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:

Friday, December 18, 2020

Square Space at Athens

Filed under: General — Tags: — m759 @ 2:37 AM

(A sequel to the previous post, Square Space at Wikipedia)

Cyberpunk and the Dude

Related remarks: A Dec. 16 Wikipedia revision by Quack5quack,
and posts in this  journal tagged Helsinki Math.

Thursday, December 17, 2020

Square Space at Wikipedia

Filed under: General — Tags: — m759 @ 11:45 PM

The State of Square-Space Art at Wikipedia  as of December 16, 2020,
after a revision by an anonymous user on that date:

See also Square Space at Squarespace.

Monday, September 14, 2020

Space People Puzzle

Filed under: General — Tags: — m759 @ 9:59 AM

How about puzzles?

Space People Lightbulb Puzzle

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

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.

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)

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.

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)

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.

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.

Sunday, April 16, 2017

Art Space Paradigm Shift

Filed under: General,Geometry — Tags: — 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 

Monday, September 26, 2016

Myspace China …

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


Tuesday, May 31, 2016

Galois Space

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

A very brief introduction:

Seven is Heaven...

Friday, April 8, 2016

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.

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.

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

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, August 14, 2015

Space Station 2015

Filed under: General,Geometry — m759 @ 9:20 AM

(A sequel to Space Station 1976)

For Kathleen Gibbons* —

'Sacred Space' at Chautauqua Institution

* Note Gibbons's work on "Discrete phase space based on finite fields."

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, 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.

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.)

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

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.)

Saturday, February 18, 2012


Filed under: General,Geometry — m759 @ 7:35 PM

From the current Wikipedia article "Symmetry (physics)"—

"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.

A family of particular transformations may be continuous  (such as rotation of a circle) or discrete  (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….

"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."

Note the confusion here between continuous (or discontinuous) transformations  and "continuous" (or "discontinuous," i.e. "discrete") groups .

This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.

For an attempt to forestall such confusion, see Noncontinuous Groups.

For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory

Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.

Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself  be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous  groups, as opposed to so-called discontinuous  (or discrete ) symmetry groups. See Weyl's Symmetry .)

For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)

(Version first archived on March 27, 2002)

Update of Sunday, February 19, 2012—

The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—

IMAGE- Brading and Castellani, 'Symmetries in Physics'- Four main sections include 'Continuous Symmetries' and 'Discrete Symmetries.'

Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous  transformations.

This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
(Nirmala Prakash, Imperial College Press)—

"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous  (discrete ) symmetry ." — Pp. 235, 236

[* Associated how?]

Monday, August 29, 2005

Monday August 29, 2005

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

The image “http://www.log24.com/log/pix05B/050829-GeorgeAndEsther2.jpg” cannot be displayed, because it contains errors.

George and Esther Szekeres

From the weblog of
David Michael Brown, Jr.:

Date:     Sun, 28 Aug 2005
             12:30:40 -0400
From:    Alf van der Poorten AM
Subject: Vale George Szekeres and
             Esther Klein Szekeres

Members of the Number Theory List will be sad to learn that George and Esther Szekeres both died this morning.  George, 94, had been quite ill for the last 2-3 days, barely conscious, and died first at 06:30.  Esther, 95, died a half hour later.

Both George Szekeres and Esther Klein will be recalled by number theorists as members of the group of young Hungarian mathematicians of the 1930s including Turan and Erdos.  George and Esther's coming to Australia in the late 40s played an important role in the invigoration of Australian Mathematics.  George was also an expert in group theory and relativity; he was my PhD supervisor.

Emeritus Professor
Alf van der Poorten AM
Centre for Number Theory Research
1 Bimbil Place, Killara NSW


Related material:


3:09 PM EDT Thursday, Aug. 25, 2005:

  "Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one." 


  "A very short space of time through very short times of space….
   Am I walking into eternity along Sandymount strand?"

   — James Joyce, Ulysses, Proteus chapter

A very short space of time through very short times of space….

   "It is demonstrated that space-time should possess a discrete structure on Planck scales."

   — Peter Szekeres, abstract of Discrete Space-Time

Peter Szekeres is the son of George and Esther Szekeres.


"At present, such relationships can at best be heuristically described in terms that invoke some notion of an 'intelligent user standing outside the system.'"

Gian-Carlo Rota in Indiscrete Thoughts, p. 152

Related material:
High Concept and
Nothing Nothings (Again).

Thursday, August 25, 2005

Thursday August 25, 2005

Filed under: General,Geometry — m759 @ 3:09 PM
Train of Thought

Part I: The 24-Cell

From S. H. Cullinane,
 Visualizing GL(2,p),
 March 26, 1985–

Visualizing the
binary tetrahedral group
(the 24-cell):

The image “http://www.log24.com/theory/images/VisuBinaryTetGrp.jpg” cannot be displayed, because it contains errors.

Another representation of
the 24-cell

The image “http://www.log24.com/theory/images/24-cell.jpg” cannot be displayed, because it contains errors.

 From John Baez,
This Week’s Finds in
Mathematical Physics (Week 198)
September 6, 2003: 

Noam Elkies writes to John Baez:

Hello again,

You write:


“I’d like to wrap up with a few small comments about last Week.  There I said a bit about a 24-element group called the ‘binary tetrahedral group’, a 24-element group called SL(2,Z/3), and the vertices of a regular polytope in 4 dimensions called the ’24-cell’.  The most important fact is that these are all the same thing! And I’ve learned a bit more about this thing from here:”


Here’s yet another way to see this: the 24-cell is the subgroup of the unit quaternions (a.k.a. SU(2)) consisting of the elements of norm 1 in the Hurwitz quaternions – the ring of quaternions obtained from the Z-span of {1,i,j,k} by plugging up the holes at (1+i+j+k)/2 and its <1,i,j,k> translates. Call this ring A. Then this group maps injectively to A/3A, because for any g,g’ in the group |g-g’| is at most 2 so g-g’ is not in 3A unless g=g’. But for any odd prime p the (Z/pZ)-algebra A/pA is isomorphic with the algebra of 2*2 matrices with entries in Z/pZ, with the quaternion norm identified with the determinant. So our 24-element group injects into SL2(Z/3Z) – which is barely large enough to accommodate it. So the injection must be an isomorphism.

Continuing a bit longer in this vein: this 24-element group then injects into SL2(Z/pZ) for any odd prime p, but this injection is not an isomorphism once p>3. For instance, when p=5 the image has index 5 – which, however, does give us a map from SL2(Z/5Z) to the symmetric group of order 5, using the action of SL2(Z/5Z) by conjugation on the 5 conjugates of the 24-element group. This turns out to be one way to see the isomorphism of PSL2(Z/5Z) with the alternating group A5.

Likewise the octahedral and icosahedral groups S4 and A5 can be found in PSL2(Z/7Z) and PSL2(Z/11Z), which gives the permutation representations of those two groups on 7 and 11 letters respectively; and A5 is also an index-6 subgroup of PSL2(F9), which yields the identification of that group with A6.


The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – Gian-Carlo Rota

Like footprints erased in the sand….

Part II: Discrete Space

The James Joyce School
 of Theoretical Physics

Log24, May 27, 2004

  “Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.” 

  “A very short space of time through very short times of space….
   Am I walking into eternity along Sandymount strand?”

   — James Joyce, Ulysses, Proteus chapter

A very short space of time through very short times of space….

   “It is demonstrated that space-time should possess a discrete structure on Planck scales.”

   — Peter Szekeres, abstract of Discrete Space-Time

   “A theory…. predicts that space and time are indeed made of discrete pieces.”

   — Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004

   “… a fundamental discreteness of spacetime seems to be a prediction of the theory….”

   — Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity

   “Theories of discrete space-time structure are being studied from a variety of perspectives.”

   — Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London


The above speculations by physicists
are offered as curiosities.
I have no idea whether
 any of them are correct.

Related material:

Stephen Wolfram offers a brief
History of Discrete Space.

For a discussion of space as discrete
by a non-physicist, see John Bigelow‘s
Space and Timaeus.

Part III: Quaternions
in a Discrete Space

Apart from any considerations of
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
 (Aug. 4, 2005):

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Saturday, December 19, 2020


Filed under: General — Tags: — m759 @ 12:43 PM

An Internet search for the anonymous author Quack5quack yields . . .

Friday, December 18, 2020

Notes Towards the Redefinition of Culture

Filed under: General — Tags: — m759 @ 11:18 PM

In the altered headline above, ” Q******* ” may, if you like,
be interpreted as ” Quellers ,” an invented term for scholars
who investigate the origins of Christianity.

See the Log24 post “Q is for Quelle ” (November 7, 2020).

Dan Brown, like the earlier novelist who wrote The Source ,
is such an investigator  (of sorts), though not a scholar .

(For an example of actual scholarship , see the webpage
That  page may be interpreted as putting the “hit” in “s***.”)

Monday, September 14, 2020

Shades (Of London Bondage continues)

Filed under: General — Tags: , — m759 @ 8:59 AM

Loitering in Lara’s dressing room, she tries on
the faux-bondage harness she picked up in London….”

See as well . . .

Socrates in the Marketplace

Filed under: General — Tags: — m759 @ 7:39 AM

Plato's diamond in Jowett's version of the Meno dialogue

Diamond Matrix slide template at presentationgo.com

“The 2×2 matrix is commonly used in business strategy
as a representational tool to show conflicting concepts and
for decision making. This four-quadrant matrix diagram
is perfect to be used for business or marketing matrices
like BCG, SWOT, Ansoff, risk assessment…

Additionally, it will also be suitable to illustrate 4 ideas or
concepts.” [Link on “illustrate” added.]

See also a Log24 search for “Resplendent.”

Thursday, August 6, 2020

Structure and Mutability . . .

Filed under: General — Tags: , — m759 @ 8:33 PM


See a Log24 search for Beadgame Space.

This  post might be regarded as a sort of “checked cell”
for the above concepts listed as tags . . .

Related material from a Log24 search for Structuralism

IMAGE- Stella Octangula and Claude Levi-Strauss

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 .

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 .


Filed under: General — Tags: , , — m759 @ 8:16 AM

See as well the previous post and Pacific Science Institute.

Tuesday, December 10, 2019

Eternal Color

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

For the above title, see posts tagged Eternal Color.

From this evening's online New York Times

Related imaterial —

A scene from the film of the above book —

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).

Thursday, June 13, 2019

Seeing the Seing

Filed under: General — Tags: , , , — m759 @ 2:30 PM

The phrase "experimental metaphysics" appeared in Peter Woit's weblog on June 11.
Google reveals that . . .

" 'experimental metaphysics' is a term coined by Abner Shimony …."

Shimony reportedly died on August 8, 2015.  Also on that date —

Tuesday, June 4, 2019

Zen and the Art

Filed under: General — Tags: , , — m759 @ 6:13 PM

Or:  Burning Bright

A post in memory of Chicago architect Stanley Tigerman,
who reportedly died at 88 on Monday.

Inside Out

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

For fans of Space Fleet  and of "reclusive but gifted" programmers

“Hello the Camp”

Filed under: General — Tags: , — m759 @ 12:20 AM

The title is a quotation from the 2015 film "Mojave."

Monday, June 3, 2019

Jar Story

Filed under: General — Tags: , , — m759 @ 3:41 PM


  “. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”

— T. S. Eliot, Four Quartets

From Writing Chinese Characters:

“It is practical to think of a character centered
within an imaginary square grid . . . .
The grid can be subdivided, usually to
9 or 16 squares. . . .

The image “http://www.log24.com/log/pix04B/041119-ZhongGuo.jpg” cannot be displayed, because it contains errors.

These “Chinese jars” (as opposed to their contents)
are as follows:

Grids, 3x3 and 4x4 .

See as well Eliot’s 1922 remarks on “extinction of personality”
and the phrase “ego-extinction” in Weyl’s Philosophy of Mathematics

Monday, May 20, 2019

The Bond with Reality

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

"The bond with reality is cut."

— Hans Freudenthal, 1962

Indeed it is.

Tuesday, May 7, 2019


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

"Honored in the Breach:
Graham Bader on Absence as Memorial"

Artforum International , April 2012 

. . . .

"In the wake of a century marked by inconceivable atrocity, the use of emptiness as a commemorative trope has arguably become a standard tactic, a default style of public memory. The power of the voids at and around Ground Zero is generated by their origin in real historical circumstance rather than such purely commemorative intent: They are indices as well as icons of the losses they mark.

Nowhere is the negotiation between these two possibilities–on the one hand, the co-optation of absence as tasteful mnemonic trope; on the other, absence's disruptive potential as brute historical scar–more evident than in Berlin, a city whose history, as Andreas Huyssen has argued, can be seen as a 'narrative of voids.' Writing in 1997, Huyssen saw this tale culminating in Berlin's post-wall development, defined equally by an obsessive covering-over of the city's lacunae–above all in the elaborate commercial projects then proliferating in the miles-long stretch occupied until 1989 by the Berlin Wall–and a carefully orchestrated deployment of absence as memorial device, particularly in the 'voids' integrated by architect Daniel Libeskind into his addition to the Berlin Museum, now known as the Jewish Museum Berlin."
. . . .

See also Breach  in this  journal. 

Monday, May 6, 2019

One Stuff

Filed under: General — Tags: , , , , — m759 @ 1:17 PM

Building blocks?

From a post of May 4

Structure of the eightfold cube

See also Espacement  and The Thing and I.

Saturday, May 4, 2019

Inside the White Cube

Filed under: General — Tags: , , , , — m759 @ 8:48 PM

Structure of the eightfold cube

See also Espacement  and The Thing and I.

Friday, May 3, 2019

“As a Chinese jar” — T. S. Eliot

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


Wednesday, April 24, 2019

Critical Visibility

Filed under: General — Tags: , , , , — m759 @ 8:24 AM

Correction — "Death has 'the whole spirit sparkling…'"
should be "Peace after death has 'the whole spirit sparkling….'" 
The page number, 373, is a reference to Wallace Stevens:
Collected Poetry and Prose
, Library of America, 1997.

See also the previous post, "Critical Invisibility."

Tuesday, April 23, 2019

Critical Invisibility

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

From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens  54, 59-79 (1992):

" what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."

On "The Emperor's New Clothes" —

Andersen’s weavers, as one commentator points out, are merely insisting that “the value of their labor be recognized apart from its material embodiment.” The invisible cloth they weave may never manifest itself in material terms, but the description of its beauty (“as light as spiderwebs” and “exquisite”) turns it into one of the many wondrous objects found in Andersen’s fairy tales. It is that cloth that captivates us, making us do the imaginative work of seeing something beautiful even when it has no material reality. Deeply resonant with meaning and of rare aesthetic beauty—even if they never become real—the cloth and other wondrous objets d’art have attained a certain degree of critical invisibility.

—  Maria Tatar, The Annotated Hans Christian Andersen  (W. W. Norton & Company, 2007). Kindle Edition. 

A Certain Dramatic Artfulness

Filed under: General,Geometry — Tags: , , , , — m759 @ 11:43 AM

See also a book found in a Log24 search for Tillich

Saturday, March 9, 2019

Weapons of Mass Distraction

Filed under: General — Tags: — m759 @ 12:46 PM

"Back to the Future" and . . .

I prefer another presentation from the above 
Universal Pictures date — June 28, 2018 —


Space Man from Plato


Friday, March 8, 2019

Photo Opportunity

Filed under: General — Tags: — m759 @ 12:30 PM

"I need a photo opportunity . . . ." — Paul Simon

A Logo for Sheinberg

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

Thursday, February 28, 2019

Previn’s Wake

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

A search for Previn in this  journal yields . . . 

"whyse Salmonson set his seel on a hexengown,"
Finnegans Wake Book II, Episode 2, pp. 296-297


Filed under: General — m759 @ 10:12 AM

Galois (i.e., finite) fields described as 'deep modern algebra'

IMAGE- History of Mathematics in a Nutshell

The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.

Note: There is no Galois (i.e., finite) field with six elements, but
the theory  of finite fields underlies applications of six-set geometry.

Monday, January 21, 2019

Fall, 1939

Filed under: General — Tags: — m759 @ 8:48 AM

She'll always have Paris.

Meditation for the Champ de Mors

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

"his onesidemissing for an allblind alley
leading to an Irish plot in the Champ de Mors"
— James Joyce, Finnegans Wake

Sunday, January 20, 2019

Scope Resolution

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

Wikipedia on a programming term

The scope resolution operator helps to identify
and specify the context to which an identifier refers,
particularly by specifying a namespace. The specific
uses vary across different programming languages
with the notions of scoping.  In many languages
the scope resolution operator is written


In a completely different context, these four dots might represent
a geometric object  — the four-point plane .

Saturday, January 19, 2019

Spinning the Wake

Filed under: General — Tags: — m759 @ 11:32 PM

Friday, January 18, 2019

Location, Location, Location

Filed under: General — Tags: — m759 @ 9:01 PM

See also, from a post of November 1, 2018

The Woke Grids …

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

… as opposed to The Dreaming Jewels .

A July 2014 Amsterdam master's thesis on the Golay code
and Mathieu group —

"The properties of G24 and M24 are visualized by
four geometric objects:  the icosahedron, dodecahedron,
dodecadodecahedron, and the cubicuboctahedron."

Some "geometric objects"  — rectangular, square, and cubic arrays —
are even more fundamental than the above polyhedra.

A related image from a post of Dec. 1, 2018

Tuesday, December 18, 2018

Shadowhunter Tales

Filed under: General — Tags: , — m759 @ 12:59 PM

The recent post "Tales from Story Space," about the 18th birthday
of the protagonist in the TV series "Shadowhunters" (2016-),
suggests a review of the actual  18th birthday of actress Lily Collins.

Collins is shown below warding off evil with a magical rune as
a shadowhunter in the 2013 film "City of Bones" —

She turned 18 on March 18, 2007.  A paper on symmetry and logic
referenced here on that date displays the following "runes" of 
philosopher Charles Sanders Peirce

See also Adamantine Meditation  (Log24, Oct. 3, 2018)
and the webpage Geometry of the I Ching.

Sunday, December 16, 2018


Filed under: General,Geometry — Tags: , — m759 @ 2:30 PM


'The Eddington Song'

Tuesday, November 6, 2018

Bait and Tackle

Filed under: General — Tags: — m759 @ 1:13 PM

The Bait —

For the tackle, see a 1988 album from The Residents.

Monday, November 5, 2018

High Life at Sils Maria

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

Related art —

Sunday, November 4, 2018

“Look Up” — The Breakthrough Prize* Theme This Evening

Filed under: General — Tags: — m759 @ 10:45 PM

Looking up images for "The Space Theory of Truth" this evening —

Detail  (from the post "Logos" of Oct. 14)


Saturday, November 3, 2018

For St. Anselm

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

"… at his home in San Anselmo . . . ."

See also Anselm in this journal, as well as the Devil's Night post Ojos.

Tuesday, October 30, 2018


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

From the Hulu series 'The Path,' the Eye logo

A better term than “phase space” might be “story space.”

Sunday, September 9, 2018

Plan 9 Continues.

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

"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.

Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."

— From p. 192 of "The Phenomenology of Mathematical Proof,"
by Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics
(May, 1997), pp. 183-196. Published by: Springer.

Stable URL: https://www.jstor.org/stable/20117627.

Related figures —

Note the 3×3 subsquare containing the triangles ABC, etc.

"That in which space itself is contained" — Wallace Stevens

Friday, July 6, 2018


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

"… Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."

— T. S. Eliot, "Burnt Norton," 1936

"Read something that means something."

Advertising slogan for The New Yorker

The previous post quoted some mystic meditations of Octavio Paz
from 1974. I prefer some less mystic remarks of Eddington from
1938 (the Tanner Lectures) published by Cambridge U. Press in 1939 —

"… we have sixteen elements with which to form a group-structure" —

See as well posts tagged Dirac and Geometry.

Tuesday, May 1, 2018


Filed under: General — Tags: , — m759 @ 2:29 PM

Remarks on space from 1998 by sci-fi author Robert J. Sawyer quoted
here on Sunday (see the tag "Sawyer's Space") suggest a review of
rather similar remarks on space from 1977 by sci-fi author M. A. Foster
(see the tag "Foster's Space"):

Quoted here on September 26, 2012

"All she had to do was kick off and flow."

— The Gameplayers of Zan

"I'se so silly to be flowing but I no canna stay."

— Finnegans Wake

Another work by Sawyer —

Sunday, April 29, 2018


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

From the online New York Times  this afternoon:

Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.

From this  journal this morning:

"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to 
make a fancy amusement-park ride
There had to be more—"

Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168

"The sensation of folding space . . . ."

Or unfolding:

Click the above unfolded space for some background.


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

'Imprisoned in a tesseract' in a 1998 science fiction novel

Thursday, April 19, 2018

Something to Behold

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

From a review of a Joyce Carol Oates novel
at firstthings.com on August 23, 2013 —

"Though the Curse is eventually exorcised,
it is through an act of wit and guile,
not an act of repentance or reconciliation.
And so we may wonder if Oates has put this story
to rest, or if it simply lays dormant. A twenty-first
century eruption of the 'Crosswicks Curse
would be something to behold." [Link added.]

Related material —

A film version of A Wrinkle in Time

The Hamilton watch from "Interstellar" (2014) —

See also a post, Vacant Space, from 8/23/13 (the date
of the above review), and posts tagged Space Writer.

Saturday, April 7, 2018


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

The FBI holding cube in "The Blacklist" —

" 'The Front' is not the whole story . . . ."

— Vincent Canby, New York Times  film review, 1976,
     as quoted in Wikipedia.

See also Solomon's Cube in this  journal.

IMAGE- 'Solomon's Cube'

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

Monday, March 12, 2018

“Quantum Tesseract Theorem?”

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

Remarks related to a recent film and a not-so-recent film.

For some historical background, see Dirac and Geometry in this journal.

Also (as Thas mentions) after Saniga and Planat —

The Saniga-Planat paper was submitted on December 21, 2006.

Excerpts from this  journal on that date —

A Halmos tombstone and the tale of HAL and the pod bay doors

     "Open the pod bay doors, HAL."

Wednesday, March 7, 2018


Filed under: General — Tags: — m759 @ 5:48 PM

"How do you  get young people excited about space?"

— Megan Garber in The Atlantic , Aug. 16, 2012

The above quote is from this  journal  on 9/11, 2014.

Related material —

Synchronology for the above date — 9/11, 2014 —

A BuzzFeed article with that date, and in reply

"A Personal Statement from Michael Shermer" with that date.

Wednesday, February 28, 2018

A Girl’s Guide to Chaos

Filed under: General — Tags: — m759 @ 9:55 AM

The title is that of a play mentioned last night in
a New York Times  obituary .

Related recent film lines —

  • Thor:  How do I escape?
  • Heimdall:  You're on a planet surrounded by doorways.
    Go through one.
  • Thor:  Which one?
  • Heimdall:  The big one!

Related material from this  journal on Jan. 20, 2018 —

The Chaos Symbol of Dan Brown.

Mathematics and Narrative

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


Excerpts from a post of May 25, 2005 —

Above is an example I like of mathematics….

Here is an example I like of narrative:

Kate felt quite dizzy. She didn't know exactly what it was
that had just happened, but she felt pretty damn  certain  that
it  was  the  sort of experience that her mother would not have
approved of on a first date.
     "Is this all part of what we have to do to go to  Asgard?"
she said. "Or are you just fooling around?"
     "We will go to Asgard...now," he said.
     At that moment he raised his hand as if to pluck an apple,
but instead of plucking he made a tiny, sharp turning movement.
The effect  was as if he had twisted the entire world through a
billionth part of a billionth  part  of  a  degree.  Everything
shifted,  was  for  a  moment  minutely  out of focus, and then
snapped back again as a suddenly different world.

— Douglas Adams, The Long Dark Tea-Time of the Soul

Image from a different  different world —

Hat-tip to a related Feb. 26 weblog post
at the American Mathematical Society.

Thursday, January 25, 2018

Beware of Analogical Extension

Filed under: General,Geometry — Tags: — m759 @ 11:29 AM

"By an archetype  I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."

— Max Black in Models and Metaphors 
    (Cornell, 1962, p. 241)

"Others … spoke of 'ultimate frames of reference' …."

A "frame of reference" for the concept  four quartets

A less reputable analogical extension  of the same
frame of reference

Madeleine L'Engle in A Swiftly Tilting Planet :

"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."

See also the phrase Galois tesseract .

Thursday, November 23, 2017

Lévi-Strauss vs. Propp

Filed under: General,Geometry — Tags: , , — m759 @ 12:25 PM
​Claude Lévi-Strauss

From his Structure and Form:
Reflections on a Work by Vladimir Propp
” *

To maintain. as I have done. that the permutability of contents is not arbitrary amounts to saying that, if the analysis is carried to a sufficiently deep level, behind diversity we will discover constancy. And, of course. the avowed constancy of form must not hide from us that functions are also permutable.

The structure of the folktale as it is illustrated by Propp presents a chronological succession of qualitatively distinct functions. each constituting an independent genre. One can wonder whether—as with dramatis personae and their attributes— Propp does not stop too soon, seeking the form too close to the level of empirical observation. Among the thirty-one functions that he distinguishes, several are reducible to the same  function reappearing at different  moments of the narrative but after undergoing one or a number of transformations . I have already suggested that this could be true of the false hero (a transformation of the villain), of assigning a difficult task (a transformation of the test), etc. (see p. 181 above), and that in this case the two parties  constituting the fundamental tale would themselves be transformations of each other.

Nothing prevents pushing this reduction even further and analyzing each separate partie  into a small number of recurrent functions, so that several of Propp’s functions would constitute groups of transformations of one and the same function. We could treat the “violation” as the reverse of the “prohibition” and the latter as a negative transformation of the “injunction.” The “departure” of the hero and his “return” would appear as the negative and positive expressions of the same disjunctive function. The “quest” of the hero (hero pursues someone or something) would become the opposite of “pursuit” (hero is pursued by something or someone), etc.

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).

These critical remarks are certainly valid for the method used by Propp and for his conclusions. However. it cannot be stressed enough that Propp envisioned them and in several places formulated with perfect clarity the solutions I have just suggested. Let us take up again from this viewpoint the two essential themes of our discussion: constancy of the content (in spite of its permutability) and permutability of functions (in spite of their constancy).

* Translated from a 1960 work in French.  It appeared in English as Chapter VIII
of Structural Anthropology, Volume 2  (U. of Chicago Press, 1976).  Chapter VIII
was originally published in Cahiers de l’Institut de Science 
Économique Appliquée , 
No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).

See also “Lévi-Strauss” + Formula  in this journal.

Some background related to the previous post

Wednesday, October 25, 2017

To the Egress

Filed under: General — Tags: — m759 @ 9:24 PM

The New York Times  at 8:22 PM ET

"Knight Landesman, a longtime publisher of Artforum magazine
and a power broker in the art world, resigned on Wednesday
afternoon, hours after a lawsuit was filed in New York accusing
him of sexually harassing at least nine women in episodes that
stretched back almost a decade."

See as well, in this  journal, Way to the Egress.

The Palo Alto Edge

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

From Stanford — The death on October 9, 2017, of a man who
“always wanted to be at the most cutting of cutting-edge technology.”

Related material from Log24 on April 26, 2017

A sketch, adapted from Girl Scouts of Palo Alto —

Click the sketch for further details.

Saturday, September 23, 2017

The Turn of the Frame

Filed under: General,Geometry — Tags: , , — m759 @ 2:19 AM

"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."

— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies  No. 55/56 (1977), pp. 94-207.
Published by Yale University Press.

See also the previous post and The Galois Tesseract.

Sunday, August 27, 2017

Black Well

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

The “Black” of the title refers to the previous post.
For the “Well,” see Hexagram 48.

Related material —

The Galois Tesseract and, more generally, Binary Coordinate Systems.

Friday, June 16, 2017

Chalkroom Jungle

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

At MASS MoCA, the installation "Chalkroom" quotes a lyric —

Oh beauty in all its forms
funny how hatred can also be a beautiful thing
When it's as sharp as a knife
as hard as a diamond


— From "One Beautiful Evening," by Laurie Anderson.

See also the previous post and "Smallest Perfect" in this journal.

Thursday, June 15, 2017

Building Six

Filed under: General — Tags: , , — m759 @ 7:47 PM

Berkshire tales of May 25, 2017 —

See also, in this  journal from May 25 and earlier, posts now tagged
"The Story of Six."

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