Wednesday, January 23, 2013

DNA and a Galois Field

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

From Ewan Birney's weblog today:

WEDNESDAY, 23 JANUARY 2013

Using DNA as a digital archive media

Today sees the publication in Nature of “Toward practical high-capacity low-maintenance storage of digital information in synthesised DNA,” a paper spearheaded by my colleague Nick Goldman and in which I played a major part, in particular in the germination of the idea.

Birney appeared in Log24 on Dec. 30, 2012, quoted as follows:

"It is not often anyone will hear the phrase 'Galois field' and 'DNA' together…."

— Birney's weblog on July 3, 2012, "Galois and Sequencing."

Birney's widespread appearance in news articles today about the above Nature publication suggests a review of the "Galois-field"-"DNA" connection.

See, for instance, the following papers:

Gail Rosen and Jeff Moore. "Investigation of Coding Structure in DNA," IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Hong Kong, April 2003. [pdf]
Gail Rosen. "Finding Near-Periodic DNA Regions using a Finite-Field Framework," 2nd IEEE Genomic Signal Processing Workshop (GENSIPS), Baltimore, MD, May 2004. [pdf]
Gail Rosen. "Examining Coding Structure and Redundancy in DNA," IEEE Engineering in Medicine and Biology Magazine, Volume 25, Issue 1, January/February 2006. [pdf]

A Log24 post of Sept. 17, 2012, also mentions the phrases "Galois field" and "DNA" together.

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Saturday, November 6, 2010

Galois Field of Dreams, continued

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

Hollywood Reporter Exclusive

Martin Sheen Caught in
Spider-Man's Web

“… lux in tenebris lucet…"

Sally Field is in early talks
to play Aunt May.

Related material:

Birthdays in this journal,
Galois Field of Dreams,
and Class of 64.

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Sunday, March 21, 2010

Galois Field of Dreams

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

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

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

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

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

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

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

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

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

* February 27 and March 13

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

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

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

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Saturday, August 29, 2026

Turning 63 . . . Points, Not Years

Filed under: General — Tags: Galois 63-Cycle, Strange Change, Turner Wheel — m759 @ 12:38 am

Compare and contrast . . .

See as well yesterday's St. Augustine Day post , and . . .

The Eternal

Symbol of necessity,
immutability, and
atemporal eternity:

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

For details, see
finite geometry of
the square and cube.

The Temporal

Symbol of the
God of Abraham,
Isaac, and Jacob:

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

For details, see
Under God
(Aug. 11, 2006)

The eternal
combined with
the temporal:

Singer 63-cycle in the Galois field GF(64) used to order the I Ching hexagrams

Comments (0)

Friday, August 7, 2026

The Field

Filed under: General — Tags: Art Space, Lost Horizon, Planes of Reality — m759 @ 2:18 pm

A remark from an earlier Log24 post today . . .

"Hard to tell who's on lifeguard duty."
— January Jones on July 16, 2026

Synchronology check of this journal on that date —

"Time for you to see the field." — Bagger Vance

If the field in question is the Galois field GF(16) . . .

a related Log24 image that is also from July 16, 2026 —

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Wednesday, January 14, 2026

Today Is World Logic Day 2026

Filed under: General — Tags: Art Space, Cullinane AI, Group 384 — m759 @ 12:34 pm

"The date chosen to celebrate World Logic Day, 14 January, corresponds to
the date of death of Kurt Gödel and the date of birth of Alfred Tarski,.
two of the most prominent logicians of the twentieth century." — Wikipedia

Some "presentatonal symbols" — The Gray Lotus.

Diamond Theory
by NotebookLM

94 sources

The Cullinane diamond theorem establishes a profound link between visual 4×4 patterns and advanced mathematical structures such as finite geometry and group theory. By analyzing symmetry-preserving transformations within a group of 322,560 permutations, the research connects simple grid designs to the complex projective space PG(3,2). These geometric frameworks further extend into coding theory, illustrating how "diamond" rings are isomorphic to matrix algebras over the finite field GF(4). The sources also highlight the Miracle Octad Generator (MOG) and its relationship to the Mathieu group M24, which is essential for understanding error-correcting codes and sphere packings. Beyond pure mathematics, the text explores "brick space" and the Klein correspondence, suggesting that these abstract symmetries represent a "universal" order found in both art and nature. Ultimately, this body of work bridges the gap between ancient motifs and modern computational science through the elegant study of structural invariants.

How does the diamond theorem bridge visual tile patterns and abstract group theory?

In what ways does PG(3,2) serve as a foundation for Diamond Theory?

How do symmetry and automorphism groups link 4×4 arrays to the Mathieu group?

__________________________________________________________________

Thursday, November 20, 2025

How are algebraic, combinatorial, and geometric structures unified within the Brick Space model?

The unification of algebraic, combinatorial, and geometric structures within the Brick Space model is achieved through its role as a graphic, isomorphic representation of key finite geometries, particularly the projective spaces PG(5,2) and PG(3,2), and their fundamental relationship via the Klein correspondence.

————————————————————————————.

The term "Brick Space" originates from the

2 \times 4

arrays, or "bricks," coined by R. T. Curtis as component parts of his Miracle Octad Generator (MOG)

. These physical or graphic arrangements serve as a concrete visualization (or model) of highly abstract mathematical structures.

1. Geometric Unification

The Brick Space model provides a geometric foundation for combinatorial objects through direct visualization of finite geometries defined over the two-element field, GF(2)

• Modeling Projective Space: The Brick Space model is the natural geometric setting for the MOG's components, which may be regarded as forming both the 5-dimensional projective space

PG (5, 2)

and the 3-dimensional projective space

PG (3, 2)

• Visualizing PG(5,2) and the Klein Quadric: The model visually represents the

PG (5, 2)

space, which has 63 points, by partitioning these points into 28 points lying off the Klein quadric and 35 points lying on the Klein quadric

. The geometry of $PG (5, 2)$ is linked to the concept of the 8-set

• The Klein Correspondence: The geometric relationship between

PG (3, 2)

and

PG (5, 2)

is formalized by the Klein correspondence

. The $PG (5, 2)$ model, developed as a brick space, is analogous to the "line diagrams" construction of $PG (3, 2)$ found in the Cullinane diamond theorem

• Lines and Points: The Brick Space implicitly models how the 35 lines of

PG (3, 2)

correspond to the 35 points on the Klein quadric in

PG (5, 2)

2. Combinatorial Unification

Combinatorial structures, primarily partitions of sets and block designs, are mapped directly onto geometric entities within the Brick Space framework

• Partitions and Lines: The central combinatorial equivalence involves mapping partitions of sets to geometric objects

. The 35 combinatorial structures arising from permutations of $4 \times 4$ diamond tiles (related to the $2 \times 4$ bricks) are isomorphic to the 35 lines of $PG (3, 2)$ . These 35 structures also mirror the 35 square patterns within the original MOG

• Partitions of the 8-Set: The

PG (3, 2)

lines correspond to the 35 partitions of an 8-set into two 4-sets

. This same framework connects to the 105 partitions of an 8-set into four 2-sets, which are essentially the same as the 105 lines contained in the Klein quadric

• Latin Squares: The combinatorial property of orthogonality of Latin squares (specifically, order-4 Latin squares) is unified geometrically by corresponding exactly to the property of skewness of lines (non-intersecting lines) in

PG (3, 2)

. This provides a geometric dictionary for abstract algebraic combinatorics

3. Algebraic Unification

Algebraic structures, rooted in finite field theory and group theory, govern the behavior and symmetries observed in the combinatorial and geometric models

• Binary Field

GF (2)

: The foundation of the geometry is the Galois field of two elements,

GF (2)

(binary arithmetic:

1 + 1 = 0

)

. The algebraic rules of $GF (2)$ dictate the structure of the affine and projective spaces represented by the bricks

• Binary Addition (XOR): The line diagrams used to formalize the

4 \times 4

patterns (related to the bricks) can be "added" via component-wise binary addition (XOR)

. The closure property of lines in $PG (3, 2)$ is represented algebraically by the fact that the three line diagrams corresponding to a pattern always sum to zero ( $D 1 \oplus D 2 \oplus D 3 = 0$ )

• Group Theory and Symmetry: The massive permutation group

G

(of order 322,560) that transforms the visual patterns is isomorphic to the affine group

AGL (4, 2)

. This group, which explains the pervasive symmetry found in all $G$ -images, is the automorphism group of the underlying finite geometry, establishing the algebraic control over the system's geometric and combinatorial properties

• Ring Theory: The

4 \times 4

patterns can be endowed with algebraic structure to generate an ideal within a ring of symmetric patterns, leading to an infinite family of "diamond rings" isomorphic to rings of matrices over

GF (4)

. This links the geometry to the machinery of abstract algebra

• M24 Connection: The affine group

AGL (4, 2)

is identified as the octad stabilizer subgroup of the Mathieu group

M_{24}

, linking the visual patterns and the underlying geometries to the abstract world of sporadic simple groups, coding theory (Golay code), and lattice theory (Leech lattice)

The Brick Space model, therefore, functions as a visual tapestry where graphic arrangements (geometry/visualization) encode set partitions (combinatorics) that behave according to the rules of binary algebra (

GF (2)

and affine groups)

. The entire structure is unified by powerful isomorphisms relating these three mathematical domains through finite geometries like $PG (3, 2)$ and $PG (5, 2)$

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Monday, December 1, 2025

“Diamond Theory” at NotebookLM Today

Filed under: General — Tags: Cullinane AI, NotebookLM Summaries — m759 @ 8:29 am

NotebookLM — Dec. 1, 2025 — "A Unifying Framework"

Diamond Theory by NotebookLM

92 sources

These documents comprehensively examine the tight relationships among abstract algebra, combinatorics, and finite geometry, primarily through the lens of the projective spaces PG(3,2) and PG(5,2). A central focus is the Cullinane Diamond Theory, which utilizes highly symmetric 4×4 arrays over the Galois field GF(2) to model affine space, whose transformation group is the extensive Affine Group AGL(4,2). This visual and geometric structure connects directly to classic combinatorial objects, such as Conwell's Heptads and Dye's Partitions, which define specific sets and orbits within the geometry. Critically, these underlying symmetries are fundamental to the operation of the Miracle Octad Generator (MOG), establishing a direct link between the patterns of the 4×4 arrays and the intricate structure of the sporadic simple Mathieu group M24. The established correspondence provides crucial geometric interpretations for abstract algebraic problems, including the classification of reguli and the construction of Mutually Orthogonal Latin Squares. Collectively, the material demonstrates a unifying mathematical framework that spans theoretical geometry, error-correcting codes, quantum information science, and visual art.

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Thursday, November 20, 2025

On Brick Space

Filed under: General — Tags: Brick Space, Cullinane AI, Group 384 — m759 @ 6:24 pm

A question suggested by NotebookLM in the "Diamond Theory" notebook,
and its answer today by NotebookLM —

How are algebraic, combinatorial, and geometric structures unified within the Brick Space model?

The term "Brick Space" originates from the

2 \times 4

arrays, or "bricks," coined by R. T. Curtis as component parts of his Miracle Octad Generator (MOG)

. These physical or graphic arrangements serve as a concrete visualization (or model) of highly abstract mathematical structures

1. Geometric Unification

The Brick Space model provides a geometric foundation for combinatorial objects through direct visualization of finite geometries defined over the two-element field, GF(2)

• Modeling Projective Space: The Brick Space model is the natural geometric setting for the MOG's components, which may be regarded as forming both the 5-dimensional projective space

PG (5, 2)

and the 3-dimensional projective space

PG (3, 2)

• Visualizing PG(5,2) and the Klein Quadric: The model visually represents the

PG (5, 2)

space, which has 63 points, by partitioning these points into 28 points lying off the Klein quadric and 35 points lying on the Klein quadric

. The geometry of $PG (5, 2)$ is linked to the concept of the 8-set

• The Klein Correspondence: The geometric relationship between

PG (3, 2)

and

PG (5, 2)

is formalized by the Klein correspondence

. The $PG (5, 2)$ model, developed as a brick space, is analogous to the "line diagrams" construction of $PG (3, 2)$ found in the Cullinane diamond theorem

• Lines and Points: The Brick Space implicitly models how the 35 lines of

PG (3, 2)

correspond to the 35 points on the Klein quadric in

PG (5, 2)

2. Combinatorial Unification

Combinatorial structures, primarily partitions of sets and block designs, are mapped directly onto geometric entities within the Brick Space framework

• Partitions and Lines: The central combinatorial equivalence involves mapping partitions of sets to geometric objects

• Partitions of the 8-Set: The

PG (3, 2)

lines correspond to the 35 partitions of an 8-set into two 4-sets

. This same framework connects to the 105 partitions of an 8-set into four 2-sets, which are essentially the same as the 105 lines contained in the Klein quadric

PG (3, 2)

. This provides a geometric dictionary for abstract algebraic combinatorics

3. Algebraic Unification

Algebraic structures, rooted in finite field theory and group theory, govern the behavior and symmetries observed in the combinatorial and geometric models

• Binary Field

GF (2)

: The foundation of the geometry is the Galois field of two elements,

GF (2)

(binary arithmetic:

1 + 1 = 0

)

. The algebraic rules of $GF (2)$ dictate the structure of the affine and projective spaces represented by the bricks

• Binary Addition (XOR): The line diagrams used to formalize the

4 \times 4

patterns (related to the bricks) can be "added" via component-wise binary addition (XOR)

. The closure property of lines in $PG (3, 2)$ is represented algebraically by the fact that the three line diagrams corresponding to a pattern always sum to zero ( $D 1 \oplus D 2 \oplus D 3 = 0$ )

• Group Theory and Symmetry: The massive permutation group

G

(of order 322,560) that transforms the visual patterns is isomorphic to the affine group

AGL (4, 2)

• Ring Theory: The

4 \times 4

patterns can be endowed with algebraic structure to generate an ideal within a ring of symmetric patterns, leading to an infinite family of "diamond rings" isomorphic to rings of matrices over

GF (4)

. This links the geometry to the machinery of abstract algebra

• M24 Connection: The affine group

AGL (4, 2)

is identified as the octad stabilizer subgroup of the Mathieu group

M_{24}

, linking the visual patterns and the underlying geometries to the abstract world of sporadic simple groups, coding theory (Golay code), and lattice theory (Leech lattice)

GF (2)

and affine groups)

. The entire structure is unified by powerful isomorphisms relating these three mathematical domains through finite geometries like $PG (3, 2)$ and $PG (5, 2)$ .

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Wednesday, September 24, 2025

Annals of Dimensional Reduction:
“Six Dimensions into Three”

Filed under: General — Tags: Brick Space, Klein Space, Sturmfels, The Flagstaff Contract, Zadie Smith — m759 @ 5:57 am

http://m759.net/wordpress/?s="six+dimensions+into+three"

The above link is for fans of Richard J. Trudeau's "Story Theory of Truth."

And then from pure mathematics, there is the reduction from eight dimensions
into six of Diamond Theory, in passing from the eight-dimensional affine space
over the two-element Galois field to the six-dimensional affine space used in
Diamond Theory to represent the five-dimensional projective space PG(5,2).

See other posts tagged Klein Space.

Some less demanding reading —

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Monday, February 10, 2025

Brick Space: Points with Parts

Filed under: General — Tags: 64 Art, Brick Space, Harvard Munch, Klein Correspondence, Points with Parts — m759 @ 3:47 pm

This post's "Points with Parts" title may serve as an introduction to
what has been called "the most powerful diagram in mathematics" —
the "Miracle Octad Generator" (MOG) of Robert T. Curtis.

The Miracle Octad Generator (MOG) of R. T. Curtis

Curtis himself has apparently not written on the geometric background
of his diagram — the finite projective spaces PG(5,2) and PG(3,2), of
five and of three dimensions over the two-element Galois field GF(2).

The component parts of the MOG diagram, the 2×4 Curtis "bricks,"
may be regarded* as forming both PG(5,2) and PG(3,2) . . .
Pace Euclid, points with parts. For more on the MOG's geometric
background, see the Klein correspondence in the previous post.

For a simpler example of "points with parts, see
http://m759.net/wordpress/?s=200229.

* Use the notions of Galois (XOR, or "symmetric-difference") addition
of even subsets, and such addition "modulo complementation," to
decrease the number of dimensions of the spaces involved.

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Sunday, June 23, 2024

Marks

Filed under: General — Tags: Abstraction and Structure, Art Space — m759 @ 2:12 pm

Some marks I find more interesting . . . Those of a Galois field.

See a June 5 post on cultural appropriation .

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Tuesday, December 13, 2022

In Memory of a Mississippi Coach

Filed under: General — Tags: Baez Monoids, Boole vs. Galois — m759 @ 1:10 pm

The "Boolean exclusive or" is the same as addition
in the two-element Galois field GF(2).

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Saturday, March 26, 2022

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

Filed under: General — Tags: Affine Cube, Cubehenge, Galois Tesseract, Six-Set — m759 @ 2:06 am

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

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

**Finite Geometry Notes — Summary Chart**
Name Tag	.Space	.Group	.Art
Box4	2×2 square representing the four-point finite affine geometry AG(2,2). (Box4.space)	S₄ = AGL(2,2) (Box4.group)	(Box4.art)
Box6	3×2 (3-row, 2-column) rectangular array representing the elements of an arbitrary 6-set.	S₆
Box8	2x2x2 cube or 4×2 (4-row, 2-column) array.	S₈ or A₈or AGL(3,2) of order 1344, or GL(3,2) of order 168
Box9	The 3×3 square.	AGL(2,3) or GL(2,3)
Box12	The 12 edges of a cube, or a 4×3 array for picturing the actions of the Mathieu group M₁₂.	Symmetries of the cube or elements of the group M₁₂
Box13	The 13 symmetry axes of the cube.	Symmetries of the cube.
Box15	The 15 points of PG(3,2), the projective geometry of 3 dimensions over the 2-element Galois field.	Collineations of PG(3,2)
Box16	The 16 points of AG(4,2), the affine geometry of 4 dimensions over the 2-element Galois field.	AGL(4,2), the affine group of 322,560 permutations of the parts of a 4×4 array (a Galois tesseract)
Box20	The configuration representing Desargues's theorem.
Box21	The 21 points and 21 lines of PG(2,4).
Box24	The 24 points of the Steiner system S(5, 8, 24).
Box25	A 5×5 array representing PG(2,5).
Box27	The 3-dimensional Galois affine space over the 3-element Galois field GF(3).
Box28	The 28 bitangents of a plane quartic curve.
Box32	Pair of 4×4 arrays representing orthogonal Latin squares.	Used to represent elements of AGL(4,2)
Box35	A 5-row-by-7-column array representing the 35 lines in the finite projective space PG(3,2)	PGL(3,2), order 20,160
Box36	Eurler's 36-officer problem.
Box45	The 45 Pascal points of the Pascal configuration.
Box48	The 48 elements of the group AGL(2,3).	AGL(2,3).
Box56	The 56 three-sets within an 8-set or 56 triangles in a model of Klein's quartic surface or the 56 spreads in PG(3,2).
Box60	The Klein configuration.
Box64	Solomon's cube.

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

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Saturday, May 8, 2021

A Tale of Two Omegas

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

The Greek capital letter Omega, Ω, is customarily
used to denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois field,
the appropriate Ω is the 4×4 grid above.

See the Cullinane diamond theorem .

If the group is the large Mathieu group of
244,823,040 permutations of 24 things,
the appropriate Ω is the 4×6 grid below.

See the Miracle Octad Generator of R. T. Curtis.

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Sunday, December 6, 2020

“Binary Coordinates”

Filed under: General — Tags: Boole vs. Galois — m759 @ 3:09 pm

The title phrase is ambiguous and should be avoided.
It is used indiscriminately to denote any system of coordinates
written with 0 ‘s and 1 ‘s, whether these two symbols refer to
the Boolean-algebra truth values false and true , to the absence
or presence of elements in a subset , to the elements of the smallest
Galois field, GF(2) , or to the digits of a binary number .

Related material from the Web —

Some related remarks from “Geometry of the 4×4 Square:
Notes by Steven H. Cullinane” (webpage created March 18, 2004) —

A related anonymous change to Wikipedia today —

The deprecated “binary coordinates” phrase occurs in both
old and new versions of the “Square representation” section
on PG(3,2), but at least the misleading remark about Steiner
quadruple systems has been removed.

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Wednesday, November 11, 2020

Qube

Filed under: General — Tags: Affine Cube, Fields, Qubes — m759 @ 8:30 pm

The new domain qube.link forwards to . . .
http://finitegeometry.org/sc/64/solcube.html .

More generally, qubes.link forwards to this post,
which defines qubes .

Definition: A qube is a positive integer that is
a prime-power cube , i.e. a cube that is the order
of a Galois field. (Galois-field orders in general are
customarily denoted by the letter q .)

Examples: 8, 27, 64. See qubes.site.

Update on Nov. 18, 2020, at about 9:40 PM ET —

Problem:

For which qubes, visualized as n×n×n arrays,
is it it true that the actions of the two-dimensional
galois-geometry affine group on each n×n face, extended
throughout the whole array, generate the affine group
on the whole array? (For the cases 8 and 64, see Binary
Coordinate Systems and Affine Groups on Small
Binary Spaces.)

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Monday, November 2, 2020

Q Bits

Filed under: General — Tags: Q-Bits — m759 @ 6:34 pm

The new domain name q-bits.space does not refer to
the q in “quantum ,” but rather to the q that symbolizes
the order of a Galois field .

See the Wikipedia article “Finite field.”

The “space” suffix refers to a web page on geometry.

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Thursday, January 10, 2019

Toy Story Continues.

Filed under: General — Tags: Fields, Toy Story — m759 @ 11:13 am

Friday, September 14, 2018

Denkraum

Filed under: G-Notes,General,Geometry — Tags: Chessboard.space, Interstice — m759 @ 1:00 am

Underlying the I Ching structure is the finite affine space
of six dimensions over the Galois field with two elements.

In this field, "1 + 1 = 0," as noted here Wednesday.

See also other posts now tagged Interstice.

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Saturday, February 17, 2018

The Binary Revolution

Filed under: General,Geometry — Tags: Boole vs. Galois, Geometric Theology, Shaw and Finite Geometry, Trinity Dice — m759 @ 5:00 pm

Michael Atiyah on the late Ron Shaw —

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

“The digital revolution based on the 2 symbols (0,1)”
“The algebra of George Boole”
“Binary codes”
“Dirac’s spinors, with their up/down dichotomy”

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

The 2×2 square
The 2x2x2 cube
The 4×4 square
The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

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Friday, February 16, 2018

Two Kinds of Symmetry

Filed under: General,Geometry — Tags: Kummerhenge, Six-Set — m759 @ 11:29 pm

The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter
revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below.

In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)

The adjective "modular" might aptly be applied to . . .

The adjective "affine" might aptly be applied to . . .

The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.

Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but
did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For Macdonald's own use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms,"
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.

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Tuesday, September 5, 2017

Annals of Critical Epistemology

Filed under: General,Geometry — Tags: BU Logic, Springer — m759 @ 5:36 pm

"But unlike many who left the Communist Party, I turned left
rather than right, and returned—or rather turned for the first time—
to a critical examination of Marx's work. I found—and still find—
that his analysis of capitalism, which for me is the heart of his work,
provides the best starting point, the best critical tools, with which—
suitably developed—to understand contemporary capitalism.
I remind you that this year is also the sesquicentennial of the
Communist Manifesto , a document that still haunts the capitalist world."

— From "Autobiographical Reflections," a talk given on June 5, 1998, by
John Stachel at the Max Planck Institute for the History of Science in Berlin
on the occasion of a workshop honoring his 70th birthday,
"Space-Time, Quantum Entanglement and Critical Epistemology."

From a passage by Stachel quoted in the previous post —

From the source for Stachel's remarks on Weyl and coordinatization —

Note that Stachel distorted Weyl's text by replacing Weyl's word
"symbols" with the word "quantities." —

This replacement makes no sense if the coordinates in question
are drawn from a Galois field — a field not of quantities , but rather
of algebraic symbols .

"You've got to pick up every stitch… Must be the season of the witch."
— Donovan song at the end of Nicole Kidman's "To Die For"

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Florence 2001

Filed under: General,Geometry — Tags: BU Logic — m759 @ 4:44 am

Or: Coordinatization for Physicists

This post was suggested by the link on the word "coordinatized"
in the previous post.

I regret that Weyl's term "coordinatization" perhaps has
too many syllables for the readers of recreational mathematics —
for example, of an article on 4×4 magic squares by Conway, Norton,
and Ryba to be published today by Princeton University Press.

Insight into the deeper properties of such squares unfortunately
requires both the ability to learn what a "Galois field" is and the
ability to comprehend seven-syllable words.

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Thursday, August 31, 2017

A Conway-Norton-Ryba Theorem

Filed under: General,Geometry — Tags: Fields, Zero Sum — m759 @ 1:40 pm

In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —

A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.

See also Magic Square and Diamond Theorem in this journal.

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Wednesday, July 5, 2017

Imaginarium of a Different Kind

Filed under: General,Geometry — Tags: Brick Space, Klein Correspondence — m759 @ 9:00 pm

The title refers to that of the previous post, "The Imaginarium."

In memory of a translator who reportedly died on May 22, 2017,
a passage quoted here on that date —

Related material — A paragraph added on March 15, 2017,
to the Wikipedia article on Galois geometry —

George Conwell gave an early demonstration of Galois geometry in 1910 when he characterized a solution of Kirkman's schoolgirl problem as a partition of sets of skew lines in PG(3,2), the three-dimensional projective geometry over the Galois field GF(2).^[3] Similar to methods of line geometry in space over a field of characteristic 0, Conwell used Plücker coordinates in PG(5,2) and identified the points representing lines in PG(3,2) as those on the Klein quadric.

— User Rgdboer

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Tuesday, January 3, 2017

Cultist Space

Filed under: General,Geometry — Tags: Conceptualist Minimalism, Fields, The Coxeter Aleph — m759 @ 6:29 pm

The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —

The previous post illustrated a 3×3 grid. That cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.

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Thursday, September 15, 2016

The Smallest Perfect Number/Universe

Filed under: General,Geometry — Tags: Six-Set, Tetrahedron vs. Square, Whitehead 1906 — m759 @ 6:29 am

The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).

* For the definition of "perfect number," see any introductory
number-theory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
the projective 3-space over the 2-element Galois field GF(2),
was coined by math writer Burkard Polster. Cullinane's square
model of PG(3,2) differs from the earlier tetrahedral model
discussed by Polster.

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Tuesday, September 13, 2016

Parametrizing the 4×4 Array

Filed under: General,Geometry — Tags: Geometry, Kummerhenge, Octad, Octad Generator, Priority, Six-Set — m759 @ 10:00 pm

The previous post discussed the parametrization of
the 4×4 array as a vector 4-space over the 2-element
Galois field GF(2).

The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2-subsets of a 6-set, as in Hudson's
1905 classic Kummer's Quartic Surface —

Hudson in 1905:

These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2-element sets — were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator," what turned out to be 15 of Hudson's
1905 "Göpel tetrads":

A recap by Cullinane in 2013:

Click images for further details.

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Monday, May 30, 2016

Perfect Universe

Filed under: General,Geometry — Tags: Six-Set — m759 @ 7:00 pm

(A sequel to the previous post, Perfect Number)

Since antiquity, six has been known as
"the smallest perfect number." The word "perfect"
here means that a number is the sum of its
proper divisors — in the case of six: 1, 2, and 3.

The properties of a six-element set (a "6-set")
divided into three 2-sets and divided into two 3-sets
are those of what Burkard Polster, using the same
adjective in a different sense, has called
"the smallest perfect universe" — PG(3,2), the projective
3-dimensional space over the 2-element Galois field.

A Google search for the phrase "smallest perfect universe"
suggests a turnaround in meaning , if not in finance,
that might please Yahoo CEO Marissa Mayer on her birthday —

The semantic turnaround here in the meaning of "perfect"
is accompanied by a model turnaround in the picture of PG(3,2) as
Polster's tetrahedral model is replaced by Cullinane's square model.

Further background from the previous post —

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

Filed under: General,Geometry — Tags: Dirac and Geometry, Facets, Kummerhenge, Octad, Octad Generator, Quantum Tesseract Theorem, Rosenhain and Göpel, The Omega Matrix, The Rosenhain Symmetry — m759 @ 8:23 am

The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .

"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."

— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com

Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).

Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .

Some related work of my own (click images for related posts)—

Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

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Diamond Theory
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