“Finite Relativity” « Search Results

Tuesday, June 17, 2014

Finite Relativity

Filed under: General,Geometry — Tags: Finite Relativity, Mathieu Raumproblem, Octad, Octad Generator — m759 @ 11:00 am

Anyone tackling the Raumproblem described here
on Feb. 21, 2014 should know the history of coordinatizations
of the 4×6 Miracle Octad Generator (MOG) array by R. T. Curtis
and J. H. Conway. Some documentation:

The above two images seem to contradict a statement by R. T. Curtis
in a 1989 paper. Curtis seemed in that paper to be saying, falsely, that
his original 1973 and 1976 MOG coordinates were those in array M below—

This seemingly false statement involved John H. Conway's supposedly
definitive and natural canonical coordinatization of the 4×6 MOG
array by the symbols for the 24 points of the projective line over GF(23)—
{∞, 0, 1, 2, 3… , 21, 22}:

An explanation of the apparent falsity in Curtis's 1989 paper:

By "two versions of the MOG" Curtis seems to have meant merely that the
octads , and not the projective-line coordinates , in his earlier papers were
mirror images of the octads that resulted later from the Conway coordinates,
as in the images below.

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Thursday, November 22, 2012

Finite Relativity

Filed under: General,Geometry — Tags: Figurate Geometry, Triangle.graphics — m759 @ 10:48 pm

(Continued from 1986)

S. H. Cullinane
The relativity problem in finite geometry.
Feb. 20, 1986.

This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: A 4×4 array.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.

Fifteen partitions of a 4x4 array into two 8-sets

A representative coordinatization:

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

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

S. H. Cullinane
The relativity problem in finite geometry.
Nov. 22, 2012.

This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.

Fifteen partitions of an array of 16 triangles into two 8-sets

A representative coordinatization:

Coordinates for a triangular finite geometry

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.

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Saturday, January 4, 2020

Welcome to the Uncanny Valley Country Club

Filed under: General — Tags: Uncanny Valley — m759 @ 12:44 pm

The previous post suggests a look at some Robot Apocalypse remarks:

Related material —

This journal on the above Stranger Dimensions date —

"Thus the theory of description matters most.
It is the theory of the word for those

For whom the word is the making of the world…."

— Wallace Stevens, "Description Without Place, VII"

See also Finite Relativity (St. Cecilia's Day, 2012).

Some other lines from "Description Without Place" —

"An age is green or red. An age believes

Or it denies. An age is solitude
Or a barricade against the singular man

By the incalculably plural."

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Monday, June 3, 2019

No Serious Difficulty

Filed under: General — m759 @ 11:13 pm

See as well a Log24 search for "FInite Relativity."

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Wednesday, March 6, 2019

The Relativity Problem and Burkard Polster

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

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

For some context, see Relativity Problem in this journal.

In the case of PG(3,2), there is a choice of geometric models
to be coordinatized: two such models are the traditional
tetrahedral model long promoted by Burkard Polster, and
the square model of Steven H. Cullinane.

The above Wikipedia section tacitly (and unfairly) assumes that
the model being coordinatized is the tetrahedral model. For
coordinatization of the square model, see (for instance) the webpage
Finite Relativity.

For comparison of the two models, see a figure posted here on
May 21, 2014 —

Labeling the Tetrahedral Model (Click to enlarge) —

"Citation needed" —

The anonymous characters who often update the PG(3,2) Wikipedia article
probably would not consider my post of 2014, titled "The Tetrahedral
Model of PG(3,2)," a "reliable source."

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Thursday, July 28, 2016

The Giglmayr Foldings

Filed under: General,Geometry — Tags: The Building-Block Metaphor — m759 @ 1:44 pm

Giglmayr's transformations (a), (c), and (e) convert
his starting pattern

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

to three length-16 sequences. Putting these resulting
sequences back into the 4×4 array in normal reading
order, we have

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

(a) (c) (e)

Four length-16 basis vectors for a Galois 4-space consisting
of the origin and 15 weight-8 vectors over GF(2):

0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1
0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1
1 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1
1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 .

(See "Finite Relativity" at finitegeometry.org/sc.)

The actions of Giglmayr's transformations on the above
four basis vectors indicate the transformations are part of
the affine group (of order 322,560) on the affine space
corresponding to the above vector space.

For a description of such transformations as "foldings,"
see a search for Zarin + Folded in this journal.

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Friday, October 31, 2014

Structure

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

On Devil’s Night

Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square

The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.

(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)

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Saturday, October 25, 2014

Foundation Square

Filed under: General,Geometry — Tags: 5x5, Fields, Octad, Octad Generator — m759 @ 2:56 pm

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean geometry or of Galois geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids. This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ₅).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

"The Hoffman-Singleton Graph and its Automorphisms," by
Paul R. Hafner, Journal of Algebraic Combinatorics , 18 (2003), 7–12, and
the Web pages "Hoffman-Singleton Graph" and "Higman-Sims Graph"
of A. E. Brouwer.

Hafner's abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ₅.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

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Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: Brick Space, Depth, Geometry, Octad, Octad Generator, Priority — m759 @ 12:24 pm

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by "Dr. Parker" — Apparently Richard A. Parker —
Lecture 4, "Discovering M₂₄," in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on "Sporadic and Related Groups."
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis's 35 4×6 1976 MOG arrays would use
Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.

See The Diamond Theorem, Finite Relativity, Galois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35 4×6 arrays of the 1976
Curtis MOG would then reveal* the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S₃ on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35 MOG arrays. For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.

* "Then a miracle occurs," as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

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Monday, November 25, 2013

Figurate Numbers

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

The title refers to a post from July 2012:

The above post, a new description of a class of figurate
numbers that has been studied at least since Pythagoras,
shows that the "triangular numbers" of tradition are not
the only triangular numbers.

"Thus the theory of description matters most.
It is the theory of the word for those
For whom the word is the making of the world…."

— Wallace Stevens, "Description Without Place"

See also Finite Relativity (St. Cecilia's Day, 2012).

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Saturday, December 29, 2012

Mapping Problem

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

A mapping problem posed (informally) in 1985
and solved 27 years later, in 2012:

(A note for fans of the recent film Looper (see previous post)—

Hunter S. Thompson in this journal on February 22, 2005 …

Hunter S. Thompson, photos from The New York Times

… and on March 3, 2009.)

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Sunday, October 2, 2011

Symmetric Generation Illustrated

Filed under: General,Geometry — Tags: Geometry, Symmetric Generation — m759 @ 7:20 pm

R.T. Curtis in a 1990 paper* discussed his method of "symmetric generation" of groups as applied to the Mathieu groups M₁₂ and M₂₄.

See Finite Relativity and the Log24 posts Relativity Problem Revisited (Sept. 20) and Symmetric Generation (Sept. 21).

Here is some exposition of how this works with M₁₂.

* "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26.

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Tuesday, September 20, 2011

Relativity Problem Revisited

Filed under: General,Geometry — Tags: Aitchison, Alperin, Finite Relativity, Geometry, Symmetric Generation — m759 @ 4:00 am

A footnote was added to Finite Relativity—

Background:

Weyl on what he calls the relativity problem—

“The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time.”

– Hermann Weyl, 1949, “Relativity Theory as a Stimulus in Mathematical Research“

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

– Hermann Weyl, 1946, The Classical Groups , Princeton University Press, p. 16

…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M₂₄ (containing the original group), acts on the larger array. There is no obvious solution to Weyl’s relativity problem for M₂₄. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M₂₄. ….

Footnote of Sept. 20, 2011:

* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction “The generators of M₂₄ are defined… as permutations of twenty-four 7-cycles in the action of PSL₂(7) on seven letters….”

See “Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups,” by R.T. Curtis, Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.

Some related articles by Curtis:

R.T. Curtis, “Natural Constructions of the Mathieu groups,” Math. Proc. Cambridge Philos. Soc. (1989), Vol. 106, pp. 423-429

R.T. Curtis. “Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M₁₂ and M₂₄” In Proceedings of 1990 LMS Durham Conference ‘Groups, Combinatorics and Geometry’ (eds. M. W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396

R.T. Curtis, “A Survey of Symmetric Generation of Sporadic Simple Groups,” in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57

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Thursday, June 9, 2011

Upshot

Filed under: General,Geometry — Tags: Finite Relativity — m759 @ 3:01 pm

Suggested by this afternoon’s NY Lottery number, 541—

Click for higher quality.

Related material: Finite Relativity and The Schwartz Notes.

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Sunday, February 22, 2009

Sunday February 22, 2009

Filed under: General,Geometry — m759 @ 4:07 pm

Themes and
Variations

Horace Brock with his collection at the Boston Museum of Fine Arts

The Boston Globe today on a current Museum of Fine Arts exhibit of works collected by one Horace Brock–

“Designed objects, Brock writes, can be broken down into ‘themes’ and ‘transformations.’ A theme is a motif, such as an S-curve; a transformation might see that curve appear elsewhere in the design, but stretched, rotated 90 degrees, mirrored, or otherwise reworked.

Aesthetic satisfaction comes from an apprehension of how those themes and transformations relate to each other, or of what Brock calls their ‘relative complexity.’ Basically– and this is the nub of it– ‘if the theme is simple, then we are most satisfied when its echoes are complex… and vice versa.'”

Related material:

Theme

and Variations

See also earlier tributes to
Hollywood Game Theory

and Hollywood Religion:

For some variations on the
above checkerboard theme, see
Finite Relativity and
A Wealth of Algebraic Structure.

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Sunday, August 12, 2007

Sunday August 12, 2007

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

The Geometry of Qubits

In the context of quantum information theory, the following structure seems to be of interest–

"… the full two-by-two matrix ring with entries in GF(2), M₂(GF(2))– the unique simple non-commutative ring of order 16 featuring six units (invertible elements) and ten zero-divisors."

— "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007

A 16-element affine space and a corresponding 16-element matrix ring

This ring is another way of looking at the 16 elements of the affine space A₄(GF(2)) over the 2-element field. (Arrange the four coordinates of each element– 1's and 0's– into a square instead of a straight line, and regard the resulting squares as matrices.) (For more on A₄(GF(2)), see Finite Relativity and related notes at Finite Geometry of the Square and Cube.) Using the above ring, Saniga constructs a system of 35 objects (not unlike the 35 lines of the finite geometry PG(3,2)) that he calls a "projective line" over the ring. This system of 35 objects has a subconfiguration isomorphic to the (2,2) generalized quadrangle W₂ (which occurs naturally as a subconfiguration of PG(3,2)– see Inscapes.)

Saniga concludes:

"We have demonstrated that the basic properties of a system of two interacting spin-1/2 particles are uniquely embodied in the (sub)geometry of a particular projective line, found to be equivalent to the generalized quadrangle of order two. As such systems are the simplest ones exhibiting phenomena like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous applications like quantum cryptography, quantum coding, quantum cloning/teleportation and/or quantum computing to mention the most salient ones, our discovery thus

not only offers a principally new geometrically-underlined insight into their intrinsic nature,

but also gives their applications a wholly new perspective

and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."

It would seem that my own
study of pure mathematics—
for instance, of the following
"diamond ring"–

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

is not without relevance to
the physics of quantum theory.

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Wednesday, March 21, 2007

Wednesday March 21, 2007

Filed under: General,Geometry — Tags: four-set — m759 @ 3:18 pm

Finite Relativity
continued

This afternoon I added a paragraph to The Geometry of Logic that makes it, in a way, a sequel to the webpage Finite Relativity:

"As noted previously, in Figure 2 viewed as a lattice the 16 digital labels 0000, 0001, etc., may be interpreted as naming the 16 subsets of a 4-set; in this case the partial ordering in the lattice is the structure preserved by the lattice's group of 24 automorphisms– the same automorphism group as that of the 16 Boolean connectives. If, however, these 16 digital labels are interpreted as naming the 16 functions from a 4-set to a 2-set (of two truth values, of two colors, of two finite-field elements, and so forth), it is not obvious that the notion of partial order is relevant. For such a set of 16 functions, the relevant group of automorphisms may be the affine group of A mentioned above. One might argue that each Venn diagram in Fig. 3 constitutes such a function– specifically, a mapping of four nonoverlapping regions within a rectangle to a set of two colors– and that the diagrams, considered simply as a set of two-color mappings, have an automorphism group of order larger than 24… in fact, of order 322,560. Whether such a group can be regarded as forming part of a 'geometry of logic' is open to debate."

The epigraph to "Finite Relativity" is:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

The added paragraph seems to fit this description.

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Friday, November 24, 2006

Friday November 24, 2006

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

Galois’s Window:

Geometry
from Point
to Hyperspace

by Steven H. Cullinane

Euclid is “the most famous
geometer ever known
and for good reason:
for millennia it has been
his window
that people first look through
when they view geometry.”

— Euclid’s Window:
The Story of Geometry
from Parallel Lines
to Hyperspace,
by Leonard Mlodinow

“…the source of
all great mathematics
is the special case,
the concrete example.
It is frequent in mathematics
that every instance of a
concept of seemingly
great generality is
in essence the same as
a small and concrete
special case.”

— Paul Halmos in
I Want To Be a Mathematician

Euclid’s geometry deals with affine
spaces of 1, 2, and 3 dimensions
definable over the field
of real numbers.

Each of these spaces
has infinitely many points.

Some simpler spaces are those
defined over a finite field–
i.e., a “Galois” field–
for instance, the field
which has only two
elements, 0 and 1, with
addition and multiplication
as follows:

+	0	1
0	0	1
1	1	0

*	0	1
0	0	0
1	0	1

We may picture the smallest
affine spaces over this simplest
field by using square or cubic
cells as “points”:

From these five finite spaces,
we may, in accordance with
Halmos’s advice,
select as “a small and
concrete special case”
the 4-point affine plane,
which we may call

Galois's Window

Galois’s Window.

The interior lines of the picture
are by no means irrelevant to
the space’s structure, as may be
seen by examining the cases of
the above Galois affine 3-space
and Galois affine hyperplane
in greater detail.

For more on these cases, see

The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
Latin-Square Geometry, and
Geometry of the 4×4 Square.

(These documents assume that
the reader is familar with the
distinction between affine and
projective geometry.)

These 8- and 16-point spaces
may be used to
illustrate the action of Klein’s
simple group of order 168
and the action of
a subgroup of 322,560 elements
within the large Mathieu group.

The view from Galois’s window
also includes aspects of
quantum information theory.
For links to some papers
in this area, see
Elements of Finite Geometry.

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Wednesday, February 25, 2004

Wednesday February 25, 2004

Filed under: General — Tags: Modernism — m759 @ 2:00 pm

Modernism as a Religion

In light of the controversy over Mel Gibson's bloody passion play that opens today, some more restrained theological remarks seem in order. Fortunately, Yale University Press has provided a framework — uniting physics, art, and literature in what amounts to a new religion — for making such remarks. Here is some background.

From a review by Adam White Scoville of Iain Pears's novel titled An Instance of the Fingerpost:

"Perhaps we are meant to see the story as a cubist retelling of the crucifixion, as Pilate, Barabbas, Caiaphas, and Mary Magdalene might have told it. If so, it is sublimely done so that the realization gradually and unexpectedly dawns upon the reader. The title, taken from Sir Francis Bacon, suggests that at certain times, 'understanding stands suspended' and in that moment of clarity (somewhat like Wordsworth's 'spots of time,' I think), the answer will become apparent as if a fingerpost were pointing at the way."

Recommended related material —

By others:

Inside Modernism: Relativity Theory, Cubism, Narrative, Thomas Vargish and Delo E. Mook, Yale University Press, 1999

Signifying Nothing: The Fourth Dimension in Modernist Art and Literature

Corpus Hypercubus,
by Dali. Not cubist,
perhaps "hypercubist."

By myself:

Finite Relativity

The Crucifixion of John O'Hara

Block Designs

The Da Vinci Code and Symbology at Harvard

The Crimson Passion

Material that is related, though not recommended —

The Aesthetics of the Machine

Connecting Physics and the Arts

Comments (1)

Friday, February 20, 2004

Friday February 20, 2004

Filed under: General,Geometry — Tags: Finite Relativity — m759 @ 3:24 pm

Finite Relativity

Today is the 18th birthday of my note

“The Relativity Problem in Finite Geometry.”

That note begins with a quotation from Weyl:

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, Symmetry, Princeton University Press, 1952, p. 138

This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.

Comments (1)

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: Change Logos, Eightfold Cube, four-set, Map Systems, Octad, Octad Generator, Strange Change — 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