Friday, April 3, 2020

Galois Space

Filed under: General — m759 @ 7:58 PM

Galois space” is now a domain name:  galois.space.

Sunday, November 19, 2017

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

This is a sequel to yesterday's post Cube Space Continued.

Tuesday, May 31, 2016

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM

A very brief introduction:

Seven is Heaven...

Tuesday, January 12, 2016

Harmonic Analysis and Galois Spaces

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM

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.

Sunday, March 10, 2013

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM


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 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

Filed under: General,Geometry — Tags: — m759 @ 3:00 AM

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

Filed under: General,Geometry — Tags: — m759 @ 6:00 PM


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."

Thomas Pynchon


(Click to enlarge.)

Thursday, July 12, 2012

Galois Space

Filed under: General,Geometry — Tags: , — m759 @ 6:01 PM

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional Galois projective space PG(3,2)—

(Click to enlarge.)

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  
(Cambridge U. Press, 1960, 488-499.)

(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)

Wednesday, May 2, 2018

Galois’s Space

Filed under: General,Geometry — Tags: , , — m759 @ 2:20 PM

(A sequel to Foster's Space and Sawyer's Space)

See posts now tagged Galois's Space.

Tuesday, May 26, 2020

Introduction to Cyberspace

Filed under: General — m759 @ 1:38 PM

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

Snow White’s Time in Space

Filed under: General — Tags: , — m759 @ 2:32 PM

Kristen Stewart — 'Freezin' my ass off'

See as well Pi Day 2020.

Wednesday, January 1, 2020

Exploring Inner Space* at The New York Times

Filed under: General — Tags: — m759 @ 11:35 AM

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 long-sought 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/telescopes-magellan-hawaii-astronomy.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

Filed under: General — Tags: , — m759 @ 3:00 PM

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.”

— Richard Evan Schwartz

Sunday, June 16, 2019

Master Plan from Outer Space

Filed under: General — Tags: , — m759 @ 12:00 PM

IMAGE- The large Desargues configuration and Desargues's theorem in light of Galois geometry

Monday, June 3, 2019

Art Wars for Spaceheads

Filed under: General — Tags: — m759 @ 4:52 PM

From a post of May 23

From the annals of Space Fleet

See as well the previous post.

Friday, May 3, 2019

The Structure of Story Space

Filed under: General — Tags: , , — m759 @ 11:11 AM

T. S. Eliot

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. 97-98. 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 quarter-sections.

Thursday, May 2, 2019

Story Space

Filed under: General — Tags: , — m759 @ 9:27 PM

"Let the Wookiee win." — C-3PO

See as well the April 8, 2019, post

Misère Play.

Monday, March 25, 2019


Filed under: General — Tags: , , , , — m759 @ 1:46 PM

(Continued from the previous post.)

In-Between "Spacing" and the "Chôra "
in Derrida: A Pre-Originary Medium?

By Louise Burchill

(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: Jean-Luc 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, Jean-Luc 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): 207-227.

. . . .

… "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 'in-between' more 'originary' than the play of differences it in-forms. 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 pre-ontological 'opening' – or, shall we say, 'in-between.'

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).


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

Filed under: General — Tags: , , , , — m759 @ 3:28 AM

"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


Parrhesia  No. 3 • 2007 • 22–32

(Up) Against the (In) Between: Interstitial Spatiality
in Genet and Derrida

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 non-contradiction 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 becoming-space of time or the becoming-time 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., 
Chora L Works. Jacques Derrida and Peter Eisenman  
(New York: The Monacelli Press, 1997), 15.

26. Ibid, 25.

27. Derrida, Margins of Philosophy.
(Brighton: The Harvester Press, 1982), 6 and 13.

28. Derrida, Chora L Works , 19 and 10.

29. Ibid, 203.

30. Derrida, Positions , 94.

Thursday, January 24, 2019

Name Space

Filed under: General,Geometry — Tags: — m759 @ 1:10 AM

A correction at Wikipedia  (Click to enlarge.) —

That this correction is needed indicates that the phrase 
"Cullinane space" might be useful. (Click to enlarge.)

On a 16-point space with some remarkable properties

Tuesday, January 22, 2019


Filed under: General — Tags: — m759 @ 1:21 AM

See other posts now tagged Namespace.

Monday, December 17, 2018

Tales from Story Space

Filed under: General — Tags: , — m759 @ 2:45 PM

"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

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 2:00 PM

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 —

Related material —

Iain Aitchison on the 'symmetric generation' of R. T. Curtis

See as well the two Log24 posts of December 1st, 2018 —

Character and In Memoriam.

A Small Space

Filed under: General — Tags: — m759 @ 1:00 PM

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Sunday, November 4, 2018

Kristen vs. the Space Witch*

Filed under: General — Tags: — m759 @ 11:59 PM

* We know the former. There is no shortage of candidates for the latter.

Saturday, November 3, 2018

The Space Theory of Truth

Filed under: General — Tags: — m759 @ 10:00 PM

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 Non-Euclidean 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

Filed under: General — Tags: , — m759 @ 1:00 PM

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

Plan 9 from Inner Space

Filed under: G-Notes,General,Geometry — m759 @ 9:57 AM

Click the image for some context.

Monday, October 22, 2018

Story Space

Filed under: General — Tags: — m759 @ 4:48 PM

A better term than "phase space" might be "story space."

See as well Expanding the Spielraum.

Sunday, July 22, 2018


Filed under: General,Geometry — Tags: , , — m759 @ 10:29 AM

See also interality in the eightfold cube.

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

Saturday, May 5, 2018

Galois Imaginary

Filed under: General,Geometry — m759 @ 9:00 PM

" 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
    April 21, 2015, book on Lovelace and Babbage

A related passage —

From The French Mathematician
by Tom Petsinis (Nov. 30, 1998) —


I had foreseen it all in precise detail.
One step led inevitably to the next,
like the proof of a shining theorem,
down to the conclusive shot that still echoes
through time and space
Facedown in the damp pine needles,
I embraced that fatal sphere
with my whole body. Dreams, memories,
even the mathematics I had cherished
and set down in my last will and testament–
all receded. I am reduced to
a singular point; in an instant
I am transformed to .

i = an imaginary being

Here, on this complex space,
i  am no longer the impetuous youth
who wanted to change the world
first with a formula and then with a flame.
Having learned the meaning of infinite patience,
i  now rise to the text whenever anyone reads 
about Evariste Galois, preferring to remain 
just below the surface, 
like a goldfish nibbling the fringe of a floating leaf.
Ink is more mythical than blood
(unless some ancient poet slit his 
vein and wrote an epic in red):
The text is a two-way mirror 
that allows me to look into
the life and times of the reader. 
Who knows, someday i  may rise
to a text that will compel me 
to push through to the other side.
Do you want proof that i  exist? Where am ?
Beneath every word, behind each letter, 
on the side of a period that will never see the light.

Sunday, March 4, 2018

The Square Inch Space: A Brief History

Filed under: General,Geometry — Tags: , — m759 @ 11:21 AM

1955  ("Blackboard Jungle") —

1976 —

2009 —

2016 —

 Some small Galois spaces (the Cullinane models)

Friday, September 15, 2017

Space Art

Filed under: General,Geometry — Tags: — m759 @ 2:05 PM

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 — 

'The Art of Space Art' in The Paris Review, Sept. 14, 2017

See also Galois Space  in this  journal.

Saturday, July 8, 2017

Desargues and Galois in Japan

Filed under: General,Geometry — m759 @ 1:00 AM

Related material now available online —

A less business-oriented sort of virtual reality —

Link to 'Desargues via Galois' in Japan

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)—

'Desargues via Galois' in Japan (via Pinterest)

Saturday, May 20, 2017

van Lint and Wilson Meet the Galois Tesseract*

Filed under: General,Geometry — Tags: — m759 @ 12:12 AM

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

Filed under: General,Geometry — m759 @ 2:00 PM

The title is from Don McLean's classic "American Pie."

A Finite Projective Space

A Non-Finite Projective Space

Sunday, April 16, 2017

Art Space Paradigm Shift

Filed under: General,Geometry — m759 @ 1:00 AM

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):

IMAGE- History of Mathematics in a Nutshell

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 Comic-Con . . . .

'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


Filed under: General — Tags: — m759 @ 7:00 PM

See "Smallest Perfect" in this journal.

Wednesday, March 29, 2017

Art Space Illustrated

Filed under: General,Geometry — Tags: , — m759 @ 10:45 AM

Another view of the previous post's art space  —

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

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).

IMAGE- Current math.stackexchange.com logo and a 1984 figure from 'Notes on Groups and Geometry, 1978-1986'

Art Space, Continued

Filed under: General — Tags: , — m759 @ 4:35 AM

"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 —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Tuesday, January 3, 2017

Cultist Space

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM

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

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

Monday, September 26, 2016

Myspace China …

Filed under: General — Tags: — m759 @ 12:00 PM


Thursday, June 30, 2016

Rubik vs. Galois: Preconception vs. Pre-conception

Filed under: General,Geometry — Tags: — m759 @ 1:20 PM

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
a prelude to action and itself a kind of action;
the meeting of psycho-analyst and analysand
is itself an example of this language.29.

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
And every moment is a new and shocking
Valuation of all we have ever been.31.

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

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

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

Filed under: General,Geometry — Tags: — m759 @ 11:00 PM

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 young-adult novel

Ogdoads: A Space Odyssey

Filed under: General — Tags: , — m759 @ 5:01 AM

"Like the Valentinian Ogdoad— a self-creating 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. 58-82, Duke University
Press (quarterly, January 2001)

See also Two by Four  in this  journal.

Wednesday, February 17, 2016

“Blank Space” Accolades

Filed under: General,Geometry — m759 @ 9:00 PM

A post in memory of British theatre director Peter Wood,
who reportedly died on February 11, 2016.

The Album of the Year Grammy:

From the date of the director's death —

"Leave a space." — Tom Stoppard

Monday, January 11, 2016

Space Oddity

Filed under: General,Geometry — Tags: — m759 @ 3:15 PM

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 42-27 , 1975

D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, 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 young-adult novel

Saturday, October 24, 2015

Two Views of Finite Space

Filed under: General,Geometry — Tags: , — m759 @ 10:00 AM

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

Filed under: General,Geometry — Tags: , — m759 @ 7:59 AM

"Perhaps an insane conceit …."    Perhaps.

Related remarks on algebra and space

"The Quality Without a Name" (Log24, August 26, 2015).

Friday, September 4, 2015

Space Program

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

Galois via Boole

(Courtesy of Intel)

Friday, August 14, 2015

Discrete Space

Filed under: General,Geometry — Tags: — m759 @ 7:24 AM

(A review)

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

Wednesday, May 13, 2015


Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

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

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Yesterday's post suggests a review of the following —

Andries Brouwer, preprint, 1982:

"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)

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
form a self-complementary 3-(16,8,3) design, namely 

the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F2.


… (iv) We have AG(4, 2).

(Proof: invoke your favorite characterization of AG(4, 2) 
or PG(3, 2), say 
Dembowski-Wagner or Veblen & Young. 

An explicit construction of the vector space is also easy….)

Related material:  Posts tagged Priority.

Tuesday, November 25, 2014

Euclidean-Galois Interplay

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

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.

Oxley's 2004 drawing of the 13-point projective plane

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.

Monday, September 22, 2014


Filed under: General,Geometry — Tags: — m759 @ 11:17 AM

Review of an image from a post of May 6, 2009:

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Thursday, March 27, 2014

Diamond Space

Filed under: General,Geometry — Tags: — m759 @ 2:28 PM


Definition:  A diamond space  — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.

The reason for the name:

IMAGE - The Diamond Theorem, including the 4x4x4 'Solomon's Cube' case

Click to enlarge.

Thursday, February 27, 2014

Sacred Space, continued

Filed under: General — Tags: — m759 @ 11:00 AM

"An image comes to mind of a white, ideal space
​that, more than any single picture, may be the
archetypal image of 20th-century art."

— Brian O'Doherty, "Inside the White Cube"

Cube  spaces exist also in mathematics.

Tuesday, December 3, 2013

Diamond Space

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM

A new website illustrates its URL.
See DiamondSpace.net.

IMAGE- Site with keywords 'Galois space, Galois geometry, finite geometry' at DiamondSpace.net

Tuesday, July 16, 2013

Space Itself

Filed under: General,Geometry — Tags: — m759 @ 10:18 AM

"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 AtlanticAug. 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

Filed under: General,Geometry — Tags: , — m759 @ 10:30 PM

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 24 .

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group 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

Filed under: General,Geometry — m759 @ 3:33 AM

 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

Filed under: General — Tags: , — m759 @ 11:00 PM


"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
to mean 'what/where the four mahābhūtas are not',
or example, the cavities in the body are called ākāsa
M.62—Vol. I, p. 423). This, clearly, is the everyday
'space' we all experience—roughly, 'What I can move
bout in', the empty part of the world. 'What you can't
ouch.' It is the 'space' of what Miss Lounsberry has so
appily described as 'the visible world of our five
senses'. I think you agree with this. And, of course, if
this is the only meaning of the word that we are
going to use, my 'superposition of several spaces' is
disqualified. So let us say 'superposition of several
extendednesses'. But when all these
extendednesses have been superposed, we get
'space'—i.e. our normal space-containing visible
world 'of the five senses'. But now there is another
point. Ākāsa is the negative of the four mahābhūtas,
certainly, but of the four mahābhūtas understood
in the same everyday sense—namely, solids (the
solid parts of the body, hair, nails, teeth, etc.),
liquids (urine, blood, etc.), heat and processes
(digestion) and motion or wind (N.B. not 'air').
These four, together with space, are the normal
furniture of our visible world 'of the five senses',
and it is undoubtedly thus that they are intended
in many Suttas. But there is, for example, a Sutta
(I am not sure where) in which the Ven. Sariputta
Thera is said to be able to see a pile of logs
successively as paṭhavi, āpo, tejo, and vāyo; and
it is evident that we are not on the same level.
On the everyday level a log of wood is solid and
therefore pathavi (like a bone), and certainly not
āpo, tejo, or vāyo. I said in my last letter that I
think that, in this second sense—i.e. as present in,
or constitutive of, any object (i.e. = rupa)—they
are structural and strictly parallel to nama and can
be defined exactly in terms of the Kummer
triangle. But on this fundamental level ākāsa has
no place at all, at least in the sense of our normal
everyday space. If, however, we take it as equivalent
to extendedness then it would be a given arbitrary
content—defining one sense out of many—of which
the four mahābhūtas (in the fundamental sense) are
the structure. In this sense (but only in this sense—
and it is probably an illegitimate sense of ākāsa)
the four mahābhūtas are the structure of space
(or spatial things). Quite legitimately, however, we
can say that the four mahābhūtas are the structure
of extended things—or of coloured things, or of smells,
or of tastes, and so on. We can leave the scientists'
space (full of right angles and without reference to the
things in it) to the scientists. 'Space' (= ākāsa) is the
space or emptiness of the world we live in; and this,
when analyzed, is found to depend on a complex
superposition of different extendednesses (because
all these extendednesses define the visible world
'of the five senses'—which will include, notably,
tangible objects—and this world 'of the five
senses' is the four mahābhūtas [everyday space]
and ākāsa).

Your second letter seems to suggest that the space
of the world we live in—the set of patterns
(superimposed) in which “we” are—is scientific space.
This I quite disagree with—if you do suggest it—,
since scientific space is a pure abstraction, never
experienced by anybody, whereas the superimposed
set of patterns is exactly what I experience—the set
is different for each one of us—, but in all of these
sets 'space' is infinite and undifferentiable, since it is,
by definition, in each set, 'what the four mahābhūtas
are not'. 

A simpler metaphysical system along the same lines—

The theory, he had explained, was that the persona
was a four-dimensional figure, a tessaract in space,
the elementals Fire, Earth, Air, and Water permutating
and pervolving upon themselves, making a cruciform
(in three-space projection) figure of equal lines and
ninety degree angles.

The Gameplayers of Zan ,
a 1977 novel by M. A. Foster

"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

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM


The three parts of the figure in today's earlier post "Defining Form"—

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

— 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):


The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.

Tuesday, July 10, 2012

Euclid vs. Galois

Filed under: General,Geometry — Tags: — m759 @ 11:01 AM


Euclidean square and triangle

Galois square and triangle


This journal on the date of Hilton Kramer's death,
The Galois Tesseract, and The Purloined Diamond.

Wednesday, October 26, 2011

Erlanger and Galois

Filed under: General,Geometry — m759 @ 8:00 PM

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

Saturday, September 3, 2011

The Galois Tesseract (continued)

Filed under: General,Geometry — Tags: , — m759 @ 1:00 PM

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—

IMAGE- An AMS abstract from 1979 showing how the affine group AGL(4,2) of 322,560 transformations acts on a 4x4 square

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978

IMAGE- Projective-space structure and Latin-square orthogonality in a set of 35 square arrays

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:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Update of Sept. 4— This post is now a page at finitegeometry.org.

Thursday, September 1, 2011

The Galois Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 7:11 PM

Click to enlarge

IMAGE- The Galois Tesseract, 1979-1999

IMAGE- Review of Conway and Sloane's 'Sphere Packings...' by Rota

Friday, April 22, 2011

Romancing the Hyperspace

Filed under: General,Geometry — m759 @ 7:59 PM

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—

Image-- examples from Galois affine geometry

Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"–

"… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:

(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.

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 non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory."

Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds.

* "Non-Euclidean" 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

Filed under: General,Geometry — Tags: , — m759 @ 5:01 AM

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.

Monday, June 21, 2010

Cube Spaces

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

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

2x2x2 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—

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 
note by Cullinane

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
group-actions theorem has not yet been ripped off.

Sunday, March 21, 2010

Galois Field of Dreams

Filed under: General,Geometry — m759 @ 10:01 AM

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

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

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

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

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

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

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

Number and Time, by Marie-Louise von Franz

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

* February 27 and March 13

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

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

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

Saturday, March 13, 2010

Space Cowboy

Filed under: General,Geometry — m759 @ 9:00 AM

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…

Plan 9 from Outer Space

Rebecca Goldstein and a Cullinane quaternion

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–

The 13 symmetry axes of the cube

Thursday, March 5, 2020

“Generated by Reflections”

Filed under: General — m759 @ 8:42 PM

See the title in this journal.

Such generation occurs both in Euclidean space 

Order-8 group generated by reflections in midplanes of cube parallel to faces

… and in some Galois spaces

Generating permutations for the Klein simple group of order 168 acting on the eightfold cube .

In Galois spaces, some care must be taken in defining "reflection."

Monday, December 2, 2019

Aesthetics at Harvard

Filed under: General — Tags: , , , — m759 @ 4:05 PM

"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 —

Some small Galois spaces (the Cullinane models)

Context:  The Trinity Stone  (Log24, June 4, 2018).

Monday, October 15, 2018

For Zingari Shoolerim*

Filed under: General,Geometry — Tags: , — m759 @ 12:19 PM

IMAGE- Site with keywords 'Galois space, Galois geometry, finite geometry' at DiamondSpace.net

The structure at top right is that of the
in the previous post.

* "Zingari shoolerim" is from
    Finnegans Wake .

Saturday, September 29, 2018

“Ikonologie des Zwischenraums”

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 9:29 AM

The title is from Warburg. The Zwischenraum  lines and shaded "cuts"
below are to be added together in characteristic two, i.e., via the
set-theoretic symmetric difference  operator.

Some small Galois spaces (the Cullinane models)

Monday, August 27, 2018

Geometry and Simplicity

Filed under: General,Geometry — Tags: — m759 @ 9:27 PM


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 705-710, 735). Kindle Edition.

For some much simpler spaces of various
dimensions, see Galois Space in this journal.

Some small Galois spaces (the Cullinane models)


Monday, June 4, 2018

The Trinity Stone Defined

Filed under: General,Geometry — Tags: , — m759 @ 8:56 PM

“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 .

An Approach to Symmetric Generation of the Simple Group of Order 168

Some small Galois spaces (the Cullinane models)

Tuesday, May 2, 2017

Image Albums

Filed under: General,Geometry — Tags: — m759 @ 1:05 PM

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Saturday, June 18, 2016

Midnight in Herald Square

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 AM

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 length-16 sequence."
— Saying adapted from a young-adult novel.

Sunday, May 8, 2016

The Three Solomons

Filed under: General,Geometry — Tags: , , , — m759 @ 11:00 PM

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:

by Sol LeWitt

"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 three-dimensional 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 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)"

See also the Cullinane models of some small Galois spaces

 Some small Galois spaces (the Cullinane models)

Friday, May 6, 2016


Filed under: General,Geometry — Tags: — m759 @ 9:48 PM

 Some small Galois spaces (the Cullinane models)

Monday, January 5, 2015


Filed under: General,Geometry — Tags: , — m759 @ 2:00 PM

Wednesday, March 13, 2013

Blackboard Jungle

Filed under: Uncategorized — m759 @ 8:00 AM 

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 (2009-04-01). 
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.

"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
grown-ups, and asked them whether
the drawing frightened them.

But they answered: 'Why should
anyone be frightened by a hat?'"

The Little Prince

* For the title, see Plato Thanks the Academy (Jan. 3).

Sunday, September 14, 2014


Filed under: General,Geometry — Tags: , — m759 @ 9:26 AM

Structured gray matter:

Graphic symmetries of Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine  Galois space —

symmetries of the underlying projective  Galois space:

Sunday, August 31, 2014

Sunday School

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

The Folding

Cynthia Zarin in The New Yorker , issue dated April 12, 2004—

“Time, for L’Engle, is accordion-pleated. 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
3-dimensional 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:

Filed under: General,Geometry — Tags: — m759 @ 11:01 AM

Continuous Euclidean space to discrete Galois space*

Euclidean space:

Point, line, square, cube, tesseract

From a page by Bryan Clair

Counting symmetries in Euclidean space:

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

* For related remarks, see posts of May 26-28, 2012.

Wednesday, July 16, 2014

Finite Jest

Filed under: General,Geometry — m759 @ 2:01 PM

(Continued from a private post of May 27, 2012)

Wednesday, May 21, 2014

Through the Vanishing Point*

Filed under: General,Geometry — Tags: , , — m759 @ 9:48 AM

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)—

IMAGE- Whitehead on Fano's construction of the 15-point projective Galois space over GF(2)

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

Filed under: General,Geometry — Tags: , , — m759 @ 12:24 PM

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, 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 Turyn-Curtis construction
from the University of Cambridge —

— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M24,” in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on “Sporadic and Related Groups.”
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis’s 35  4×6  1976 MOG arrays would use
Cullinane’s analysis of the 4×4 subarrays’ affine and projective structure,
and point out the fact that Conwell’s 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1’s, all other squares as 0’s) of the 35  4×6 arrays of the 1976
Curtis MOG would then reveal*  the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35  MOG arrays.  For more
details of this “by-hand” construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction,  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 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Friday, March 7, 2014

Kummer Varieties

Filed under: General,Geometry — Tags: , , — m759 @ 11:20 AM

The Dream of the Expanded Field continues

Image-- The Dream of the Expanded Field

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 —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

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 64-point vector space and
to the Weyl group of type E7(E7):

The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik  87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .

Wednesday, February 5, 2014

Mystery Box II

Filed under: General,Geometry — Tags: — m759 @ 4:07 PM

Continued from previous post and from Sept. 8, 2009.

Box containing Froebel's Third Gift-- The Eightfold Cube

Examination of the box's contents does not solve
the contents' real mystery. That requires knowledge
of the non-Euclidean 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

Filed under: General,Geometry — Tags: — m759 @ 10:31 AM

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 non-mathematical 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 non-mathematical 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

Filed under: General,Geometry — Tags: — m759 @ 2:21 PM

"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

Filed under: General,Geometry — Tags: , — m759 @ 4:00 PM


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 4-dimensional hypercube that
(as pointed out by Coxeter in 1950) may also 
be viewed as a 4×4 array (with opposite edges

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 15-point 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

Filed under: General,Geometry — m759 @ 8:00 AM

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 (2009-04-01). 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*

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

"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


Filed under: General,Geometry — Tags: — m759 @ 12:24 PM

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
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius 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 re-describing 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 Pisanski-Servatius 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


Filed under: General,Geometry — m759 @ 11:00 AM


IMAGE- Golomb and Mazur awarded National Medals of Science


IMAGE- The Leibniz medal

Click medal for some background. The medal may be regarded
as illustrating the 16-point Galois space. (See previous post.)

Related material: Jews in Hyperspace.

Saturday, January 5, 2013

Vector Addition in a Finite Field

Filed under: General,Geometry — Tags: , — m759 @ 10:18 AM

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 4-space over GF(2)—

The same field, again disguised as an affine 4-space,
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 vector-space 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
Self-Dual Configurations and Regular Graphs.”

Monday, August 29, 2011

Many = Six.

Filed under: General,Geometry — m759 @ 7:20 PM

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.

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Click image for some background.

Tuesday, May 10, 2011

Groups Acting

Filed under: General,Geometry — Tags: , — m759 @ 10:10 AM

The LA Times  on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented 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 Spider-Man).

A high-minded— if not Shakespearean— non-Nordic 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. 392-394

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions

Not so high-minded— The Cosmic Cube


One way of blending high and low—

The high-minded 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 Nordic-averse, some background on the Hebrew connection—

Saturday, August 7, 2010

The Matrix Reloaded

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

   For aficionados of mathematics and narrative

Illustration from
"The Galois Quaternion— A Story"

The Galois Quaternion

This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—
Coxeter's 1950 representation in the Euclidean plane of the 9-point affine plane over GF(3)

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

Filed under: General,Geometry — Tags: , , , , — m759 @ 6:00 PM

Image-- The Four-Diamond Tesseract

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.

Four-Part Tesseract Divisions


The above figure shows how four-part partitions
of the 16 vertices  of a tesseract in an infinite
Euclidean  space are related to four-part 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."
— Jamie James in The Music of the Spheres  (1993)

Another picture related to philosophy and religion—

Jung's Four-Diamond 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 156-157—



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).


  Paul Valéry, Oeuvres  (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

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 multi-dimensionally* 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 multi-dimensional symbols beyond our grasp.

Related material:

Imago Creationis

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—

Frame of Reference


The Diamond Theorem


Some context by a British mathematician —



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

Filed under: General,Geometry — m759 @ 11:07 AM

“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 

“I laugh because I dare not cry.
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.)

Ad for a movie of the book 'Flatland'

Points All Her Own,

Part II:

(For the backstory, see
Galois Geometry:
The Simplest Examples

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.)

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

In simpler terms:


Garfield on May 6, 2009: Smackdown!

Wednesday, June 3, 2020

The Ghost Writer

Filed under: General — Tags: , , — m759 @ 3:14 PM

See also Litsky’s obituary from All Saints’ Day, 2018.

Litsky reportedly died on October 30, 2018 — Devil’s Night.

Sunday, May 17, 2020

Classics Illustrated

Filed under: General — Tags: — m759 @ 2:32 PM

Related image —

Thursday, April 16, 2020

A Four-Color Epic

Filed under: General — Tags: , — m759 @ 4:15 PM

A love story of epic, epic, epic proportion” — Kristen Stewart

See also the following letter to Knuth on four-color enthusiast
Spencer-Brown, as well as Tim Robinson on the same subject
in his book My Time in Space .

The Dreaming

Filed under: General — Tags: — m759 @ 1:19 PM

Wednesday, April 15, 2020

Death Warmed Over

Filed under: General — Tags: — m759 @ 4:01 PM

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

Compare and Contrast.

Filed under: General — Tags: , , — m759 @ 9:38 PM

Logos —

See also Devs .

Saturday, March 7, 2020

The “Octad Group” as Symmetries of the 4×4 Square

Filed under: General — m759 @ 6:32 PM

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 Gi  is
particularly well-adapted 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 so-called octad group

G = (Z2)4  A8 .

It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
‘reference octad’ in the Golay code, which we take
to be O= {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
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 4-space 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 79T-A37, “Symmetry invariance in a
diamond ring
,” by Steven H. Cullinane in Notices of the American Mathematical
, February 1979, pages A-193, 194.

* The Galois tesseract .

Update of March 15, 2020 —

Conway and Sloane on the “octad group” in 1993 —

Tuesday, January 28, 2020

Very Stable Kool-Aid

Filed under: General — Tags: — m759 @ 2:16 PM

Two of the thumbnail previews
from yesterday's 1 AM  post

"Hum a few bars"

"For 6 Prescott Street"

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
Department of Social Relations
Center for Research in Personality
Morton Prince House
5 Divinity Avenue
Cambridge 38, Massachusetts

July 17, 1961

Dr. Thomas S. Szasz
c/o Upstate Medical School
Irving Avenue
Syracuse 10, New York

Dear Dr. Szasz:

Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun.

The Myth of Mental Illness is the most important book in the history of psychiatry.

I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century.

It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty.

. . . .

The small Morton Prince House in the above letter might, according to
the above-quoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.

Related "jewel box" material for those who
prefer narrative to mathematics —

"In The Electric Kool-Aid Acid Test , Tom Wolfe writes about encountering 
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three I-Ching coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster-
in-Chief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."

— Tim Doody in The Morning News  web 'zine on July 26, 2012**

Oracular advice related to yesterday evening's
"jewel box" post …

A 4-dimensional hypercube H (a tesseract ) has 24 square
2-dimensional faces
.  In its incarnation as a Galois  tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4-space over the finite (i.e., Galois) two-element
field GF(2)), the 24 faces transform into 140 4-point "facets." The Galois 
version of H has a group of 322,560 automorphisms. Therefore, by the
orbit-stabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of  2,304 affine transformations.

Similar remarks apply to the I Ching  In its incarnation as  
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6-dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.

* The volume Wolfe mentions was, according to Fadiman, the I Ching.

** See also this  journal on that date — July 26, 2012.

Friday, January 3, 2020

Valhalla Requiem

Filed under: General — Tags: , , — m759 @ 9:59 PM

See also this  journal on Monday, the day of Kupfer's reported death


Spectral Woo

Filed under: General — Tags: , , — m759 @ 12:14 PM

 " 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


Filed under: General — Tags: — m759 @ 8:25 PM

Structure of the eightfold cube

Nada for Hemingway

Filed under: General — Tags: , — m759 @ 11:47 AM

See Nada + Hemingway  in this journal.

'Inner Space' YouTube upload on March 15, 2014

The above upload date suggests a look at 
other posts now tagged Red to Green.

Wednesday, January 1, 2020


Filed under: General — Tags: — m759 @ 11:30 PM
"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/

See also Aguila de Oro.

Le Mot Juste

Filed under: General — Tags: — m759 @ 3:23 PM

Related art

Monday, December 23, 2019


Filed under: General — Tags: , — m759 @ 7:34 PM

"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 —

Counting symmetries with the orbit-stabilizer theorem

In the above, 322,560 is the order 
of the octad stabilizer group .

Older Posts »

Powered by WordPress