"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
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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."
“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
See also Relativity Problem and Diamonds and Whirls.
A sequel to last night's post The 4×4 Relativity Problem —
In other words, how should the triangle corresponding to
the above square be coordinatized ?
See also a post of July 8, 2012 — "Not Quite Obvious."
Context — "Triangles Are Square," a webpage stemming
from an American Mathematical Monthly item published
in 1984.
The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.
“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 Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)
See also a 2011 publication of the Mathematical Association of America —
In the Beginning…
"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)
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
Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—
"… the successive powers of the primitive root λ or 10 are
λ = 10, λ2 = 21, λ3 = 22, λ4 = 02,
λ5 = 20, λ6 = 12, λ7 = 11, λ8 = 01.
These are the proper coordinate symbols….
(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"
Coxeter's Figure 10 yields...
The Aleph
The details:
Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).
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, M 24 (containing the original group), acts on the larger array. There is no obvious solution to Weyl's relativity problem for M 24. 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 M 24. ….
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 M 24 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 M 12 and M 24" 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
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
Twenty-four years ago 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 coordinates (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.
There is, however, an assignment of symbol-strings that yields a family of sets with automorphism group M24.
R.D. Carmichael in 1931 on his construction of the Steiner system S(5,8,24)–
"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."
— R. D. Carmichael, 1931, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240
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Finite Geometry: A Unifying Framework by NotebookLM on September 18, 2025 Finite geometry serves as a powerful unifying framework that bridges diverse mathematical disciplines with artistic and practical applications, primarily by revealing deep underlying symmetries and structures. Here's how finite geometry achieves this unification:
In essence, finite geometry, exemplified by the Cullinane diamond theorem, acts as a "portal" that unveils profound mathematical structures underlying seemingly simple patterns, demonstrating the interconnectedness of geometry, algebra, combinatorics, and visual art, with significant implications for fields ranging from error-correcting codes to experimental design and signal processing. * This AI statement is false. See "Origin," Sept. 20, 2025. |
For those who prefer narratives to mathematics . . .
Abacus Conundrums (Monday, Sept. 15, 2025).
For the significance of the date "February 20," see . . .
|
Google One: Gemini Advanced AI on 13 May 2024. Prompt: How can Weyl's 1946 remarks on coordinatization in "The Classical Groups" be applied to models of finite geometries? Response (after several drafts): In "The Classical Groups," Hermann Weyl discusses the concept of coordinatization, which is the process of assigning numerical coordinates to points in a geometric space. This process is fundamental in connecting abstract geometric concepts with concrete numerical representations, allowing for algebraic manipulation and analysis. Weyl's remarks on coordinatization have significant implications for models of finite geometries. Finite geometries are geometric systems with a finite number of points and lines, and they often exhibit different properties compared to traditional Euclidean geometry. Here's how Weyl's ideas can be applied to finite geometries: 1. **Relativity of Coordinatization:** Weyl emphasizes that the choice of coordinate system is arbitrary and that different coordinate systems can be used to describe the same geometric object. This relativity of coordinatization is particularly relevant in finite geometries, where multiple coordinate systems may exist, each with its own advantages and disadvantages. 2. **Equivalent Coordinatizations:** Weyl introduces the concept of "equivalent coordinatizations," which are different coordinate systems that can be transformed into one another through a group of transformations. In finite geometries, identifying and understanding these equivalent coordinatizations can provide deeper insights into the underlying structure and symmetries of the geometry. 3. **Objective Class of Coordinatizations:** Weyl discusses the problem of fixing an "objective class" of coordinatizations that capture essential features of the geometry while being independent of arbitrary choices. In finite geometries, this could involve identifying coordinatizations that best reflect the specific properties and relationships within the geometry. 4. **Group of Transformations:** Weyl emphasizes the role of the "group of transformations" that mediates between equivalent coordinatizations. In finite geometries, understanding these transformation groups can help classify different types of finite geometries and reveal their underlying symmetries. By applying Weyl's insights, researchers can gain a deeper understanding of finite geometries, including their classification, properties, and relationships. Additionally, exploring different coordinatization schemes can lead to new algebraic and geometric tools for studying finite geometries. For further exploration, you can delve into the works of other mathematicians like Steven H. Cullinane, who has written extensively on finite geometry and its connections to relativity and other areas of mathematics. I hope this explanation is helpful! Please let me know if you have any further questions. |
Update at about 5 PM ET the same day . . .
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Prompt: Correction: Cullinane has NOT written on finite geometry's "connections to relativity." He has, instead, quoted Weyl's remarks on what Weyl himself called "the relativity problem," but what was in fact his own term for problems involving coordinatization. Such problems are of interest both in physics and in finite geometry, but the latter is a field of pure mathematics, NOT of physics. Response: You are absolutely correct. I apologize for the error in my previous response. Steven H. Cullinane has indeed quoted Weyl's remarks on "the relativity problem" in the context of finite geometry. However, as you rightly pointed out, this term refers to Weyl's own terminology for problems involving coordinatization and not the theory of relativity in physics. Finite geometry is a field of pure mathematics and does not directly connect to the physical theories of relativity. While both areas involve concepts of space and transformations, they operate in fundamentally different domains. Cullinane's work focuses on applying Weyl's insights on coordinatization to the specific challenges and structures within finite geometries. This involves exploring different coordinatization schemes, understanding their equivalences, and identifying transformation groups that reveal the underlying symmetries of finite geometries. Thank you for pointing out this important distinction. It highlights the importance of precise language and accurate attribution when discussing complex mathematical concepts. |
From The Man Who Knew Infinity to The Man Who Knew Zero.
Related mathematics: The Diamond Theorem Correlation, which
results from interchanging infinity and zero in the figure below.
"Read something that means something."
— New Yorker ad
.
Click image for
related posts.
Background — Relativity Problem in Log24.
"The class is objectively characterized, but not
the individual coordinate assignment."
Tell it to Watchduck, Hermann.
See a related remark by Quack5quack in Raiders of the Lost Coordinates.
Related material —
“The message was clear: having a finite frame of reference
creates the illusion of a world, but even the reference frame itself
is an illusion. Observers create reality, but observers aren’t real.
There is nothing ontologically distinct about an observer, because
you can always find a frame in which that observer disappears:
the frame of the frame itself, the boundary of the boundary.”
— Amanda Gefter in 2014, quoted here on Mayday 2020.
See as well, in a post from the date of Hunter Thompson’s death :
“Today, February 20, is the 19th anniversary of my note
The Relativity Problem in Finite Geometry.”
“No serious difficulty is encountered as long as one deals
with a domain consisting of a finite number of points only,
which can be ‘called up’ one after the other.” — Weyl
Background — The relativity problem in this journal.
See also this journal on November 29, 2011 —The Flight from Ennui.
Related illustration from earlier in 2011 —
See also this journal on 20 Sept. 2011 — Relativity Problem Revisited —
as well as other posts tagged Congregated Light.
The Hudson array mentioned above is as follows —
See also Whitehead and the
Relativity Problem (Sept. 22).
For coordinatization of a 4×4
array, see a note from 1986
in the Feb. 26 post Citation.
The exercise in the previous post was suggested by a passage
purporting to "use standard block design theory" that was written
by some anonymous author at Wikipedia on March 1, 2019:
Here "rm OR" apparently means "remove original research."
Before the March 1 revision . . .
The "original research" objected to and removed was the paragraph
beginning "To explain this further." That paragraph was put into the
article earlier on Feb. 28 by yet another anonymous author (not by me).
An account of my own (1976 and later) original research on this subject
is pictured below, in a note from Feb. 20, 1986 —
"János Bolyai was a nineteenth-century mathematician who
set the stage for the field of non-Euclidean geometry."
— Transylvania Now , October 26, 2018
From Coxeter and the Relativity Problem —
Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,
The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.
See as well, in this journal, Deathly Hallows, Relativity Problem, and Space Cross.
A related quote: "This is not mathematics; this is theology."
"Composed in light of both Hiroshima and
Einstein’s general theory of relativity,
Dali’s crucifixion . . . ."
— "The Crucified God: A Death in Pictures,"
by Ed Simon, April 11, 2017,
http://marginalia.lareviewofbooks.org/…
See as well Log24, The Relativity Problem at Hiroshima (Dec. 3, 2018).
Related material —
Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —
In finite geometry and combinatorics,
an inscape is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:
Related material — the phrase
"Quantum Tesseract Theorem" and …
A. An image from the recent
film "A Wrinkle in Time" —
B. A quote from the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
From the former date above —
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Saturday, September 17, 2016 |
From the latter date above —
|
Tuesday, October 18, 2016
Parametrization
|
From March 2018 —
From mathoverflow.net on Dec. 7, 2016 —
The exceptional isomorphism between
PGL(3,2) and PSL(2,7): geometric origin?
Essentially the same question was asked earlier at
math.stackexchange.com on Aug. 2, 2010.
See also this journal in November 2017 —
"Read something that means something."
— New Yorker ad
.
Background — Relativity Problem in Log24.
For example, Plato's diamond as an object to be transformed —
Versions of the transformed object —
See also The 4×4 Relativity Problem in this journal.
The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space nature.
Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by
a blank plus the 15 line diagrams of the diamond theorem.
Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann 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
Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space coordinates. He describes it
as simply a mapping to a set of reproducible symbols .
(But Weyl does imply that these symbols should, like vector-space
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)
For illustrations based on the above equations, see
Coxeter and the Relativity Problem and Singer 7-Cycles .
The art above is by the Copenhagen studio
Hvass & Hannibal. For a photo of the artists,
see a webpage on Beijing Design Week 2011.
Hvass and Hannibal were apparently in Beijing
for the "open workshop," Sept. 17-23, 2011.
Gestalt-related material from this journal that week —
* Title suggested by that of a book by Quine.
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 —
For the late mathematics educator Zoltan Dienes.
“There comes a time when the learner has identified
the abstract content of a number of different games
and is practically crying out for some sort of picture
by means of which to represent that which has been
gleaned as the common core of the various activities.”
— Article by “Melanie” at Zoltan Dienes’s website
Dienes reportedly died at 97 on Jan. 11, 2014.
From this journal on that date —
A star figure and the Galois quaternion.
The square root of the former is the latter.
Update of 5:01 PM ET Feb. 6, 2014 —
An illustration by Dienes related to the diamond theorem —
See also the above 15 images in …
… and versions of the 4×4 coordinatization in The 4×4 Relativity Problem
(Jan. 17, 2014).
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
|
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) |
Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
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