Rubik core:
Non- Rubik core:
Central structure from a Galois plane
(See image below.)
Sunday, June 4, 2023
The Galois Core
Comments Off on The Galois Core
Wednesday, October 26, 2022
The Hunt for Galois October
"… Évariste was born on October 25, 1811."
— Eric Temple Bell, Men of Mathematics
Related material —
But seriously . . .
Comments Off on The Hunt for Galois October
Tuesday, March 22, 2022
Esprit for Pascal and Galois: Finesse vs. Geometrie
Finesse —
|
Sunday December 10, 2006 m759 @ 9:00 PM
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
Geometrie —
Comments Off on Esprit for Pascal and Galois: Finesse vs. Geometrie
Saturday, December 18, 2021
Desargues, Galois, Veblen, and Young
This post was suggested by a review in the Jan. 2022
Notices of the American Mathematical Society :
My own sympathies are with Veblen.
Comments Off on Desargues, Galois, Veblen, and Young
Sunday, November 22, 2020
The Galois-Fano Plane
A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —
Related material — The Eightfold Cube.
Update at 10:51 PM ET the same day —
Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .
DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.
Comments Off on The Galois-Fano Plane
Saturday, October 24, 2020
The Galois Tesseract
Stanley E. Payne and J. A. Thas in 1983* (previous post) —
“… a 4×4 grid together with
the affine lines on it is AG(2,4).”
Payne and Thas of course use their own definition
of affine lines on a grid.
Actually, a 4×4 grid together with the affine lines on it
is, viewed in a different way, not AG(2,4) but rather AG(4,2).
For AG(4,2) in the proper context, see
Affine Groups on Small Binary Spaces and
The Galois Tesseract.
* And 26 years later, in 2009.
Comments Off on The Galois Tesseract
Monday, June 1, 2020
A Graveyard Smash: Galois Geometry Meets Nordic Aliens
Comments Off on A Graveyard Smash: Galois Geometry Meets Nordic Aliens
Friday, April 3, 2020
Saturday, March 2, 2019
Saturday, May 5, 2018
Galois Imaginary
|
" Lying at the axis of everything, zero is both real and imaginary. Lovelace was fascinated by zero; as was Gottfried Leibniz, for whom, like mathematics itself, it had a spiritual dimension. It was this that let him to imagine the binary numbers that now lie at the heart of computers: 'the creation of all things out of nothing through God's omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing.' He also wrote, 'The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and nonbeing.' "
— A footnote from page 229 of Sydney Padua's |
A related passage —
|
From The French Mathematician 0
I had foreseen it all in precise detail. i = an imaginary being
Here, on this complex space, |
Comments Off on Galois Imaginary
Wednesday, May 2, 2018
Galois’s Space
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
Comments Off on Galois’s Space
Sunday, November 19, 2017
Galois Space
This is a sequel to yesterday's post Cube Space Continued.
Comments Off on Galois Space
Saturday, July 8, 2017
Desargues and Galois in Japan
Related material now available online —
A less business-oriented sort of virtual reality —
For example, "A very important configuration is obtained by
taking the plane section of a complete space five-point."
(Veblen and Young, 1910, p. 39)—
Comments Off on Desargues and Galois in Japan
Saturday, May 20, 2017
van Lint and Wilson Meet the Galois Tesseract*
Click image to enlarge.
The above 35 projective lines, within a 4×4 array —
The above 15 projective planes, within a 4×4 array (in white) —
* See Galois Tesseract in this journal.
Comments Off on van Lint and Wilson Meet the Galois Tesseract*
Sunday, August 14, 2016
The Boole-Galois Games
Continued from earlier posts on Boole vs. Galois.
From a Google image search today for “Galois Boole.”
Click the image to enlarge it.
Comments Off on The Boole-Galois Games
Thursday, June 30, 2016
Rubik vs. Galois: Preconception vs. Pre-conception
From Psychoanalytic Aesthetics: The British School ,
by Nicola Glover, Chapter 4 —
|
In his last theoretical book, Attention and Interpretation (1970), Bion has clearly cast off the mathematical and scientific scaffolding of his earlier writings and moved into the aesthetic and mystical domain. He builds upon the central role of aesthetic intuition and the Keats's notion of the 'Language of Achievement', which
… includes language that is both Bion distinguishes it from the kind of language which is a substitute for thought and action, a blocking of achievement which is lies [sic ] in the realm of 'preconception' – mindlessness as opposed to mindfulness. The articulation of this language is possible only through love and gratitude; the forces of envy and greed are inimical to it.. This language is expressed only by one who has cast off the 'bondage of memory and desire'. He advised analysts (and this has caused a certain amount of controversy) to free themselves from the tyranny of the past and the future; for Bion believed that in order to make deep contact with the patient's unconscious the analyst must rid himself of all preconceptions about his patient – this superhuman task means abandoning even the desire to cure . The analyst should suspend memories of past experiences with his patient which could act as restricting the evolution of truth. The task of the analyst is to patiently 'wait for a pattern to emerge'. For as T.S. Eliot recognised in Four Quartets , 'only by the form, the pattern / Can words or music reach/ The stillness'.30. The poet also understood that 'knowledge' (in Bion's sense of it designating a 'preconception' which blocks thought, as opposed to his designation of a 'pre -conception' which awaits its sensory realisation), 'imposes a pattern and falsifies'
For the pattern is new in every moment The analyst, by freeing himself from the 'enchainment to past and future', casts off the arbitrary pattern and waits for new aesthetic form to emerge, which will (it is hoped) transform the content of the analytic encounter. 29. Attention and Interpretation (Tavistock, 1970), p. 125 30. Collected Poems (Faber, 1985), p. 194. 31. Ibid., p. 199. |
See also the previous posts now tagged Bion.
Preconception as mindlessness is illustrated by Rubik's cube, and
"pre -conception" as mindfulness is illustrated by n×n×n Froebel cubes
for n= 1, 2, 3, 4.
Suitably coordinatized, the Froebel cubes become Galois cubes,
and illustrate a new approach to the mathematics of space .
Comments Off on Rubik vs. Galois: Preconception vs. Pre-conception
Tuesday, May 31, 2016
Tuesday, January 12, 2016
Harmonic Analysis and Galois Spaces
The above sketch indicates, in a vague, hand-waving, fashion,
a connection between Galois spaces and harmonic analysis.
For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.
Comments Off on Harmonic Analysis and Galois Spaces
Wednesday, January 6, 2016
Galois.io
The title is a new URL.
Midrash on the URL suffix —
" 'I/O' is a computer term of very long standing
that means 'input/output,' i.e. the means by which
a computer communicates with the outside world.
In a domain name, it's a shibboleth that implies
that the intended audience for a site is other
programmers."
— Phil Darnowsky on Dec. 18, 2014
Remarks for a wider audience —
See some Log24 posts related to Dec. 18, 2014.
Comments Off on Galois.io
Tuesday, March 24, 2015
Brouwer on the Galois Tesseract
Yesterday's post suggests a review of the following —
|
Andries Brouwer, preprint, 1982:
"The Witt designs, Golay codes and Mathieu groups" Pages 8-9: Substructures of S(5, 8, 24) An octad is a block of S(5, 8, 24). Theorem 5.1
Let B0 be a fixed octad. The 30 octads disjoint from B0
the design of the points and affine hyperplanes in AG(4, 2), Proof…. … (iv) We have AG(4, 2).
(Proof: invoke your favorite characterization of AG(4, 2) An explicit construction of the vector space is also easy….) |
Related material: Posts tagged Priority.
Comments Off on Brouwer on the Galois Tesseract
Tuesday, November 25, 2014
Euclidean-Galois Interplay
For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3-space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
Comments Off on Euclidean-Galois Interplay
Saturday, October 18, 2014
Tuesday, June 3, 2014
Galois Matrices
The webpage Galois.us, on Galois matrices , has been created as
a starting point for remarks on the algebra (as opposed to the geometry)
underlying the rings of matrices mentioned in AMS abstract 79T-A37,
“Symmetry invariance in a diamond ring.”
See also related historical remarks by Weyl and by Atiyah.
Comments Off on Galois Matrices
Thursday, March 20, 2014
Classical Galois
Comments Off on Classical Galois
Sunday, November 24, 2013
Tuesday, August 6, 2013
Desargues via Galois
The following image gives a brief description
of the geometry discussed in last spring's
Classical Geometry in Light of Galois Geometry.
Update of Aug. 7, 2013: See also an expanded PDF version.
Comments Off on Desargues via Galois
Monday, June 10, 2013
Galois Coordinates
Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."
A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."
A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galois-field coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory monograph.
But such a survey might not find any such pre-1976
coordinatization of a 4×4 array by the 16 elements
of the vector 4-space over the Galois field with two
elements, GF(2).
Such coordinatizations are important because of their
close relationship to the Mathieu group M 24 .
See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M 24 ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.
Related material:
Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—
* A rather abstract 2011 paper that uses the phrase
"Galois coordinates" may have some implications
for the naive form of the relativity problem
related to square and cubical arrays.
Comments Off on Galois Coordinates
Saturday, April 13, 2013
Veblen and Young in Light of Galois
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
Comments Off on Veblen and Young in Light of Galois
Sunday, March 10, 2013
Galois Space
The 16-point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.
Some closely related material:
- Wolfgang Kühnel,
"Minimal Triangulations of Kummer Varieties,"
Abh. Math. Sem. Univ. Hamburg 57, 7-20 (1986).For the first two pages, click here.
- Jonathan Spreer and Wolfgang Kühnel,
"Combinatorial Properties of the K 3 Surface:
Simplicial Blowups and Slicings,"
preprint, 26 pages. (2009/10) (pdf).
(Published in Experimental Math. 20,
issue 2, 201–216 (2011).)
Comments Off on Galois Space
Monday, March 4, 2013
Occupy Galois Space
Continued from February 27, the day Joseph Frank died…
"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review in 1945, propelled him
to prominence as a theoretician."
— Bruce Weber in this morning's print copy
of The New York Times (p. A15, NY edition)
That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:
See also Galois Space and Occupy Space in this journal.
Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:
"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."
Frank is survived by, among others, his wife, a mathematician.
Comments Off on Occupy Galois Space
Thursday, February 21, 2013
Galois Space
The previous post suggests two sayings:
"There is such a thing as a Galois space."
— Adapted from Madeleine L'Engle
"For every kind of vampire, there is a kind of cross."
Illustrations—
Comments Off on Galois Space
Wednesday, January 23, 2013
DNA and a Galois Field
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.
Comments Off on DNA and a Galois Field
Sunday, October 14, 2012
Sunday, July 29, 2012
The Galois Tesseract
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vector-space structure:
| 0 | c | d | c + d |
| a | a + c | a + d | a + c + d |
| b | b + c | b + d | b + c + d |
| a + b | a + b + c | a + b + d | a + b + c + d |
(This vector-space a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)
The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)
A 1982 discussion of a more abstract form of AG(4, 2):
Source:
The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.
Comments Off on The Galois Tesseract
Thursday, July 12, 2012
Galois Space
An example of lines in a Galois space * —
The 35 lines in the 3-dimensional Galois projective space PG(3,2)—
_lines_as_arrays.jpg)
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3-set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.
The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.
* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh].
(Cambridge U. Press, 1960, 488-499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
Comments Off on Galois Space
Tuesday, July 10, 2012
Euclid vs. Galois
Euclidean square and triangle—
Galois square and triangle—
Background—
This journal on the date of Hilton Kramer's death,
The Galois Tesseract, and The Purloined Diamond.
Comments Off on Euclid vs. Galois
Wednesday, October 26, 2011
Erlanger and Galois
Peter J. Cameron yesterday on Galois—
"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."
Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.
Group theory is an essential part of modern geometry as well as of modern algebra—
"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."
— Felix Christian Klein, Erlanger Programm , 1872
("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))
Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—
"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity Σ try to determine is group of automorphisms , the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way."
For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.
* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2
Comments Off on Erlanger and Galois
For Galois
Tue Oct 25, 2011 08:26 AM [London time]
from the weblog of Peter Cameron—
Today is Évariste Galois’ 200th birthday.
The event will be celebrated with the publication of a new transcription
and translation of Galois’ works (edited by Peter M. Neumann)
by the European Mathematical Society. The announcement is here.
Cameron's further remarks are also of interest.
Comments Off on For Galois
Friday, September 9, 2011
Galois vs. Rubik
(Continued from Abel Prize, August 26)
The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.
The coffee scene from "Bleu"
Related material from this journal:
The Dream of
the Expanded Field
Comments Off on Galois vs. Rubik
Saturday, September 3, 2011
The Galois Tesseract (continued)
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Comments Off on The Galois Tesseract (continued)
Thursday, September 1, 2011
Monday, June 27, 2011
Galois Cube Revisited
The 3×3×3 Galois Cube
This cube, unlike Rubik's, is a
purely mathematical structure.
Its properties may be compared
with those of the order-2 Galois
cube (of eight subcubes, or
elements ) and the order-4 Galois
cube (of 64 elements). The
order-3 cube (of 27 elements)
lacks, because it is based on
an odd prime, the remarkable
symmetry properties of its smaller
and larger cube neighbors.
Comments Off on Galois Cube Revisited
Saturday, November 6, 2010
Galois Field of Dreams, continued
Hollywood Reporter Exclusive
Martin Sheen Caught in
Spider-Man's Web
Sally Field is in early talks
to play Aunt May.
Related material:
Birthdays in this journal,
Galois Field of Dreams,
and Class of 64.
Comments Off on Galois Field of Dreams, continued
Friday, September 17, 2010
The Galois Window
Yesterday's excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.
That approach will appeal to few mathematicians, so here is another.
Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace is a book by Leonard Mlodinow published in 2002.
More recently, Mlodinow is the co-author, with Stephen Hawking, of The Grand Design (published on September 7, 2010).
A review of Mlodinow's book on geometry—
"This is a shallow book on deep matters, about which the author knows next to nothing."
— Robert P. Langlands, Notices of the American Mathematical Society, May 2002
The Langlands remark is an apt introduction to Mlodinow's more recent work.
It also applies to Martin Gardner's comments on Galois in 2007 and, posthumously, in 2010.
For the latter, see a Google search done this morning—
Here, for future reference, is a copy of the current Google cache of this journal's "paged=4" page.
Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron's web journal. Following the link, we find…
For n=4, there is only one factorisation, which we can write concisely as 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S4, and acts as S3 on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.
This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.
See also, in this journal, Window and Window, continued (July 5 and 6, 2010).
Gardner scoffs at the importance of Galois's last letter —
"Galois had written several articles on group theory, and was
merely annotating and correcting those earlier published papers."
— Last Recreations, page 156
For refutations, see the Bulletin of the American Mathematical Society in March 1899 and February 1909.
Comments Off on The Galois Window
Monday, June 14, 2010
Birkhoff on the Galois “Theory of Ambiguity”
The Principle of Sufficient Reason
from "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940
EXCERPT 1—
My primary purpose will be to show how a properly formulated
Principle of Sufficient Reason plays a fundamental
role in scientific thought and, furthermore, is to be regarded
as of the greatest suggestiveness from the philosophic point
of view.2
In the preceding lecture I pointed out that three branches
of philosophy, namely Logic, Aesthetics, and Ethics, fall
more and more under the sway of mathematical methods.
Today I would make a similar claim that the other great
branch of philosophy, Metaphysics, in so far as it possesses
a substantial core, is likely to undergo a similar fate. My
basis for this claim will be that metaphysical reasoning always
relies on the Principle of Sufficient Reason, and that
the true meaning of this Principle is to be found in the
“Theory of Ambiguity” and in the associated mathematical
“Theory of Groups.”
If I were a Leibnizian mystic, believing in his “preestablished
harmony,” and the “best possible world” so
satirized by Voltaire in “Candide,” I would say that the
metaphysical importance of the Principle of Sufficient Reason
and the cognate Theory of Groups arises from the fact that
God thinks multi-dimensionally3 whereas men can only
think in linear syllogistic series, and the Theory of Groups is
2 As far as I am aware, only Scholastic Philosophy has fully recognized and ex-
ploited this principle as one of basic importance for philosophic thought
3 That is, uses multi-dimensional symbols beyond our grasp.
______________________________________________________________________
the appropriate instrument of thought to remedy our deficiency
in this respect.
The founder of the Theory of Groups was the mathematician
Evariste Galois. At the end of a long letter written in
1832 on the eve of a fatal duel, to his friend Auguste
Chevalier, the youthful Galois said in summarizing his
mathematical work,4 “You know, my dear Auguste, that
these subjects are not the only ones which I have explored.
My chief meditations for a considerable time have been
directed towards the application to transcendental Analysis
of the theory of ambiguity. . . . But I have not the time, and
my ideas are not yet well developed in this field, which is
immense.” This passage shows how in Galois’s mind the
Theory of Groups and the Theory of Ambiguity were
interrelated.5
Unfortunately later students of the Theory of Groups
have all too frequently forgotten that, philosophically
speaking, the subject remains neither more nor less than the
Theory of Ambiguity. In the limits of this lecture it is only
possible to elucidate by an elementary example the idea of a
group and of the associated ambiguity.
Consider a uniform square tile which is placed over a
marked equal square on a table. Evidently it is then impossible
to determine without further inspection which one
of four positions the tile occupies. In fact, if we designate
its vertices in order by A, B, C, D, and mark the corresponding
positions on the table, the four possibilities are for the
corners A, B, C, D of the tile to appear respectively in the
positions A, B, C, D; B, C, D, A; C, D, A, B; and D, A, B, C.
These are obtained respectively from the first position by a
4 My translation.
5 It is of interest to recall that Leibniz was interested in ambiguity to the extent
of using a special notation v (Latin, vel ) for “or.” Thus the ambiguously defined
roots 1, 5 of x2-6x+5=0 would be written x = l v 5 by him.
______________________________________________________________________
null rotation ( I ), by a rotation through 90° (R), by a rotation
through 180° (S), and by a rotation through 270° (T).
Furthermore the combination of any two of these rotations
in succession gives another such rotation. Thus a rotation R
through 90° followed by a rotation S through 180° is equivalent
to a single rotation T through 270°, Le., RS = T. Consequently,
the "group" of four operations I, R, S, T has
the "multiplication table" shown here:
This table fully characterizes the group, and shows the exact
nature of the underlying ambiguity of position.
More generally, any collection of operations such that
the resultant of any two performed in succession is one of
them, while there is always some operation which undoes
what any operation does, forms a "group."
__________________________________________________
EXCERPT 2—
Up to the present point my aim has been to consider a
variety of applications of the Principle of Sufficient Reason,
without attempting any precise formulation of the Principle
itself. With these applications in mind I will venture to
formulate the Principle and a related Heuristic Conjecture
in quasi-mathematical form as follows:
PRINCIPLE OF SUFFICIENT REASON. If there appears
in any theory T a set of ambiguously determined ( i e .
symmetrically entering) variables, then these variables can themselves
be determined only to the extent allowed by the corresponding
group G. Consequently any problem concerning these variables
which has a uniquely determined solution, must itself be
formulated so as to be unchanged by the operations of the group
G ( i e . must involve the variables symmetrically).
HEURISTIC CONJECTURE. The final form of any
scientific theory T is: (1) based on a few simple postulates; and
(2) contains an extensive ambiguity, associated symmetry, and
underlying group G, in such wise that, if the language and laws
of the theory of groups be taken for granted, the whole theory T
appears as nearly self-evident in virtue of the above Principle.
The Principle of Sufficient Reason and the Heuristic Conjecture,
as just formulated, have the advantage of not involving
excessively subjective ideas, while at the same time
retaining the essential kernel of the matter.
In my opinion it is essentially this principle and this
conjecture which are destined always to operate as the basic
criteria for the scientist in extending our knowledge and
understanding of the world.
It is also my belief that, in so far as there is anything
definite in the realm of Metaphysics, it will consist in further
applications of the same general type. This general conclu-
sion may be given the following suggestive symbolic form:
While the skillful metaphysical use of the Principle must
always be regarded as of dubious logical status, nevertheless
I believe it will remain the most important weapon of the
philosopher.
___________________________________________________________________________
A more recent lecture on the same subject —
by Jean-Pierre Ramis (Johann Bernoulli Lecture at U. of Groningen, March 2005)
Comments Off on Birkhoff on the Galois “Theory of Ambiguity”
Monday, May 31, 2010
Memorial for Galois
… and for Louise Bourgeois
"The épateurs were as boring as the bourgeois,
two halves of one dreariness."
— D. H. Lawrence, The Plumed Serpent
Comments Off on Memorial for Galois
Sunday, May 30, 2010
A Post for Galois
Evariste Galois, 1811-1832 (Vita Mathematica, V. 11)
- Paperback: 168 pages
- Publisher: Birkhäuser Basel; 1 edition (December 6, 1996)
- Language: English
- ISBN-10: 3764354100
- ISBN-13: 978-3764354107
- Product Dimensions: 9.1 x 6 x 0.4 inches
- Shipping Weight: 9.1 ounces
- Average Customer Review: 5.0 out of 5 stars (1 customer review)
- Amazon Bestsellers Rank: #933,939 in Books
Awarded 5 stars by Christopher G. Robinson (Cambridge, MA USA).
See also other reviews by Robinson.
Galois was shot in a duel on today's date, May 30, in 1832. Related material for those who prefer entertainment to scholarship—
"It is a melancholy pleasure that what may be [Martin] Gardner’s last published piece, a review of Amir Alexander’s Duel at Dawn: Heroes, Martyrs & the Rise of Modern Mathematics, will appear next week in our June issue." —Roger Kimball of The New Criterion, May 23, 2010.
Today is, incidentally, the feast day of St. Joan of Arc, Die Jungfrau von Orleans. (See "against stupidity" in this journal.)
Comments Off on A Post for Galois
Sunday, March 21, 2010
Galois Field of Dreams
It is well known that the seven
Similarly, recent posts* have noted that the thirteen
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
* 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)
Comments Off on Galois Field of Dreams
Wednesday, October 23, 2024
Ingathering Scriptures
I recommend . . .
https://www.amazon.com/
Ingathering-Complete-People-Stories-Henderson-ebook/
dp/B0878RMD5D/ .
Some will prefer Exodus 23:16.
Related philosophy —
Descartes and the Tents of Armageddon (Galois's birthday, 2007).
Related art —
https://claude.site/artifacts/3c1ce26b-4664-448c-87df-c20802053eea .
Update on Oct. 28, 2024 —
For a newer version by a different AI, Websim, see Diamond Theorem Studio.
Comments Off on Ingathering Scriptures
Wednesday, September 18, 2024
Flatiron Building Timelessness . . .
Directed by Quine, not Hitchcock
Click the "timelessness" quote below for the "Bell, Book and Candle" scene
with Kim Novak and James Stewart atop the Flatiron Building.
"Before time began . . . ." — Optimus Prime
Comments Off on Flatiron Building Timelessness . . .
Directed by Quine, not Hitchcock
Directed by Quine, not Hitchcock
Saturday, September 7, 2024
Annals of Cultural History
Bloomsday, and then Galois's birthday, and then . . . Square Space!
"Ride a painted pony, let the spinning wheel spin."
Comments Off on Annals of Cultural History
Tuesday, August 27, 2024
For Rubik Worshippers
The above is six-dimensional as an affine space, but only five-dimensional
as a projective space . . . the space PG(5, 2).
As the domain of the smallest model of the Klein correspondence and the
Klein quadric, PG (5,2) is not without mathematical importance.
See Chess Bricks and Ovid.group.
This post was suggested by the date July 6, 2024 in a Warren, PA obituary
and by that date in this journal.
Comments Off on For Rubik Worshippers
Monday, August 19, 2024
Saturday, August 17, 2024
Space Odyssey . . . 2001 to 2024
-Representations-Wikipedia.jpg)
As for what Polster called "God's fingerprint" . . .
A version for Hollywood —
Comments Off on Space Odyssey . . . 2001 to 2024
Sunday, June 23, 2024
Marks
Some marks I find more interesting . . . Those of a Galois field.
See a June 5 post on cultural appropriation .
Comments Off on Marks
Saturday, June 22, 2024
O Carina
From a 2016 graphic novel by Douglas Rushkoff —
See as well . . .
https://zelda.fandom.com/wiki/Triforce and Galois Rhyme.
Related reading:
In that 2016 Rushkoff book, vide the foreword, dated March 1, 2016 …
and, from that same date in this journal, posts tagged Buttressed.
Comments Off on O Carina
Tuesday, May 14, 2024
Sunday, May 5, 2024
Opus Pistoris*
* Title derived from the original title, Opus Pistorum , of
Under the Roofs of Paris, by Henry Miller. In Latin, that
title means Opus "of the millers," or "of the bakers."
This post was suggested by the April 25 death of a French cultural figure
reported today by The New York Times .
Related reading —
Comments Off on Opus Pistoris*
Thursday, April 25, 2024
Space Trace
The title "Space Trace" was suggested yesterday by Claude.ai.
How classical space leaves a Galois trace:
Comments Off on Space Trace
Tuesday, March 26, 2024
Pour les petites
"This is by way of a pep talk, mes petites . . ."
Musical and theatrical accompaniment . . .
Musical: Valse pour les petites , reportedly recorded on March 2, 2019;
Theatrical: A Log24 post from that same date, Schoolgirls for Galois.
Comments Off on Pour les petites
Tuesday, January 2, 2024
Mathematics Made Absurd: Domain and Range
"… the dominant discourse limits the range
of discussion in each domain…."
— https://americanaffairsjournal.org/2023/11/
the-stagnant-science-mainstream-economics-in-america/
See as well Boole vs. Galois and …
Comments Off on Mathematics Made Absurd: Domain and Range
Friday, December 29, 2023
A Wrinkle in Logic
Also on March 8, 2018 —
This post was suggested by the word "entanglement' in the previous post.
See as well "Galois (Xor) addition."
Comments Off on A Wrinkle in Logic
Tuesday, December 19, 2023
Copilot Report
|
From Microsoft Copilot (with "Precise" setting)
on Dec. 18, 2023 —
How is the 1910 work of G. M. Conwell on PG(3,2) related to the Miracle Octad Generator of R. T. Curtis?
Searching for:
Searching for: Generating answers for you… The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics12. Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space1. On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M242. The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator3. These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG3. For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references123. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help. Learn more
5
_________________________________________________________
__________________________________________________________
Here is a summary of the document:
Reference:
|
Correction:
"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."
Comments Off on Copilot Report
Thursday, December 7, 2023
“Wheel Turnin’ ’Round and ’Round” — Song Lyric
Image by A. A. ("Bert") Jagers from a search in this journal for Galois sequence —
Comments Off on “Wheel Turnin’ ’Round and ’Round” — Song Lyric
Monday, November 27, 2023
Problems with the Process
Condensed from Peter J. Cameron's weblog today —
|
“Words that tear and strange rhymes” "In his youth, Paul Simon thought of himself as a poet . . . . And surprisingly often he describes problems with the process:
For me, things were somewhat similar. Like many people, I wrote poetry in my youth. Julian Jaynes says something like 'Poems are rafts grasped at by men drowning in inadequate minds', but I think I knew from early on that one of the main reasons was to practise my writing, so that when I had something to say I could say it clearly. When Bob Dylan renounced the over-elaborate imagery of Blonde on Blonde for the clean simplicity of John Wesley Harding, I took that as a role model. Could Simon’s experience happen in mathematics? It is possible to imagine that an important mathematical truth is expressed in 'words that tear and strange rhymes'. More worryingly, an argument written in the most elegant style could be wrong, and we may be less likely to see the mistake because the writing is so good." |
The problem with the process in this case is Cameron's misheard lyrics.
From https://www.paulsimon.com/track/kathys-song-2/ —
And a song I was writing is left undone
I don’t know why I spend my time
Writing songs I can’t believe
With words that tear and strain to rhyme
A rather different artist titled a more recent song
"Strange Rhymes Can Change Minds."
See also . . .
Comments Off on Problems with the Process
Saturday, November 18, 2023
“Don’t solicit for your sister,* it’s not nice.” — Tom Lehrer
From this journal at 1:51 AM ET Thursday, September 8, 2022 —
"The pleasure comes from the illusion" . . .
Exercise:
Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).
Note that the 4-row-2-column "brick" at left is quite
different from the other two bricks, which together
show chevron variations within a Galois tesseract —
.
Further Weil remarks . . .
A Slew of Prayers
"The pleasure comes from the illusion
and the far from clear meaning;
once the illusion is dissipated,
and knowledge obtained, one becomes
indifferent at the same time;
at least in the Gitâ there is a slew of prayers
(slokas) on the subject, each one more final
than the previous ones."
* —
Comments Off on “Don’t solicit for your sister,* it’s not nice.” — Tom Lehrer
Friday, November 10, 2023
Logos
Related art —
(For some backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)
Comments Off on Logos
Tuesday, October 24, 2023
A Bond with Reality: The Geometry of Cuts
Illustrations of object and gestures
from finitegeometry.org/sc/ —
Object
Gestures
An earlier presentation of the above
seven partitions of the eightfold cube:
|
|
Related mathematics:
|
The use of binary coordinate systems
Natural physical transformations of square or cubical arrays See "The Thing and I." |
and . . .
Related entertainment:
Or Matt Helm by way of a Jedi cube.
Comments Off on A Bond with Reality: The Geometry of Cuts
Thursday, October 19, 2023
Math for Barbie
Continued from "Barbie at the Space Barn," Oct. 17.
"Open the Space Barn doors, Barbie." —
For those who prefer the Hollywood part of L.A.,
there is Barbierella —
Comments Off on Math for Barbie
Friday, October 13, 2023
Turn, Turn, Turn
The conclusion of a Hungarian political figure's obituary in
tonight's online New York Times, written by Clay Risen —
"A quietly religious man, he spent his last years translating
works dealing with Roman Catholic canon law."
This journal on the Hungarian's date of death, October 8,
a Sunday, dealt in part with the submission to Wikipedia of
the following brief article . . . and its prompt rejection.
|
The Cullinane diamond theorem is a theorem
The theorem also explains symmetry properties of the Reference
1. Cullinane diamond theorem at |
Some quotations I prefer to Catholic canon law —
|
Ludwig Wittgenstein,
97. Thought is surrounded by a halo. * See the post Wittgenstein's Diamond. Related language in Łukasiewicz (1937)—
|
See as well Diamond Theory in 1937.
Comments Off on Turn, Turn, Turn
Friday, October 6, 2023
“A Shining” — New Jon Fosse Book, Out on Halloween
Shining-related material from my own life —
Other material related to my own life, also on the date
November 25, 2009, but less personal —
From the current version —
Some background —
Comments Off on “A Shining” — New Jon Fosse Book, Out on Halloween
Monday, September 18, 2023
The Passage of Time
The figure above summarizes a new way of looking at
so-called "figurate numbers." The old way goes back
at least to the time of Pythagoras.
A more explicit presentation —
Comments Off on The Passage of Time
Sunday, August 6, 2023
Contra Gombrich
A search in this journal for Cornell + Warburg suggests
a review of the concept "iconology of the interval " . . .
Ikonologie des Zwischenraums —
"Yet if this Denkraum , this 'twilight region,' is where the artist and
emblem-maker invent, then, as Gombrich well knew, Warburg also
constantly regrets the 'loss' of this 'thought-space,' which he also
dubs the Zwischenraum and Wunschraum ."
— Memory, Metaphor, and Aby Warburg's Atlas of Images ,
Christopher D. Johnson, Cornell University Press, 2012, p. 56
Comments Off on Contra Gombrich
Monday, June 26, 2023
Tuesday, May 30, 2023
Seal 7 . . .
See John Baez this morning on Galois. Note that Baez's
report of Galois's dies natalis is in error.
Comments Off on Seal 7 . . .
Saturday, May 13, 2023
The Identity of an Entity
Comments Off on The Identity of an Entity
Tuesday, May 9, 2023
Alternate Film Title: Something, Somewhere, Sometime
"Long promised road
Flows to the source, gentle force"
— The Beach Boys
From a post of June 10, 2013 —
Comments Off on Alternate Film Title: Something, Somewhere, Sometime
Sunday, April 30, 2023
For Harlan Kane: The Walpurgisnacht Hallucination
Note that if the "compact Riemann surface" is a torus formed by
joining opposite edges of a 4×4 square array, and the phrase
"vector bundle" is replaced by "projective line," and so forth,
the above ChatGPT hallucination is not completely unrelated to
the following illustration from the webpage "galois.space" —
See as well the Cullinane diamond theorem.
Comments Off on For Harlan Kane: The Walpurgisnacht Hallucination
Friday, April 28, 2023
The Small Space Model
From the previous post, "The Large Language Model,"
a passage from Wikipedia —
"… sometimes large models undergo a 'discontinuous phase shift'
where the model suddenly acquires substantial abilities not seen
in smaller models. These are known as 'emergent abilities,' and
have been the subject of substantial study." — Wikipedia
Compare and contrast
this with the change undergone by a "small space model,"
that of the finite affine 4-space A with 16 points (a Galois tesseract ),
when it is augmented by an eight-point "octad." The 30 eight-point
hyperplanes of A then have a natural extension within the new
24-point set to 759 eight-point octads, and the 322,560 affine
automorphisms of the space expand to the 244,823,040 Mathieu
automorphisms of the 759-octad set — a (5, 8, 24) Steiner system.
For a visual analogue of the enlarged 24-point space and some remarks
on analogy by Simone Weil's brother, a mathematician, see this journal
on September 8 and 9, 2022.
Comments Off on The Small Space Model
Wednesday, April 19, 2023
New Types of Combinatorial Structure
Comments Off on New Types of Combinatorial Structure
Monday, April 10, 2023
Space
(Perspective Not as Symbolic Form)
From a post of June 8, 2014 —
See August 6, 2013 — Desargues via Galois.
Comments Off on Space
Thursday, April 6, 2023
Zero Sum
Related elementary mathematics from Google image searches —
Despite the extremely elementary nature of the above tables,
the difference between the binary addition of Boole and that
of Galois seems not to be widely known.
See "The Hunt for Galois October" and "In Memory of a Mississippi Coach."
Comments Off on Zero Sum
Tuesday, March 28, 2023
Onestone Parable
“You’re literally looking for like a one in a million thing.
You filter out the 999,999 of the boring ones, then
you’ve got something that’s weird, and then that’s worth
further exploration.”
— Quote from a mathematics story today at Gizmodo
A different "one in a million" mathematics story —
On Steiner Quadruple Systems of Order 16.
See also Galois Tesseract.
Comments Off on Onestone Parable
Wednesday, March 8, 2023
The Adapter
See Helen Mirren with a plastic 45-rpm record adapter.
Related Log24 posts — Galois Seals.
Comments Off on The Adapter
Monday, February 27, 2023
Thursday, February 23, 2023
“Where Whirled and Well”
— Conrad Aiken, Great Circle
And the light shone in darkness and
Against the Word the unstilled world still whirled
About the centre of the silent Word.
— T. S. Eliot, "Ash Wednesday"
About the Centre:
See also Dorm Room.
Comments Off on “Where Whirled and Well”
Wednesday, February 8, 2023
Text and Context: The Epiphany Shibboleth
The epigraph to Chapter 2 of Category Theory in Context by Emily Riehl —
[Maz16] Barry Mazur. Thinking about Grothendieck.
Notices of the AMS, 63(4):404–405, 2016.
The above epigraph in context, in a paper dated
January 6, 2016 (Epiphany) —
Also on Epiphany 2016 —
Wednesday, January 6, 2016
|
Comments Off on Text and Context: The Epiphany Shibboleth
Tuesday, February 7, 2023
The Graduate School of Design
The above cubic equation may also be written as
x3 – x – 1 = 0.
The equation occurred in my own work in 1985:
An architects' equation that appears also in Galois geometry.
For further details on the plastic number, see an article by
Siobhan Roberts on John Baez in The New York Times —
Comments Off on The Graduate School of Design
Interstices
Perhaps Crossan should have consulted Galois, not Piaget . . .
From Hermann Weyl's 1952 classic Symmetry —
"Galois' ideas, which for several decades remained
a book with seven seals but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."
Comments Off on Interstices
Friday, February 3, 2023
Rhyme Time
From Wednesday, St. Bridget's Day, 2023 —
Poetic meditation from The New Yorker today —
"If the tendency of rhyme, like that of desire,
is to pull distant things together
and force their boundaries to blur,
then the countervailing force in this book,
the one that makes it go, is the impulse
toward narrative, toward making sense of
the passage of time."
Comments Off on Rhyme Time
Thursday, February 2, 2023
“Here I Come Again”
From tonight's previous post —
"here I come again . . . the square root of minus one,
having terminated my humanities" —
Samuel Beckett, Stories and Texts for Nothing
(New York: Grove, 1967), 128.
|
From The French Mathematician 0
I had foreseen it all in precise detail. i = an imaginary being
Here, on this complex space, |
Related reading . . .
See also "William Lawvere, Category Theory, Hegel, Mao, and Code."
( https://www.reddit.com/r/socialistprogrammers/comments/m1oe88/
william_lawvere_category_theory_hegel_mao_and_code/ )
Also relating category theory and computation —
the interests of Lawvere and those of Davis — is
an article at something called The Topos Institute (topos.site) —
"Computation and Category Theory," by Joshua Meyers,
Wednesday, 10 Aug., 2022.
Meyers on Davis —
Comments Off on “Here I Come Again”
Wednesday, February 1, 2023
Variations in Memory of a Designer
Last updated at 22:46 PM ET on 1 February 2023.
Click for a designer's obituary.
Paraphrase for a road-sign collector:
See as well … Today's New York Times obituary
of the Harvard Business School Publishing
Director of Intellectual Property.
Comments Off on Variations in Memory of a Designer
Sunday, January 22, 2023
The Stillwell Dichotomies
| Number | Space |
| Arithmetic | Geometry |
| Discrete | Continuous |
Related literature —
From a "Finite Fields in 1956" post —
The Nutshell:
Related Narrative:
Comments Off on The Stillwell Dichotomies
Saturday, January 14, 2023
Châtelet on Weil — A “Space of Gestures”
|
From Gilles Châtelet, Introduction to Figuring Space Metaphysics does have a catalytic effect, which has been described in a very beautiful text by the mathematician André Weil: Nothing is more fertile, all mathematicians know, than these obscure analogies, these murky reflections of one theory in another, these furtive caresses, these inexplicable tiffs; also nothing gives as much pleasure to the researcher. A day comes when the illusion vanishes: presentiment turns into certainty … Luckily for researchers, as the fogs clear at one point, they form again at another.4 André Weil cuts to the quick here: he conjures these 'murky reflections', these 'furtive caresses', the 'theory of Galois that Lagrange touches … with his finger through a screen that he does not manage to pierce.' He is a connoisseur of these metaphysical 'fogs' whose dissipation at one point heralds their reforming at another. It would be better to talk here of a horizon that tilts thereby revealing a new space of gestures which has not as yet been elucidated and cut out as structure. 4 A. Weil, 'De la métaphysique aux mathématiques', (Oeuvres, vol. II, p. 408.) |
For gestures as fogs, see the oeuvre of Guerino Mazzola.
For some clearer remarks, see . . .
Illustrations of object and gestures
from finitegeometry.org/sc/ —
Object
Gestures
An earlier presentation
of the above seven partitions
of the eightfold cube:
|
|
Related material: Galois.space .
Comments Off on Châtelet on Weil — A “Space of Gestures”
Friday, December 30, 2022
Bullshit Studies: The View from East Lansing
Detail of the above screen (click to enlarge) —
See also this journal on the above date — June 10, 2021.
From this journal on May 6, 2009 —
A related picture of images that "reappear metamorphosed
in the coordinate system of the high region" —
(For the backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)
Comments Off on Bullshit Studies: The View from East Lansing
Saturday, December 24, 2022
Window as Matrix
Grids
Author: Rosalind Krauss
Source: October , Vol. 9 (Summer, 1979), pp. 50-64
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/778321
From page 59:
"Flowing and freezing; glace in French means glass, mirror, and ice;
transparency, opacity, and water. In the associative system
of symbolist thought this liquidity points in two directions.
First, towards the flow of birth-the amniotic fluid, the 'source'-
but then, towards the freezing into stasis or death-
the unfecund immobility of the mirror. For Mallarmé, particularly,
the window functioned as this complex, polysemic sign by which
he could also project the 'crystallization of reality into art.' 5
Mallarmé's Les Fenêtres dates from 1863;
Redon's most evocative window, Le Jour , appeared in 1891
in the volume Songes . If the window is this matrix of
ambi- or multivalence, and the bars of the windows-the grid-
are what help us to see, to focus on, this matrix, they are
themselves the symbol of the symbolist work of art.
They function as the multilevel representation through which
the work of art can allude, and even reconstitute, the forms of Being."
5 Robert G. Cohn, "Mallarmé's Windows," Yale French Studies ,
no. 54 (1977), 23-31.
Another evocative example — See Galois Window in this journal.
Comments Off on Window as Matrix
Thursday, December 22, 2022
Wednesday, December 14, 2022
Plan 9 From Moriarty
Some related mathematical windmills —
|
For the eight-limbed star at the top of the quaternion array She drew from her handbag a pale grey gleaming implement that looked by quick turns to me like a knife, a gun, a slim sceptre, and a delicate branding iron—especially when its tip sprouted an eight-limbed star of silver wire. “The test?” I faltered, staring at the thing. “Yes, to determine whether you can live in the fourth dimension or only die in it.” — Fritz Leiber, short story, 1959 |
See as well . . .
Comments Off on Plan 9 From Moriarty
Tuesday, December 13, 2022
In Memory of a Mississippi Coach
Comments Off on In Memory of a Mississippi Coach
Friday, November 25, 2022
“Open the pod bay doors, Hal.”
"Both Hal and the students were great interlocutors
with lots of ideas and differing perspectives."
—Leah Dickerman, MoMA, May 10, 2010, at
https://www.princeton.edu/news/2010/05/10/
class-snapshot-origins-abstraction
Later . . .
See as well Desargues + Galois.
Comments Off on “Open the pod bay doors, Hal.”
Thursday, November 24, 2022
The Drum Machine
"A struggling music producer sells his soul to a 1970s drum machine."
— Summary of a short film by Kevin Ignatius, "Hook Man."
The music producer pawns his current drum device
and acquires a demonic 1970s machine.
Artistic symbolism —
The 16-pad device at left may be viewed by enthusiasts of ekphrasis
as a Galois tesseract, and the machine at right as the voice of
Hal Foster, an art theorist who graduated from Princeton in 1977.
For an example of Foster's prose style, see
the current London Review of Books.
Comments Off on The Drum Machine
Sunday, November 6, 2022
The Paris Review Opening
Comments Off on The Paris Review Opening
Saturday, October 29, 2022
Boolean Halloween
From Log24 posts tagged Boole vs. Galois —
Kauffman‘s fixation on the work of Spencer-Brown is perhaps in part
due to Kauffman’s familiarity with Boolean algebra and his ignorance of
Galois geometry. See other posts now tagged Boole vs. Galois.
See also “A Four-Color Epic” (April 16, 2020).
Comments Off on Boolean Halloween
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)
Comments Off on Orthogonal Latin Triangles
Thursday, September 8, 2022
Analogy in Mathematics: Chevron Variations
André Weil in 1940 on analogy in mathematics —
| . "Once it is possible to translate any particular proof from one theory to another, then the analogy has ceased to be productive for this purpose; it would cease to be at all productive if at one point we had a meaningful and natural way of deriving both theories from a single one. In this sense, around 1820, mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with anguish and delight, to be guided by the analogy between the division of the circle (Gauss’s problem) and the division of elliptic functions. Today, we can easily show that both problems have a place in the theory of abelian equations; we have the theory (I am speaking of a purely algebraic theory, so it is not a matter of number theory in this case) of abelian extensions. Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is an historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time; at least in the Gitâ there is a slew of prayers (slokas) on the subject, each one more final than the previous ones." |
"The pleasure comes from the illusion" . . .
Exercise:
Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).
Note that the 4-row-2-column "brick" at left is quite
different from the other two bricks, which together
show chevron variations within a Galois tesseract —
Comments Off on Analogy in Mathematics: Chevron Variations
Saturday, September 3, 2022
1984 Revisited
Note the three quadruplets of parallel edges in the 1984 figure above.
The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —
Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.
Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —
And then, more simply, there is the Galois tesseract —
For parts of my own world in June 2010, see this journal for that month.
The above Galois tesseract appears there as follows:
See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post.
Comments Off on 1984 Revisited
Sunday, August 7, 2022
Monday, August 1, 2022
Enowning
Related material — The Eightfold Cube.
See also . . .
"… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty."
— Sir Michael Atiyah, “The Art of Mathematics”
in the AMS Notices , January 2010
Comments Off on Enowning
Saturday, July 30, 2022
Modal Diamond Box
|
A mnemonic from a course titled
“Traditionally, there are two modalities, namely,
|
(square) for necessarily
(diamond) for possibly.For less rigorous remarks, search Log24 for Modal Diamond Box.
Comments Off on Modal Diamond Box
Tuesday, July 5, 2022
For Ron Howard, Tom Hanks, and Dan Brown — Symbology!
Comments Off on For Ron Howard, Tom Hanks, and Dan Brown — Symbology!
Thursday, June 23, 2022
The Nutshell Suite
The above is a summary of
Pythagorean philosophy
reposted here on . . .
Battle of the Nutshells:
From a much larger nutshell
on the above Pythagorean date—
Now let's dig a bit deeper into history . . .
Comments Off on The Nutshell Suite
Wednesday, June 22, 2022
Code Wars: “Use the Source, Luke.”
Click the above galaxy for a larger image.
"O God, I could be bounded in a nutshell
and count myself a king of infinite space,
were it not that I have bad dreams." — Hamlet
Battle of the Nutshells —
From a much larger nutshell
on the above code date—
Comments Off on Code Wars: “Use the Source, Luke.”
Tuesday, May 31, 2022
A Mad Night’s Work*
Last night's posts on triangles, and today's anniversary of the
death of Evariste Galois, suggest a review . . .
"Take triangles, perhaps" . . . as a category.
And then . . . take squares, perhaps, as another category,
and then . . . find a suitable "translation machine."
See "Square Triangles."
* Title adapted from a 2001 essay by Pierre Cartier.
Comments Off on A Mad Night’s Work*
Saturday, May 28, 2022
Grothendieck at Chapman …
Last two days of the conference, May 27 and 28, 2022 —
|
27th Friday
9:00 – 10:00 Andrés Villaveces (Univ. Nacional de Colombia):
10:00 – 11:00 Olivia Caramello (Univ. of Insubria; by Zoom): 1:00 – 11:15 Coffee Break
1:15 – 12:15 Mike Shulman (Univ. of San Diego):
12:15 – 1:15 José Gil-Ferez (Chapman Univ.) 1:15 – 2:30 Lunch
2:30 – 3:30 Oumar Wone (Chapman) :
3:30 – 4:30 Claudio Bartocci (Univ. of Genova):
4:30 – 5:30 Christian Houzel (IUFM de Paris): 28th Saturday
9:00 – 10:00 Silvio Ghilardi (Univ. degli Studi, Milano):
10:00 – 11:00 Matteo Viale (Univ. of Turin; by zoom): 11:00 – 11:15 Coffee Break
11:15 – 12:15 Benjamin Collas (RIMS, Kyoto Univ.):
12:15 – 1:15 Closing: general discussion |
Comments Off on Grothendieck at Chapman …
Friday, May 27, 2022
Great Escapes
.jpg){kind=link}
Comments Off on Great Escapes
Plan 9 from Disney
"With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series
Scholium —
Abstracting from narrative to structure, and from structure
to pure number, the Tablet of Ahkmenrah represents the
number 9 and the Cube of Rubik represents the number 27.
Returning from pure abstract numbers to concrete representations,
9 yields the structures in Log24 posts tagged Triangle.graphics,
and 27 yields a Galois cube .
Comments Off on Plan 9 from Disney
Tuesday, May 24, 2022
Playing the Palace
From a Jamestown (NY) Post-Journal article yesterday on
"the sold-out 10,000 Maniacs 40th anniversary concert at
The Reg Lenna Center Saturday" —
" 'The theater has a special place in our hearts. It’s played
a big part in my life,' Gustafson said.
Before being known as The Reg Lenna Center for The Arts,
it was formerly known as The Palace Theater. He recalled
watching movies there as a child…."
This, and the band's name, suggest some memories perhaps
better suited to the cinematic philosophy behind "Plan 9 from
Outer Space."
"With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series
The above 3×3 Tablet of Ahkmenrah image comes from
a Log24 search for the finite (i.e., Galois) field GF(3) that
was, in turn, suggested by last night's post "Making Space."
See as well a mysterious document from a website in Slovenia
that mentions a 3×3 array "relating to nine halls of a mythical
palace where rites were performed in the 1st century AD" —
Comments Off on Playing the Palace
Saturday, May 7, 2022
Interality Meets the Seven Seals
Related material — Posts tagged Interality and Seven Seals.
From Hermann Weyl's 1952 classic Symmetry —
"Galois' ideas, which for several decades remained
a book with seven seals but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."
Comments Off on Interality Meets the Seven Seals
Monday, April 25, 2022
Annals of Mathematical History
Some related remarks —
Comments Off on Annals of Mathematical History
Monday, April 18, 2022
Iconic Simplicity
Comments Off on Iconic Simplicity