“Galois space” is now a domain name: galois.space.
Friday, April 3, 2020
Sunday, November 19, 2017
Galois Space
This is a sequel to yesterday's post Cube Space Continued.
Tuesday, May 31, 2016
Tuesday, January 12, 2016
Harmonic Analysis and Galois Spaces
The above sketch indicates, in a vague, handwaving, fashion,
a connection between Galois spaces and harmonic analysis.
For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.
Sunday, March 10, 2013
Galois Space
The 16point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 16_{6} configuration.
Some closely related material:
 Wolfgang Kühnel,
"Minimal Triangulations of Kummer Varieties,"
Abh. Math. Sem. Univ. Hamburg 57, 720 (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).)
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.
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—
Thursday, July 12, 2012
Galois Space
An example of lines in a Galois space * —
The 35 lines in the 3dimensional Galois projective space PG(3,2)—
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3set 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, 488499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
Wednesday, May 2, 2018
Galois’s Space
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
Tuesday, May 26, 2020
Introduction to Cyberspace
Or approaching.
On the Threshold:
Click the search result above for the July 1982 Omni
story that introduced into fiction the term “cyberspace.”
Part of a page from the original Omni version —
For some other kinds of space, see my notes from the 1980’s.
Some related remarks on space (and illustrated clams) —
— George Steiner, “A Death of Kings,” The New Yorker ,
September 7, 1968, pp. 130 ff. The above is from p. 133.
See also Steiner on space, algebra, and Galois.
Thursday, April 16, 2020
Wednesday, January 1, 2020
Exploring Inner Space* at The New York Times
From Corrections: Jan. 1, 2020 —
The astronomy article, by Dennis Overbye, is dated Dec. 23* (a Monday).
The above reference to "Tuesday" is explained by the fine print
at the bottom of the Science Times article — "A version of this article
appears in print on [Tuesday] , Section D, Page 6 of the
New York edition with the headline: In Battle of Giant Telescopes,
Outlook for the U.S. Dims."
From the article as quoted on Thursday, Dec. 26,
at https://uclafacultyassociation.blogspot.com —
"Now, as the wheels of the academic and government bureaucracy begin to turn, many American astronomers worry that they are following in the footsteps of their physicist colleagues. In 1993, Congress canceled the Superconducting Super Collider, and the United States ceded the exploration of inner space to Europe and CERN, which built the Large Hadron Collider, 27 miles in diameter, where the longsought Higgs boson was eventually discovered. The United States no longer builds particle accelerators. There could come a day, soon, when Americans no longer build giant telescopes. That would be a crushing disappointment to a handful of curious humans stuck on Earth, thirsting for cosmic grandeur. In outer space, nobody can hear you cry." Source: https://www.nytimes.com/2019/12/23/science/telescopesmagellanhawaiiastronomy.html 
Related material from this journal on April 2, 2019 —
Cover design by Greg Stadnyk, available in an animated gif.
* See also this journal on Dec. 23.
Monday, December 2, 2019
Gray Space
See as well a search for Gray Space in this journal.
Related material: The Schwartz Omega .
“Looking carefully at Golay’s code
is like staring into the sun.”
Sunday, June 16, 2019
Monday, June 3, 2019
Art Wars for Spaceheads
Friday, May 3, 2019
The Structure of Story Space
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A Permanent Order of Wondertale Elements
In Vol. I of Structural Anthropology , p. 209, I have shown that this analysis alone can account for the double aspect of time representation in all mythical systems: the narrative is both “in time” (it consists of a succession of events) and “beyond” (its value is permanent). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 9798. p. 108). If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns). 
… Or by congruent quartersections.
Thursday, May 2, 2019
Story Space
Monday, March 25, 2019
Espacement
(Continued from the previous post.)
InBetween "Spacing" and the "Chôra " (Ch. 2 in Henk Oosterling & Ewa Plonowska Ziarek (Eds.), Intermedialities: Philosophy, Arts, Politics , Lexington Books, October 14, 2010) "The term 'spacing' ('espacement ') is absolutely central to Derrida's entire corpus, where it is indissociable from those of différance (characterized, in the text from 1968 bearing this name, as '[at once] spacing [and] temporizing' ^{1}), writing (of which 'spacing' is said to be 'the fundamental property' ^{2}) and deconstruction (with one of Derrida's last major texts, Le Toucher: JeanLuc Nancy , specifying 'spacing ' to be 'the first word of any deconstruction' ^{3})." 1 Jacques Derrida, “La Différance,” in Marges – de la philosophie (Paris: Minuit, 1972), p. 14. Henceforth cited as D . 2 Jacques Derrida, “Freud and the Scene of Writing,” trans. A. Bass, in Writing and Difference (Chicago: University of Chicago Press, 1978), p. 217. Henceforth cited as FSW . 3 Jacques Derrida, Le Toucher, JeanLuc Nancy (Paris: Galilée, 2000), p. 207. . . . . "… a particularly interesting point is made in this respect by the French philosopher, Michel Haar. After remarking that the force Derrida attributes to différance consists simply of the series of its effects, and is, for this reason, 'an indefinite process of substitutions or permutations,' Haar specifies that, for this process to be something other than a simple 'actualisation' lacking any real power of effectivity, it would need “a soubassement porteur ' – let’s say a 'conducting underlay' or 'conducting medium' which would not, however, be an absolute base, nor an 'origin' or 'cause.' If then, as Haar concludes, différance and spacing show themselves to belong to 'a pure Apollonism' 'haunted by the groundless ground,' which they lack and deprive themselves of,^{16} we can better understand both the threat posed by the 'figures' of space and the mother in the Timaeus and, as a result, Derrida’s insistent attempts to disqualify them. So great, it would seem, is the menace to différance that Derrida must, in a 'properly' apotropaic gesture, ward off these 'figures' of an archaic, chthonic, spatial matrix in any and all ways possible…." 16 Michel Haar, “Le jeu de Nietzsche dans Derrida,” Revue philosophique de la France et de l’Etranger 2 (1990): 207227. . . . . … "The conclusion to be drawn from Democritus' conception of rhuthmos , as well as from Plato's conception of the chôra , is not, therefore, as Derrida would have it, that a differential field understood as an originary site of inscription would 'produce' the spatiality of space but, on the contrary, that 'differentiation in general' depends upon a certain 'spatial milieu' – what Haar would name a 'groundless ground' – revealed as such to be an 'inbetween' more 'originary' than the play of differences it informs. As such, this conclusion obviously extends beyond Derrida's conception of 'spacing,' encompassing contemporary philosophy's continual privileging of temporization in its elaboration of a preontological 'opening' – or, shall we say, 'inbetween.' 
For permutations and a possible "groundless ground," see
the eightfold cube and group actions both on a set of eight
building blocks arranged in a cube (a "conducting base") and
on the set of seven natural interstices (espacements ) between
the blocks. Such group actions provide an elementary picture of
the isomorphism between the groups PSL(2,7) (acting on the
eight blocks) and GL(3,2) (acting on the seven interstices).
Espacements
For the Church of Synchronology —
See also, from the reported publication date of the above book
Intermedialities , the Log24 post Synchronicity.
Sunday, March 24, 2019
Espacement: Geometry of the Interstice in Literary Theory
"You said something about the significance of spaces between
elements being repeated. Not only the element itself being repeated,
but the space between. I'm very interested in the space between.
That is where we come together." — Peter Eisenman, 1982
https://www.parrhesiajournal.org/ Parrhesia No. 3 • 2007 • 22–32
(Up) Against the (In) Between: Interstitial Spatiality by Clare Blackburne Blackburne — www.parrhesiajournal.org 24 — "The excessive notion of espacement as the resurgent spatiality of that which is supposedly ‘without space’ (most notably, writing), alerts us to the highly dynamic nature of the interstice – a movement whose discontinuous and ‘aberrant’ nature requires further analysis." Blackburne — www.parrhesiajournal.org 25 — "Espacement also evokes the ambiguous figure of the interstice, and is related to the equally complex derridean notions of chora , différance , the trace and the supplement. Derrida’s reading of the Platonic chora in Chora L Works (a series of discussions with the architect Peter Eisenman) as something which defies the logics of noncontradiction and binarity, implies the internal heterogeneity and instability of all structures, neither ‘sensible’ nor ‘intelligible’ but a third genus which escapes conceptual capture.^{25} Crucially, chora , spacing, dissemination and différance are highly dynamic concepts, involving hybridity, an ongoing ‘corruption’ of categories, and a ‘bastard reasoning.’^{26} Derrida identification of différance in Margins of Philosophy , as an ‘unappropriable excess’ that operates through spacing as ‘the becomingspace of time or the becomingtime of space,’^{27} chimes with his description of chora as an ‘unidentifiable excess’ that is ‘the spacing which is the condition for everything to take place,’ opening up the interval as the plurivocity of writing in defiance of ‘origin’ and ‘essence.’^{28} In this unfolding of différance , spacing ‘insinuates into presence an interval,’^{29} again alerting us to the crucial role of the interstice in deconstruction, and, as Derrida observes in Positions , its impact as ‘a movement, a displacement that indicates an irreducible alterity’: ‘Spacing is the impossibility for an identity to be closed on itself, on the inside of its proper interiority, or on its coincidence with itself. The irreducibility of spacing is the irreducibility of the other.’^{30}"
25. Quoted in Jeffrey Kipnis and Thomas Leeser, eds., 26. Ibid, 25.
27. Derrida, Margins of Philosophy. 28. Derrida, Chora L Works , 19 and 10. 29. Ibid, 203. 30. Derrida, Positions , 94. 
Thursday, January 24, 2019
Name Space
A correction at Wikipedia (Click to enlarge.) —
That this correction is needed indicates that the phrase
"Cullinane space" might be useful. (Click to enlarge.)
Tuesday, January 22, 2019
Monday, December 17, 2018
Tales from Story Space
"Kiernan Brennan Shipka (born November 10, 1999)
is an American actress. She is best known for starring as
Sabrina Spellman on the Netflix supernatural horror series
Chilling Adventures of Sabrina (2018–present)." — Wikipedia
As noted here earlier, Shipka turned 18 on Nov. 10 last year.
From Log24 on that date —
Another 18th birthday in Story Space —
Sunday, December 9, 2018
Quaternions in a Small Space
The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.
Click to enlarge —
Three links from the above finitegeometry.org webpage on the
quaternion group —

Visualizing GL(2,p) — A 1985 note illustrating group actions
on the 3×3 (ninefold) square. 
Another 1985 note showing group actions on the 3×3 square
transferred to the 2x2x2 (eightfold) cube.  Quaternions in an Affine Galois Plane — A webpage from 2010.
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
Sunday, November 4, 2018
Kristen vs. the Space Witch*
* We know the former. There is no shortage of candidates for the latter.
Saturday, November 3, 2018
The Space Theory of Truth
Earlier posts have discussed the "story theory of truth"
versus the "diamond theory of truth," as defined by
Richard Trudeau in his 1987 book The NonEuclidean Revolution.
In a New York Times opinion piece for tomorrow's print edition,*
novelist Dara Horn touched on what might be called
"the space theory of truth."
When they return to synagogue, mourners will be greeted
with more ancient words: “May God comfort you
among the mourners of Zion and Jerusalem.”
In that verse, the word used for God is hamakom —
literally, “the place.” May the place comfort you.
[Link added.]
The Source —
See Dara Horn in this journal, as well as Makom.
* "A version of this article appears in print on ,
on Page A23 of the New York edition with the headline:
American Jews Know This Story."
Tuesday, October 30, 2018
Story Structure, Story Space
Constance Grady at Vox today on a new Netflix series —
We don’t yet have a story structure that allows witches to be powerful for long stretches of time without men holding them back. And what makes the new Sabrina so exciting is that it seems to be trying to build that story structure itself, in real time, to find a way to let Sabrina have her power and her freedom. It might fail. But if it does, it will be a glorious and worthwhile failure — the type that comes with trying to pioneer a new kind of story. 
See also Story Space in this journal.
Tuesday, October 23, 2018
Monday, October 22, 2018
Story Space
Sunday, July 22, 2018
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, 
Sunday, March 4, 2018
The Square Inch Space: A Brief History
Friday, September 15, 2017
Space Art
Silas in "Equals" (2015) —
Ever since we were kids it's been drilled into us that …
Our purpose is to explore the universe, you know.
Outer space is where we'll find …
… the answers to why we're here and …
… and where we come from.
Related material —
See also Galois Space in this journal.
Saturday, July 8, 2017
Desargues and Galois in Japan
Related material now available online —
A less businessoriented sort of virtual reality —
For example, "A very important configuration is obtained by
taking the plane section of a complete space fivepoint."
(Veblen and Young, 1910, p. 39)—
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.
Friday, April 28, 2017
A Generation Lost in Space
The title is from Don McLean's classic "American Pie."
A Finite Projective Space —
A NonFinite Projective Space —
Sunday, April 16, 2017
Art Space Paradigm Shift
This post's title is from the tags of the previous post —
The title's "shift" is in the combined concepts of …
Space and Number
From Finite Jest (May 27, 2012):
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —
"Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange"
— io9 , July 29, 2016
" 'This man comes from a binary universe
where it’s all about logic,' the actor told us
at San Diego ComicCon . . . .
'And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.' "
[Typo now corrected, except in a comment.]
Thursday, April 13, 2017
Wednesday, March 29, 2017
Art Space Illustrated
Another view of the previous post's art space —
More generally, see Solomon's Cube in Log24.
See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above
cube arrangement from "Affine groups on small binary spaces" (1984).
Art Space, Continued
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson today
From Log24 posts tagged Art Space —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —
Two such considerations —
Tuesday, January 3, 2017
Cultist Space
The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —
The previous post illustrated a 3×3 grid. That cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.
Monday, September 26, 2016
Thursday, June 30, 2016
Rubik vs. Galois: Preconception vs. Preconception
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 .
Wednesday, June 29, 2016
Space Jews
For the Feast of SS. Peter and Paul —
In memory of Alvin Toffler and Simon Ramo,
a review of figures from the midnight that began
the date of their deaths, June 27, 2016 —
The 3×3×3 Galois Cube
See also Rubik in this journal.
Friday, April 8, 2016
Space Cross
For George Orwell
Illustration from a book on mathematics —
This illustrates the Galois space AG(4,2).
For some related spaces, see a note from 1984.
"There is such a thing as a space cross."
— Saying adapted from a youngadult novel
Ogdoads: A Space Odyssey
"Like the Valentinian Ogdoad— a selfcreating theogonic system
of eight Aeons in four begetting pairs— the projected eightfold work
had an esoteric, gnostic quality; much of Frye's formal interest lay in
the 'schematosis' and fearful symmetries of his own presentations."
— From p. 61 of James C. Nohrnberg's "The Master of the Myth
of Literature: An Interpenetrative Ogdoad for Northrop Frye,"
Comparative Literature , Vol. 53 No. 1, pp. 5882, Duke University
Press (quarterly, January 2001)
See also Two by Four in this journal.
Wednesday, February 17, 2016
“Blank Space” Accolades
A post in memory of British theatre director Peter Wood,
who reportedly died on February 11, 2016.
From the date of the director's death —
"Leave a space." — Tom Stoppard
Monday, January 11, 2016
Space Oddity
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for "Galois spaces" + "Boolean spaces"
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
"Harmonic Analysis of Switching Functions" ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
"Galois Switching Functions and Their Applications,"
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 4227 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239249, Mar. 1978
"Switching functions constructed by Galois extension fields,"
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981
An illustration from the Lechner paper above —
"There is such a thing as harmonic analysis of switching functions."
— Saying adapted from a youngadult novel
Saturday, October 24, 2015
Two Views of Finite Space
The following slides are from lectures on "Advanced Boolean Algebra" —
The small Boolean spaces above correspond exactly to some small
Galois spaces. These two names indicate approaches to the spaces
via Boolean algebra and via Galois geometry .
A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —
" 'All you need to do is give me your soul: give up geometry
and you will have this marvellous machine.' (Nowadays you
can think of it as a computer!) "
Related material — The article "Diamond Theory" in the journal
Computer Graphics and Art , Vol. 2 No. 1, February 1977. That
article, despite the word "computer" in the journal's title, was
much less about Boolean algebra than about Galois geometry .
For later remarks on diamond theory, see finitegeometry.org/sc.
Wednesday, October 21, 2015
Algebra and Space
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
Friday, September 4, 2015
Friday, August 14, 2015
Discrete Space
(A review)
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Wednesday, May 13, 2015
Space
Notes on space for day 13 of May, 2015 —
The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."
Related poetic material:
The ninefold square and Apollo, as well as …
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 89: Substructures of S(5, 8, 24) An octad is a block of S(5, 8, 24). Theorem 5.1
Let B_{0} be a fixed octad. The 30 octads disjoint from B_{0}
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.
Tuesday, November 25, 2014
EuclideanGalois 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 3space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order5 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 3space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 1617 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
Monday, September 22, 2014
Thursday, March 27, 2014
Diamond Space
Definition: A diamond space — informal phrase denoting
a subspace of AG(6, 2), the sixdimensional affine space
over the twoelement Galois field.
The reason for the name:
Click to enlarge.
Thursday, February 27, 2014
Sacred Space, continued
"An image comes to mind of a white, ideal space
that, more than any single picture, may be the
archetypal image of 20thcentury art."
— Brian O'Doherty, "Inside the White Cube"
Cube spaces exist also in mathematics.
Tuesday, December 3, 2013
Diamond Space
A new website illustrates its URL.
See DiamondSpace.net.
Tuesday, July 16, 2013
Space Itself
"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space …
but in space itself?"
— Megan Garber in The Atlantic , Aug. 16, 2012
One approach:
"There is such a thing as a tesseract" and
Diamond Theory in 1937.
See, too, Baez in this journal.
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 Galoisfield 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 pre1976
coordinatization of a 4×4 array by the 16 elements
of the vector 4space 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.
Wednesday, January 16, 2013
Space Race
Japanese character
for "field"
This morning's leading
New York Times obituaries—
For other remarks on space, see
Galois + Space in this journal.
Saturday, September 22, 2012
Occupy Space
"The word 'space' has, as you suggest, a large number of different meanings."
— Nanavira Thera in [Early Letters. 136] 10.xii.1958
From that same letter (links added to relevant Wikipedia articles)—
Space (ākāsa) is undoubtedly used in the Suttas
Your second letter seems to suggest that the space 
A simpler metaphysical system along the same lines—
The theory, he had explained, was that the persona
— The Gameplayers of Zan , 
"I am glad you have discovered that the situation is comical:
ever since studying Kummer I have been, with some difficulty,
refraining from making that remark."
— Nanavira Thera, [Early Letters, 131] 17.vii.1958
Sunday, July 29, 2012
The Galois Tesseract
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vectorspace 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 vectorspace 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 "SelfDual 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 ConwaySloane diagram.
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.
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, (18921893), 215249))
Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143144)—
"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 structureendowed entity Σ try to determine is group of automorphisms , the group of those elementwise 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
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
twothirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79TA37, 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 4space over the 2element 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 4point 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
Latinsquare 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 M_{24}, 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 misnamed 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: 345353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Thursday, September 1, 2011
Friday, April 22, 2011
Romancing the Hyperspace
For the title, see Palm Sunday.
"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987
From this date (April 22) last year—
Richard J. Trudeau in The NonEuclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"– "… Plato and Kant, and most of the philosophers and scientists in the 2200year interval between them, did share the following general presumptions: (1) Diamonds– informative, certain truths about the world– exist. Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry." Trudeau's book was published in 1987. The nonEuclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory." Although nonEuclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds. * "NonEuclidean" here means merely "other than Euclidean." No violation of Euclid's parallel postulate is implied. 
Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."
Geometry, a part of pure mathematics, is not about the world. See G. H. Hardy, A Mathematician's Apology .
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 coauthor, 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 1234, 1324, 1423. Its automorphism group is the symmetric group S_{4}, and acts as S_{3} 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.
Monday, June 21, 2010
Cube Spaces
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Example 1— The 2×2×2 Cube—
also known as the eightfold cube—
Group actions on the eightfold cube, 1984—
Version by Laszlo Lovasz et al., 2003—
Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
Example 2— The 3×3×3 Cube
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
Example 3— The 4×4×4 Cube
A note from 27 years ago today—
As far as I know, this version of the
groupactions theorem has not yet been ripped off.
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 finitegeometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)
A group of collineations** of the 21point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4space over the twoelement 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
** G_{20160} in Mitchell 1910, LF(3,2^{2}) in Edge 1965
— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
of the Finite Projective Plane PG(2,2^{2}),"
Princeton Ph.D. dissertation (1910)
— Edge, W. L., "Some Implications of the Geometry of
the 21Point Plane," Math. Zeitschr. 87, 348362 (1965)
Saturday, March 13, 2010
Space Cowboy
From yesterday's Seattle Times—
According to police, employees of a Second Avenue mission said the suspect, clad in black and covered in duct tape, had come into the mission "and threatened to blow the place up." He then told staffers "that he was a vampire and wanted to eat people."
The man… also called himself "a space cowboy"….
This suggests two film titles…
and Apollo's 13—
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–
Thursday, March 5, 2020
“Generated by Reflections”
See the title in this journal.
Such generation occurs both in Euclidean space …
… and in some Galois spaces —
In Galois spaces, some care must be taken in defining "reflection."
Monday, December 2, 2019
Aesthetics at Harvard
"What the piece of art is about is the gray space in the middle."
— David Bowie, as quoted in the above Crimson piece.
Bowie's "gray space" is the space between the art and the beholder.
I prefer the gray space in the following figure —
Context: The Trinity Stone (Log24, June 4, 2018).
Monday, October 15, 2018
For Zingari Shoolerim*
The structure at top right is that of the
ROMAORAMMAROAMOR square
in the previous post.
* "Zingari shoolerim" is from
Finnegans Wake .
Saturday, September 29, 2018
“Ikonologie des Zwischenraums”
The title is from Warburg. The Zwischenraum lines and shaded "cuts"
below are to be added together in characteristic two, i.e., via the
settheoretic symmetric difference operator.
Monday, August 27, 2018
Geometry and Simplicity
From …
Thinking in Four Dimensions
By Dusa McDuff
"I’ve got the rather foolhardy idea of trying to explain
to you the kind of mathematics I do, and the kind of
ideas that seem simple to me. For me, the search
for simplicity is almost synonymous with the search
for structure.
I’m a geometer and topologist, which means that
I study the structure of space …
. . . .
In each dimension there is a simplest space
called Euclidean space … "
— In Roman Kossak, ed.,
Simplicity: Ideals of Practice in Mathematics and the Arts
(Kindle Locations 705710, 735). Kindle Edition.
For some much simpler spaces of various
dimensions, see Galois Space in this journal.
Monday, June 4, 2018
The Trinity Stone Defined
“Unsheathe your dagger definitions.” — James Joyce, Ulysses
The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .
Tuesday, May 2, 2017
Image Albums
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
Saturday, June 18, 2016
Midnight in Herald Square
In memory of New Yorker artist Anatol Kovarsky,
who reportedly died at 97 on June 1.
Note the Santa, a figure associated with Macy's at Herald Square.
See also posts tagged Herald Square, as well as the following
figure from this journal on the day preceding Kovarsky's death.
A note related both to Galois space and to
the "Herald Square"tagged posts —
"There is such a thing as a length16 sequence."
— Saying adapted from a youngadult novel.
Sunday, May 8, 2016
The Three Solomons
Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick's Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.
A quote from LeWitt indicates the depth of the word "conceptual"
in his approach to "conceptual art."
From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:
THE SQUARE AND THE CUBE "The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two and threedimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed." "Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America 55, No. 4 (JulyAugust 1967): 54. (LeWitt’s contribution was originally untitled.)" 
See also the Cullinane models of some small Galois spaces —
Friday, May 6, 2016
Monday, January 5, 2015
Gitterkrieg*
Wednesday, March 13, 2013

"I pondered deeply, then, over the
adventures of the jungle. And after
some work with a colored pencil
I succeeded in making my first drawing.
My Drawing Number One.
It looked something like this:
I showed my masterpiece to the
grownups, and asked them whether
the drawing frightened them.
But they answered: 'Why should
anyone be frightened by a hat?'"
* For the title, see Plato Thanks the Academy (Jan. 3).
Sunday, September 14, 2014
Sensibility
Structured gray matter:
Graphic symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Sunday, August 31, 2014
Sunday School
The Folding
Cynthia Zarin in The New Yorker , issue dated April 12, 2004—
“Time, for L’Engle, is accordionpleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”
The geometry of the 4×4 square array is that of the
3dimensional projective Galois space PG(3,2).
This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc. on
15 June 1974). Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.
Some history:
Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.
[Rewritten for clarity on Sept. 3, 2014.]
Thursday, July 17, 2014
Paradigm Shift:
Continuous Euclidean space to discrete Galois space*
Euclidean space:
Counting symmetries in Euclidean space:
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
* For related remarks, see posts of May 2628, 2012.
Wednesday, July 16, 2014
Wednesday, May 21, 2014
Through the Vanishing Point*
Marshall McLuhan in "Annie Hall" —
"You know nothing of my work."
Related material —
"I need a photo opportunity
I want a shot at redemption
Don't want to end up a cartoon
In a cartoon graveyard"
— Paul Simon
It was a dark and stormy night…
— Page 180, Logicomix
A photo opportunity for Whitehead
(from Romancing the Cube, April 20, 2011)—
See also Absolute Ambition (Nov. 19, 2010).
* For the title, see Vanishing Point in this journal.
Friday, March 21, 2014
Three Constructions of the Miracle Octad Generator
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 TurynCurtis construction
from the University of Cambridge —
— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M_{24},” in slides for lectures 18 from lectures
at Cambridge in 20102011 on “Sporadic and Related Groups.”
See also the Parker lectures of 20122013 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+4partitions of an 8set with the 35 lines of the projective 3space
over the 2element 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 22March 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
S_{3} 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 “byhand” 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, not by hand (such as Turyn’s), 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 —
Friday, March 7, 2014
Kummer Varieties
The Dream of the Expanded Field continues…
From Klein's 1893 Lectures on Mathematics —
"The varieties introduced by Wirtinger may be called Kummer varieties…."
— E. Spanier, 1956
From this journal on March 10, 2013 —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —
Two such considerations —
Update of 10 PM ET March 7, 2014 —
The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64point vector space and
to the Weyl group of type E_{7}, W (E_{7}):
The Cayley reference is to "Algorithm for the characteristics of the
triple ϑfunctions," Journal für die Reine und Angewandte
Mathematik 87 (1879): 165169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441445
of Volume 10 of his Collected Mathematical Papers .
Wednesday, February 5, 2014
Mystery Box II
Continued from previous post and from Sept. 8, 2009.
Examination of the box's contents does not solve
the contents' real mystery. That requires knowledge
of the nonEuclidean geometry of Galois space.
In this case, without that knowledge, prattle (as in
today's online New York Times ) about creativity and
"thinking outside the box" is pointless.
Thursday, November 7, 2013
Pattern Grammar
Yesterday afternoon's post linked to efforts by
the late Robert de Marrais to defend a mathematical
approach to structuralism and kaleidoscopic patterns.
Two examples of nonmathematical discourse on
such patterns:
1. A Royal Society paper from 2012—
Click the above image for related material in this journal.
2. A book by Junichi Toyota from 2009—
Kaleidoscopic Grammar: Investigation into the Nature of Binarism
I find such nonmathematical approaches much less interesting
than those based on the mathematics of reflection groups .
De Marrais described the approaches of Vladimir Arnold and,
earlier, of H. S. M. Coxeter, to such groups. These approaches
dealt only with groups of reflections in Euclidean spaces.
My own interest is in groups of reflections in Galois spaces.
See, for instance, A Simple Reflection Group of Order 168.
Galois spaces over fields of characteristic 2 are particularly
relevant to what Toyota calls binarism .
Thursday, July 4, 2013
Declaration of Independent
"Classical Geometry in Light of Galois Geometry"
is now available at independent.academia.edu.
Related commentary: Yesterday's post Vision
and a post of February 21, 2013: Galois Space.
Saturday, March 16, 2013
The Crosswicks Curse
From the prologue to the new Joyce Carol Oates
novel Accursed—
"This journey I undertake with such anticipation
is not one of geographical space but one of Time—
for it is the year 1905 that is my destination.
1905!—the very year of the Curse."
Today's previous post supplied a fanciful link
between the Crosswicks Curse of Oates and
the Crosswicks tesseract of Madeleine L'Engle.
The Crosswicks Curse according to L'Engle
in her classic 1962 novel A Wrinkle in Time —
"There is such a thing as a tesseract."
A tesseract is a 4dimensional hypercube that
(as pointed out by Coxeter in 1950) may also
be viewed as a 4×4 array (with opposite edges
identified).
Meanwhile, back in 1905…
For more details, see how the Rosenhain and Göpel tetrads occur naturally
in the diamond theorem model of the 35 lines of the 15point projective
Galois space PG(3,2).
See also Conwell in this journal and George Macfeely Conwell in the
honors list of the Princeton Class of 1905.
Wednesday, March 13, 2013
Blackboard Jungle
From a review in the April 2013 issue of
Notices of the American Mathematical Society—
"The author clearly is passionate about mathematics
as an art, as a creative process. In reading this book,
one can easily get the impression that mathematics
instruction should be more like an unfettered journey
into a jungle where an individual can make his or her
own way through that terrain."
From the book under review—
"Every morning you take your machete into the jungle
and explore and make observations, and every day
you fall more in love with the richness and splendor
of the place."
— Lockhart, Paul (20090401). A Mathematician's Lament:
How School Cheats Us Out of Our Most Fascinating and
Imaginative Art Form (p. 92). Bellevue Literary Press.
Kindle Edition.
Related material: Blackboard Jungle in this journal.
See also Galois Space and Solomon's Mines.
Wednesday, March 6, 2013
Midnight in Pynchon*
"It is almost as though Pynchon wishes to
repeat the grand gesture of Joyce’s Ulysses…."
— Vladimir Tasic on Pynchon's Against the Day
Related material:
Tasic's Mathematics and the Roots of Postmodern Thought
and Michael Harris's "'Why Mathematics?' You Might Ask"
*See also Occupy Galois Space and Midnight in Dostoevsky.
Tuesday, February 19, 2013
Configurations
Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.
My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010.
For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books and Amazon.com):
For a similar 1998 treatment of the topic, see Burkard Polster's
A Geometrical Picture Book (Springer, 1998), pp. 103104.
The PisanskiServatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's
symmetry planes , contradicting the usual use of of that term.
That argument concerns the interplay between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois structures as a guide to redescribing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)
Related material: Remarks on configurations in this journal
during the month that saw publication of the PisanskiServatius book.
* Earlier guides: the diamond theorem (1978), similar theorems for
2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
(1985). See also Spaces as Hypercubes (2012).
Wednesday, January 16, 2013
Medals
National…
International…
Click medal for some background. The medal may be regarded
as illustrating the 16point Galois space. (See previous post.)
Related material: Jews in Hyperspace.
Saturday, January 5, 2013
Vector Addition in a Finite Field
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4space over GF(2)—
The same field, again disguised as an affine 4space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vectorspace structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“SelfDual Configurations and Regular Graphs.”
Monday, August 29, 2011
Many = Six.
A comment today on yesterday's New York Times philosophy column "The Stone"
notes that "Augustine… incorporated Greek ideas of perfection into Christianity."
Yesterday's post here for the Feast of St. Augustine discussed the 2×2×2 cube.
Today's Augustine comment in the Times reflects (through a glass darkly)
a Log24 post from Augustine's Day, 2006, that discusses the larger 4×4×4 cube.
For related material, those who prefer narrative to philosophy may consult
Charles Williams's 1931 novel Many Dimensions . Those who prefer mathematics
to either may consult an interpretation in which Many = Six.
Click image for some background.
Tuesday, May 10, 2011
Groups Acting
The LA Times on last weekend's film "Thor"—
"… the film… attempts to bridge director Kenneth Branagh's highminded Shakespearean intentions with Marvel Entertainment's bottomlineoriented need to crank out entertainment product."
Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to SpiderMan).
A highminded— if not Shakespearean— nonNordic approach to groups acting—
"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."
— Thomas W. Tucker, review of Lyndon's Groups and Geometry in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392394
"Groups act "… For some examples, see
 The 2×2×2 Cube,
 The Diamond 16 Puzzle,
 The Diamond Theorem, and
 Finite Geometry of the Square and Cube.
Related entertainment—
Highminded— Many Dimensions—
Not so highminded— The Cosmic Cube—
One way of blending high and low—
The highminded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.
The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—
The above illustration is from undated software by Ed Pegg Jr.
For mathematical background, see a 1985 note, "Visualizing GL(2,p)."
For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).
For the Nordicaverse, some background on the Hebrew connection—
Saturday, August 7, 2010
The Matrix Reloaded
For aficionados of mathematics and narrative —
Illustration from
"The Galois Quaternion— A Story"
This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—
The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean plane, but rather with unit squares
representing points in a finite Galois affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.
See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.
Saturday, June 19, 2010
Imago Creationis
In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.
FourPart Tesseract Divisions—
The above figure shows how fourpart partitions
of the 16 vertices of a tesseract in an infinite
Euclidean space are related to fourpart partitions
of the 16 points in a finite Galois space
Euclidean spaces versus Galois spaces in a larger context—
Infinite versus Finite The central aim of Western religion — "Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
 Martha Cooley in The Archivist (1998)
The central aim of Western philosophy — Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy." 
Another picture related to philosophy and religion—
Jung's FourDiamond Figure from Aion—
This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—
Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156157—
Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science… reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896). O Paul Valéry, Oeuvres (Paris: Pléiade, 195760) C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 195761) 
Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—
… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… 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 multidimensionally^{*} whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect. * That is, uses multidimensional symbols beyond our grasp. 
Related material:
A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).
Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—
Some context by a British mathematician —
Imago by Wallace Stevens Who can pick up the weight of Britain, Who can move the German load Or say to the French here is France again? Imago. Imago. Imago. It is nothing, no great thing, nor man Of ten brilliancies of battered gold And fortunate stone. It moves its parade Of motions in the mind and heart, A gorgeous fortitude. Medium man In February hears the imagination's hymns And sees its images, its motions And multitudes of motions And feels the imagination's mercies, In a season more than sun and south wind, Something returning from a deeper quarter, A glacier running through delirium, Making this heavy rock a place, Which is not of our lives composed . . . Lightly and lightly, O my land, Move lightly through the air again. 
Wednesday, May 6, 2009
Wednesday May 6, 2009
“My pursuits are a joke
in that the universe is a joke.
One has to reflect
the universe faithfully.”
— John Frederick Michell
Feb. 9, 1933 –
April 24, 2009
This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”
— Robert A. Heinlein,
The Number of the Beast
For Marisa Tomei
(born Dec. 4, 1964) —
on the day that
Bob Seger turns 64 —
A Joke:
Points All Her Own
Points All Her Own,
Part I:
(For the backstory, see
the Log24 entries and links
on Marisa Tomei’s birthday
last year.)
Points All Her Own,
Part II:
(For the backstory, see
Galois Geometry:
The Simplest Examples.)
Points All Her Own,
Part III:
(For the backstory, see
Geometry of the I Ching
and the history of
Chinese philosophy.)
In simpler terms:
Wednesday, June 3, 2020
The Ghost Writer
See also Litsky’s obituary from All Saints’ Day, 2018.
Litsky reportedly died on October 30, 2018 — Devil’s Night.
Sunday, May 17, 2020
Thursday, April 16, 2020
A FourColor Epic
“A love story of epic, epic, epic proportion” — Kristen Stewart
See also the following letter to Knuth on fourcolor enthusiast
SpencerBrown, as well as Tim Robinson on the same subject
in his book My Time in Space .
Wednesday, April 15, 2020
Death Warmed Over
In memory of the author of My Time in Space * —
Tim Robinson, who reportedly died on April 3 —
See also an image from a Log24 post, Gray Space —
Related material from Robinson’s reported date of death —
* First edition, hardcover, Lilliput Press, Ireland, April 1, 2001.
Saturday, March 14, 2020
Saturday, March 7, 2020
The “Octad Group” as Symmetries of the 4×4 Square
From “Mathieu Moonshine and Symmetry Surfing” —
(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich
https://arxiv.org/abs/1609.09302v2 —
“This presentation of the symmetry groups G_{i} is
particularly welladapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the socalled octad group
G = (Z_{2})^{4}^{ }⋊ A_{8}_{ }.
It can be described as a maximal subgroup of M_{24}
obtained by the setwise stabilizer of a particular
‘reference octad’ in the Golay code, which we take
to be O_{9 }= {3,5,6,9,15,19,23,24} ∈ 𝒢_{24}. The octad
subgroup is of order 322560, and its index in M_{24}
is 759, which is precisely the number of
different reference octads one can choose.”
This “octad group” is in fact the symmetry group of the affine 4space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79TA37, “Symmetry invariance in a
diamond ring,” by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A193, 194.
* The Galois tesseract .
Update of March 15, 2020 —
Conway and Sloane on the “octad group” in 1993 —
Tuesday, January 28, 2020
Very Stable KoolAid
Two of the thumbnail previews
from yesterday's 1 AM post …
Further down in the "6 Prescott St." post, the link 5 Divinity Avenue
leads to …
A Letter from Timothy Leary, Ph.D., July 17, 1961
Harvard University July 17, 1961
Dr. Thomas S. Szasz Dear Dr. Szasz: Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun. The Myth of Mental Illness is the most important book in the history of psychiatry. I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century. It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty. . . . . 
The small Morton Prince House in the above letter might, according to
the abovequoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.
Related "jewel box" material for those who
prefer narrative to mathematics —
"In The Electric KoolAid Acid Test , Tom Wolfe writes about encountering
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three IChing coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster
inChief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."
— Tim Doody in The Morning News web 'zine on July 26, 2012**
Oracular advice related to yesterday evening's
"jewel box" post …
A 4dimensional hypercube H (a tesseract ) has 24 square
2dimensional faces. In its incarnation as a Galois tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4space over the finite (i.e., Galois) twoelement
field GF(2)), the 24 faces transform into 140 4point "facets." The Galois
version of H has a group of 322,560 automorphisms. Therefore, by the
orbitstabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of 2,304 affine transformations.
Similar remarks apply to the I Ching In its incarnation as
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.
* The volume Wolfe mentions was, according to Fadiman, the I Ching.
** See also this journal on that date — July 26, 2012.
Friday, January 3, 2020
Valhalla Requiem
Spectral Woo
"… during that spell between the feasts of Christmas and Epiphany
when ghosts and specters are supposed to be abroad . . . ."
Heinrich Zimmer on Sir Gawain and the Green Knight
Times Literary Supplement , January 3, 2020
Sciences  Book Review
The world is not enough:
Guessing at the game God is playing
By Samuel Graydon
See as well …
Thursday, January 2, 2020
Nada for Hemingway
See Nada + Hemingway in this journal.
The above upload date suggests a look at
other posts now tagged Red to Green.
Wednesday, January 1, 2020
Águila
"We learned so much about singing from each other because you get to sort of be them for a second when you're shadowing them in harmony. It's like getting on an eagle and getting to see the world through that eagle's experience." Read more: https://www.springfieldspringfield.co.uk/ movie_script.php?movie=lindaronstadtthesoundofmyvoice
See also Aguila de Oro.
Monday, December 23, 2019
Orbit
"December 22, the birth anniversary of India’s famed mathematician
Srinivasa Ramanujan, is celebrated as National Mathematics Day."
— Indian Express yesterday
"Orbits and stabilizers are closely related." — Wikipedia
Symmetries by Plato and R. T. Curtis —
In the above, 322,560 is the order
of the octad stabilizer group .