formidably centered presence, the still counterpoint"

A more abstract perspective:

IMAGE- Concepts of Space

See also Desargues via Galois (August 6, 2013).

Friday, March 7, 2014

Kummer Varieties

Filed under: General,Geometry — Tags: , , — m759 @ 11:20 am

The Dream of the Expanded Field continues

Image-- The Dream of the Expanded Field

From Klein's 1893 Lectures on Mathematics —

"The varieties introduced by Wirtinger may be called Kummer varieties…."
E. Spanier, 1956

From this journal on March 10, 2013 —

From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Update of 10 PM ET March 7, 2014 —

The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7(E7):

The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik  87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .

Friday, November 9, 2012

Ideas

Filed under: General,Geometry — m759 @ 6:21 am

(Continued from Deconstructing Alice)

The Dream of the Expanded Field

Image-- 4x4 square and 4x4x4 cube

"Somehow it seems to fill my head
with ideas— only I don't exactly know
what they are!"

See also Deep Play.

Friday, September 9, 2011

Galois vs. Rubik

(Continued from Abel Prize, August 26)

IMAGE- Elementary Galois Geometry over GF(3)

The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.

Image-- Sugar cube in coffee, from 'Bleu'

The coffee scene from “Bleu”

Related material from this journal:

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

Tuesday, November 23, 2010

Art Object

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

There is more than one way
to look at a cube.

http://www.log24.com/log/pix10B/101123-plain_cube_200x227.gif

 From Cambridge U. Press on Feb. 20, 2006 —

IMAGE- 'Cambridge Tracts in Mathematics 168: The Cube'

and from this journal on June 30, 2010 —

In memory of Wu Guanzhong, Chinese artist
who died in Beijing on June 25, 2010

Image-- The Dream of the Expanded Field

See also this journal on Feb. 20, 2006
(the day The Cube  was published).

Sunday, July 18, 2010

Du Sucre

Filed under: General,Geometry — m759 @ 4:19 am

http://passionforcinema.com/sapphire/ on "Bleu" —  Jan. 9, 2010 —

"An extremely long lens on an insert of a sugar cube, dipped just enough, in a small cup of coffee, so that it gradually seeps in the dark beverage. Four and a half seconds of unadulterated cinematic bliss."

Image-- Sugar cube in coffee, from 'Bleu'

Related material from this journal:

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

Friday, June 25, 2010

ART WARS continued

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

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

See The Klein Correspondence.

Monday, April 18, 2022

Iconic Simplicity

Filed under: General — Tags: , — m759 @ 11:38 am

An illustration from posts tagged Holy Field GF(3) —

IMAGE- Elementary Galois Geometry over GF(3)

See also a Log24 search for "Four Gods."

Monday, October 23, 2017

Plan 9 Continues

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

Click for some background

Another approach, for Dan Brown fans —

In the following passage, Brown claims that an eight-ray star
with arrowheads at the rays' ends is "the mathematical symbol for
entropy."  Brown may have first encountered this symbol at a 
questionable "Sacred Science" website.  Wikipedia discusses
some even less  respectable uses of the symbol.

Saturday, October 18, 2014

Educational Series

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

Barron's Educational Series (click to enlarge):

The Tablet of Ahkmenrah:

IMAGE- The Tablet of Ahkmenrah, from 'Night at the Museum'

 "With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series

Another educational series (this journal):

Image-- Rosalind Krauss and The Ninefold Square

Art theorist Rosalind Krauss and The Ninefold Square

IMAGE- Elementary Galois Geometry over GF(3)

Thursday, October 10, 2013

Continued

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

"… the walkway between here and there would be colder than a witch’s belt buckle. Or a well-digger’s tit. Or whatever the saying was. Vera had been hanging by a thread for a week now, comatose, in and out of Cheyne-Stokes respiration, and this was exactly the sort of night the frail ones picked to go out on. Usually at 4 a.m. He checked his watch. Only 3:20, but that was close enough for government work."

— King, Stephen (2013-09-24).
    Doctor Sleep: A Novel  (p. 133). Scribner. Kindle Edition. 

From Space.com, the death of an astronaut this morning —

"Carpenter passed at 5:30 a.m. MDT (7:30 a.m. EDT; 1130 GMT)."

A link, "Continued," in this journal at 3:26 a.m. EDT today led to

Thursday, February 5, 2009

Thursday February 5, 2009

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  

Saturday, July 20, 2002

Saturday July 20, 2002

 

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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Page created Jan. 6, 2006, by Steven H. Cullinane      diamondtheorem.com

 

Initial Xanga entry.  Updated Nov. 18, 2006.

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