Some strings attached . . .
Sunday, March 1, 2026
Monday, February 23, 2026
Decomposition Theory
For some group actions on simpler decompositions — in finite spaces — of
point-sets at the vertices of n-dimensional cubes into point-sets at the vertices
of the cubes' n-2-dimensional subcubes . . . See the Feb. 13, 2026, post
Cube Space as well as the post below from the date of Daverman's death —
Another finite-geometry decomposition result that can be applied to the
representation, by 8-set-four-colorings, of lines in the Klein quadric —
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Sunday, February 22, 2026
Affine Cubes: Iacta Est
The new tag "Affine Cube" was added to various Log24 posts last night.
A vocabulary check this morning yields . . .
The above magic-cube Hendricks reportedly died on July 7, 2007 . . .
Also on July 7, 2007, in this journal . . . Nymphet Witches.
My own approach to affine cubes is somewhat different:
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Saturday, February 21, 2026
Snark for Mark
W. Blaschke and K. Reidemeister (1922).
"Über die Entwicklung der Affingeometrie" (PDF).
Jahresbericht DMV. 31: 63–81 —
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Saturday, February 14, 2026
Friday, February 13, 2026
Geometry for Friday the 13th
The previous post — "Cube Space" — and today's date
suggest a review of the 13 symmetry axes of the cube.
Related geometry —
|
By NotebookLM today — Symmetry in Finite Geometry and Combinatorial Design The provided sources explore the mathematical and artistic intersections of finite geometry, specifically focusing on the Cullinane diamond theorem and its square-based representations of PG(3,2). By utilizing 4×4 and 4×6 arrays, these works illustrate how combinatorial designs, such as Latin squares and Miracle Octad Generators, relate to highly symmetric structures like the Mathieu group M24 and the binary Golay code. The texts demonstrate that properties of symmetry, such as the affine group AGL(4,2), govern both abstract group theory and visual patterns found in puzzles, quilt designs, and sphere packings. This framework extends into coding theory and quantum mechanics, where geometric "bricks" and "lines" help simplify the analysis of complex lattices and error-correcting systems. Ultimately, the collection bridges rigorous algebraic abstraction with interactive visualization, showing that the logic of finite space underpins both mathematical truth and aesthetic form. |
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Cube Space
Theorem:
Some large natural symmetry groups of the sets of 8, 16, 32, or 64 points
in Euclidean space that are located at the vertices of a cube in 3, 4, 5. or 6
dimensions are generated by, respectively, arbitrary permutations of
parallel edges or parallel faces or parallel cubes or parallel hypercubes .
(For an example, see Diamond Theory in 1937.)
Illustration of related group actions:
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Friday, July 4, 2025
1984-1985
Meanwhile . . .
|
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Wednesday, November 13, 2024
Group Theory: The Romance of Alias and Alibi*
Alias:
Victor at 194 Tower Avenue in "The Penguin"
Alibi:
Marcela_234 at Likewise.com
Romance in Numberland:
* Technical terms from pure mathematics —
For scholia on "the cube is being moved around," vide . . .
http://m759.net/wordpress/?s="Stack+Exchange" .
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Saturday, September 14, 2024
Notes on a Friday the 13th Death
A passage accessed via the new URL Starbrick.art* —
Thursday, February 25, 2021
|
A related cultural note suggested by the New York Times obituary today
of fashion designer Mary McFadden, who reportedly died yesterday
(a Friday the Thirteenth) and is described by the Times as a late-life
partner of "eightfold-way" physicist Murray Gell-Mann —
* A reference to the 2-column 4-row matrix (a "brick") that underlies
the patterns in the Miracle Octad Generator of R. T. Curtis. The only
connection of this eight-part matrix to Gell-Mann's "Eightfold Way"
that I know of is simply the number 8 itself.
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Sunday, March 3, 2024
Deep Blue Research: A Report by You.com AI
|
Cullinane Diamond Theorem Research Report by https://you.com/?chatMode=research on March 3, 2024 Overview of the Cullinane Diamond Theorem The Cullinane Diamond Theorem is a mathematical concept developed by Steven H. Cullinane that explores the symmetrical properties of certain geometric patterns. It is particularly concerned with the structure of finite projective geometry as it relates to the 35 square patterns found in R. T. Curtis's Miracle Octad Generator (MOG). The theorem asserts that every G-image of a diamond figure D possesses some form of ordinary or color-interchange symmetry. Symmetry and Group Theory The theorem is deeply rooted in group theory, with G being isomorphic to the affine group A on the linear 4-space over the finite field GF(2). This group has a significant order of 322,560 and underlies the symmetry of both the diamond-theorem figures and the square patterns of the MOG . The 35 structures of the 840 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), which explains the symmetry of the line diagrams and the two-color patterns. The Diamond Theorem in Various Dimensions The diamond theorem can be applied to different cases, such as the 4×4, 2×2, and even 3-dimensional cubes like the 2x2x2 and 4x4x4 cubes. The generalized statement of the theorem involves inscribing a white diamond in a black square and dividing it into four quadrants, each with a diagonal split into black and white halves. Algebraic Implications The theorem has algebraic implications, such as the ability to express every 4-coloring as a sum of three 2-colorings. This algebraic formulation is useful for demonstrating that certain four-colored graphic images can form a ring under multiplication. There is also an infinite family of such 'diamond' rings, isomorphic to rings of matrices over GF(4). Philosophical and Cultural Context Steven H. Cullinane has also explored the philosophical implications of the diamond theorem, referring to it in the context of Plato's Meno dialogue and opposing it to the 'Story Theory of truth' as discussed by Richard J. Trudeau. The theorem is seen as a correct statement of the relation between geometry and the physical universe. Clarification on Cullinan Diamonds It is important to note that the Cullinane Diamond Theorem is unrelated to the Cullinan diamond, which is a famous large gemstone discovered in South Africa and now part of the British Crown Jewels. The Cullinan diamond has its own historical and cultural significance, with its various cut shapes and ownership history. In summary, the Cullinane Diamond Theorem is a mathematical concept that reveals the symmetrical properties of certain geometric patterns, with applications in group theory and algebra. It also has philosophical implications, drawing connections to classical ideas about truth and geometry. |
Instagram ad for You.com AI in research mode
"Show me ALL your sources, babe."
— Line adapted from Leonardo DiCaprio
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Saturday, September 16, 2023
Sunday, January 22, 2023
“Preform an affine transformation”
"Before time began, there was the Cube." — Transformers (2007)
Related literature:
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Friday, January 13, 2023
The “Diamond Space” of Mazzola
The Source —
Some similar notions from my own work . . .
The "Digraph" of Mazzola might correspond to a directed graph
indicating the structure of a permutation, as at right below —
Mazzola's "Formula" might correpond to a matrix and translation that
transform the above "Space" of eight coordinates, and his "Gesture"
to a different way of generating affine transformations of that space . . .
as in my webpage "Cube Space, 1984-2003."
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Wednesday, October 5, 2022
Physicality
"Battles argues that 'the experience of the physicality
of the book is strongest in large libraries,' and stand
among the glass cube at the center of the British Library,
the stacks upon stacks in Harvard’s Widener Library, or
the domed portico of the Library of Congress and tell me
any differently."
— Ed Simon, Binding the Ghost: Theology, Mystery, and
the Transcendence of Literature. Hardcover – April 19, 2022.
… And back to cube:
Related meditation: Beer Summit.
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Saturday, September 10, 2022
Orthogonal Latin Triangles
From a 1964 recreational-mathematics essay —
Note that the first two triangle-dissections above are analogous to
mutually orthogonal Latin squares . This implies a connection to
affine transformations within Galois geometry. See triangle graphics
in this journal.
Update of 4:40 AM ET —
Other mystical figures —
"Before time began, there was the Cube."
— Optimus Prime in "Transformers" (Paramount, 2007)
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Saturday, March 26, 2022
Box Geometry: Space, Group, Art (Work in Progress)
| 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 A8 or 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 |
|
| 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 |
||
| Box60 | The Klein configuration. | ||
| Box64 | Solomon's cube. |
— Steven H. Cullinane, March 26-27, 2022
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Friday, December 31, 2021
Aesthetics in Academia
Related art — The non-Rubik 3x3x3 cube —
The above structure illustrates the affine space of three dimensions
over the three-element finite (i.e., Galois) field, GF(3). Enthusiasts
of Judith Brown's nihilistic philosophy may note the "radiance" of the
13 axes of symmetry within the "central, structuring" subcube.
I prefer the radiance (in the sense of Aquinas) of the central, structuring
eightfold cube at the center of the affine space of six dimensions over
the two-element field GF(2).
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Monday, July 12, 2021
Educational Series
.jpg)
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Wednesday, November 11, 2020
Qube
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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Wednesday, September 2, 2020
Space Wars: Sith Pyramid vs. Jedi Cube
For the Sith Pyramid, see posts tagged Pyramid Game.
For the Jedi Cube, see posts tagged Enigma Cube
and cube-related remarks by Aitchison at Hiroshima.
This post was suggested by two events of May 16, 2019 —
A weblog post by Frans Marcelis on the Miracle Octad
Generator of R. T. Curtis (illustrated with a pyramid),
and the death of I. M. Pei, architect of the Louvre pyramid.
That these events occurred on the same date is, of course,
completely coincidental.
Perhaps Dan Brown can write a tune to commemorate
the coincidence.
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Tuesday, January 28, 2020
Very Stable Kool-Aid
Two of the thumbnail previews
from yesterday's 1 AM post …
Further down in the "6 Prescott St." post, the link 5 Divinity Avenue
leads to …
|
A Letter from Timothy Leary, Ph.D., July 17, 1961
Harvard University July 17, 1961
Dr. Thomas S. Szasz Dear Dr. Szasz: Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun. The Myth of Mental Illness is the most important book in the history of psychiatry. I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century. It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty. . . . . |
The small Morton Prince House in the above letter might, according to
the above-quoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.
Related "jewel box" material for those who
prefer narrative to mathematics —
"In The Electric Kool-Aid Acid Test , Tom Wolfe writes about encountering
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three I-Ching coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster-
in-Chief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."
— Tim Doody in The Morning News web 'zine on July 26, 2012**
Oracular advice related to yesterday evening's
"jewel box" post …
A 4-dimensional hypercube H (a tesseract ) has 24 square
2-dimensional faces. In its incarnation as a Galois tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4-space over the finite (i.e., Galois) two-element
field GF(2)), the 24 faces transform into 140 4-point "facets." The Galois
version of H has a group of 322,560 automorphisms. Therefore, by the
orbit-stabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of 2,304 affine transformations.
Similar remarks apply to the I Ching In its incarnation as
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6-dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.
* The volume Wolfe mentions was, according to Fadiman, the I Ching.
** See also this journal on that date — July 26, 2012.
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Thursday, June 27, 2019
Group Actions on the 4x4x4 Cube
For affine group actions, see Ex Fano Appollinis (June 24)
and Solomon's Cube.
For one approach to Mathieu group actions on a 24-cube subset
of the 4x4x4 cube, see . . .
For a different sort of Mathieu cube, see Aitchison.
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Sunday, December 9, 2018
Quaternions in a Small Space
The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.
Click to enlarge —
Three links from the above finitegeometry.org webpage on the
quaternion group —
-
Visualizing GL(2,p) — A 1985 note illustrating group actions
on the 3×3 (ninefold) square. -
Another 1985 note showing group actions on the 3×3 square
transferred to the 2x2x2 (eightfold) cube. - Quaternions in an Affine Galois Plane — A webpage from 2010.
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
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Sunday, July 1, 2018
Deutsche Ordnung
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film "Triple Cross."
Related structures —
Greg Egan's animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
|
Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to 'being simple': consider Eliot's Four Quartets or Mozart's late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it's often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we've seen into the heart of things. I'll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. |
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
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Friday, June 29, 2018
Triangles in the Eightfold Cube
From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube —
“Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eight-triangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eight-triangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”
Related material from 1975 —
More recently …
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Wednesday, March 29, 2017
Art Space Illustrated
Another view of the previous post's art space —
More generally, see Solomon's Cube in Log24.
See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above
cube arrangement from "Affine groups on small binary spaces" (1984).
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Friday, December 23, 2016
Requiem for a Mathematician
From a Dec. 21 obituary posted by the
University of Tennessee at Knoxville —
"Wade was ordained as a pastor and served
at Oakwood Baptist Church in Knoxville."
Other information —
In a Log24 post, "Seeing the Finite Structure,"
of August 16, 2008, Wade appeared as a co-author
of the Walsh series book mentioned above —
Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp et al.,
Taylor & Francis, 1990
From the 2008 post —
The patterns on the faces of the cube on the cover
of Walsh Series above illustrate both the
Walsh functions of order 3 and the same structure
in a different guise, subspaces of the affine 3-space
over the binary field. For a note on the relationship
of Walsh functions to finite geometry, see
Symmetry of Walsh Functions.
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Monday, September 12, 2016
The Kummer Lattice
The previous post quoted Tom Wolfe on Chomsky's use of
the word "array."
An example of particular interest is the 4×4 array
(whether of dots or of unit squares) —
.
Some context for the 4×4 array —
The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .
Further background on the Kummer lattice:
Alice Garbagnati and Alessandra Sarti,
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action."
To appear in Rocky Mountain J. Math. —
The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite geometry, see the website
Finite Geometry of the Square and Cube.
Some further context …
"To our knowledge, the relation of the Golay code
to the Kummer lattice … is a new observation."
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 "
As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface. The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.
* Update of Sept. 14: "Uncoordinatized," but parametrized by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.
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Monday, August 1, 2016
Cube
From this journal —
See (for instance) Sacred Order, July 18, 2006 —
From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —
|
The doors open slowly. I step into a hangar. From the rafters high above, lights blaze down, illuminating a twelve-foot cube the color of gunmetal. My pulse rate kicks up. I can’t believe what I’m looking at. Leighton must sense my awe, because he says, “Beautiful, isn’t it?” It is exquisitely beautiful. At first, I think the hum inside the hangar is coming from the lights, but it can’t be. It’s so deep I can feel it at the base of my spine, like the ultralow-frequency vibration of a massive engine. I drift toward the box, mesmerized.
— Crouch, Blake. Dark Matter: A Novel |
See also Log24 on the publication date of Dark Matter .
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