Monday, June 1, 2015

Excited about Space

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

Background — A search for the title in this journal.

Grothendieck was at times excited about space:

"The notion of space  is certainly one of the oldest
in mathematics. It is fundamental to our 'geometric'
perspective on the world, and has been so tacitly
for over two millenia. It's only over the course of the
19th century that this concept has, bit-by-bit, freed
itself from the tyranny of our immediate perceptions
(that is, one and the same as the 'space' that
surrounds us), and of its traditional theoretical
treatment (Euclidean), to attain to its present
dynamism and autonomy. In our own times it has
joined the ranks of those notions that are most freely
and universally employed in mathematics, and is
familiar, I would say, to every mathematician
without exception. It has become a concept of multiple
and varied aspects, of hundreds of thousands of faces…."

— fermentmagazine.org/rands/promenade12.html

An aspect not  so familiar:  Diamond Space.

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.

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."

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.

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.

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.

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

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:

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.)

Tuesday, July 16, 2013

Child Buyers

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

The title refers to a classic 1960 novel by John Hersey.

“How do you  get young people excited about space?”

— Megan Garber in The Atlantic , Aug. 16, 2012
(Italics added.) (See previous four posts.)

Allyn Jackson on “Simplicity, in Mathematics and in Art,”
in the new August 2013 issue of Notices of the American
Mathematical Society

“As conventions evolve, so do notions of simplicity.
Franks mentioned Gauss’s 1831 paper that
established the respectability of complex numbers.”

This suggests a related image by Gauss, with a
remark on simplicity—

IMAGE- Complex Grid, by Gauss

Here Gauss’s diagram is not, as may appear at first glance,
a 3×3 array of squares, but is rather a 4×4 array of discrete
points (part of an infinite plane array).

Related material that does  feature the somewhat simpler 3×3 array
of squares, not  seen as part of an infinite array—

Marketing the Holy Field

IMAGE- The Ninefold Square, in China 'The Holy Field'

Click image for the original post.

For a purely mathematical view of the holy field, see Visualizing GL(2,p).

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

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

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.

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

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.

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, 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 .

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."

Saturday, June 3, 2017

Expanding the Spielraum (Continued*)

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

Or:  The Square

"What we do may be small, but it has
 a certain character of permanence."
— G. H. Hardy

* See Expanding the Spielraum in this journal.

Tuesday, May 23, 2017

Pursued by a Biplane

Filed under: General,Geometry — Tags: — m759 @ 9:41 PM

The Galois Tesseract as a biplane —

Saturday, May 20, 2017

The Ludicrous Extreme

Filed under: General,Geometry — Tags: — m759 @ 1:04 AM

From a review of the 2016 film "Arrival"

"A seemingly off-hand reference to Abbott and Costello
is our gateway. In a movie as generally humorless as Arrival,
the jokes mean something. Ironically, it is Donnelly, not Banks,
who initiates the joke, naming the verbally inexpressive
Heptapod aliens after the loquacious Classical Hollywood
comedians. The squid-like aliens communicate via those beautiful,
cryptic images. Those signs, when thoroughly comprehended,
open the perceiver to a nonlinear conception of time; this is
Sapir-Whorf taken to the ludicrous extreme."

Jordan Brower in the Los Angeles Review of Books

Further on in the review —

"Banks doesn’t fully understand the alien language, but she
knows it well enough to get by. This realization emerges
most evidently when Banks enters the alien ship and, floating
alongside Costello, converses with it in their picture-language.
She asks where Abbott is, and it responds — as presented
in subtitling — that Abbott 'is death process.'
'Death process' — dying — is not idiomatic English, and what
we see, written for us, is not a perfect translation but a
rendering of Banks’s understanding. This, it seems to me, is a
crucial moment marking the hard limit of a human mind,
working within the confines of human language to understand
an ultimately intractable xenolinguistic system."

For what may seem like an intractable xenolinguistic system to
those whose experience of mathematics is limited to portrayals
by Hollywood, see the previous post —

van Lint and Wilson Meet the Galois Tesseract.

The death process of van Lint occurred on Sept. 28, 2004.

See this journal on that date

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.

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

Thursday, April 27, 2017

Partner, Anchor, Decompose

Filed under: General,Geometry — Tags: — m759 @ 12:31 PM

See also a figure from 2 AM ET April 26 

" Partner, anchor, decompose. That's not math.
That's the plot to 'Silence of the Lambs.' "

Greg Gutfeld, September 2014

Wednesday, April 26, 2017


Filed under: General,Geometry — Tags: — m759 @ 6:18 PM

See also "Cornerstone" in this journal and

A sidebar from a Google search today —

'The Square of Opposition: A Cornerstone of Thought'

This suggests a review of posts now tagged Obelisk,
which include


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

From a post of March 16, 2017

"Bulk apperception" is defined in the Westworld script 
as "basically, overall intelligence."  The phrase is apparently
unique to Westworld.

These two words do, however, nearly  occur together
in at least one book — Andrew Feffer's The Chicago
Pragmatists and American Progressivism


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

A post in memory of film director Jonathan Demme

'What Goes Around Comes Around: The Films of Jonathan Demme'

A Tale Unfolded

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

A sketch, adapted tonight from Girl Scouts of Palo Alto

From the April 14 noon post High Concept

From the April 14 3 AM post Hudson and Finite Geometry

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

From the April 24 evening post The Trials of Device

Pentagon with pentagram    

Note that Hudson's 1905 "unfolding" of even and odd puts even on top of
the square array, but my own 2013 unfolding above puts even at its left.

Monday, April 24, 2017

The Trials of Device

Filed under: General,Geometry — Tags: , , — m759 @ 3:28 PM

"A blank underlies the trials of device"
— Wallace Stevens, "An Ordinary Evening in New Haven" (1950)

A possible meaning for the phrase "the trials of device" —

See also Log24 posts mentioning a particular device, the pentagram .

For instance —

Wittgenstein's pentagram and 4x4 'counting-pattern'

Related figures

Pentagon with pentagram    

Monday, April 17, 2017


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

Related art —

See also the previous post.

Saturday, April 15, 2017

Quanta Dating

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

From Quanta Magazine  —

For the Church of Synchronology

See also this  journal on July 17, 2014, and March 28, 2017.

Friday, April 14, 2017

Hudson and Finite Geometry

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

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

The above four-element sets of black subsquares of a 4×4 square array 
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .

Hudson did not  view these 35 tetrads as planes through the origin in a finite
affine 4-space (or, equivalently, as lines in the corresponding finite projective

In order to view them in this way, one can view the tetrads as derived,
via the 15 two-element subsets of a six-element set, from the 16 elements
of the binary Galois affine space pictured above at top left.

This space is formed by taking symmetric-difference (Galois binary)
sums of the 15 two-element subsets, and identifying any resulting four-
element (or, summing three disjoint two-element subsets, six-element)
subsets with their complements.  This process was described in my note
"The 2-subsets of a 6-set are the points of a PG(3,2)" of May 26, 1986.

The space was later described in the following —

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

Wednesday, April 12, 2017

Contracting the Spielraum

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

The contraction of the title is from group actions on
the ninefold square  (with the center subsquare fixed)
to group actions on the eightfold cube.

From a post of June 4, 2014

At math.stackexchange.com on March 1-12, 2013:

Is there a geometric realization of the Quaternion group?” —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).

Expanding the Spielraum

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

Cézanne's Greetings.

"Cézanne ignores the laws of classical perspective . . . ."

— Voorhies, James. “Paul Cézanne (1839–1906).”
In Heilbrunn Timeline of Art History .  New York:
The Metropolitan Museum of Art, 2000–. (October 2004)

Some others do not.

This is what I called "the large Desargues configuration
in posts of April 2013 and later.

Tuesday, April 4, 2017

White Cube

Filed under: General — Tags: , — m759 @ 12:21 PM

"Inside the White Cube" —

"We have now reached
a point where we see
not the art but the space first….
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."


"Space: what you
damn well have to see."

— James Joyce, Ulysses  

Monday, April 3, 2017

Even Core

Filed under: General,Geometry — Tags: , — m759 @ 9:16 PM

4x4x4 gray cube


Odd Core

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


3x3x3 Galois cube, gray and white

Wednesday, March 29, 2017

The Crimson Abyss

Filed under: General,Geometry — Tags: , — m759 @ 3:19 PM

"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 , March 29


IMAGE- 'Affine Groups on Small Binary Spaces,' illustration


Logo design for Stack Exchange Math by Jin Yang

Recent posts now tagged Crimson Abyss suggest
the above logo be viewed in light of a certain page 29

"… as if into a crimson abyss …." —

Update of 9 PM ET March 29, 2017:

Prospero's Children  was first published by HarperCollins,
London, in 1999. A statement by the publisher provides
an instance of the famous "much-needed gap." —

"This is English fantasy at its finest. Prospero’s Children 
steps into the gap that exists between The Lion, the Witch
and the Wardrobe
  and Clive Barker’s Weaveworld , and
is destined to become a modern classic."

Related imagery —

See also "Hexagram 64 in Context" (Log24, March 16, 2017).

Design Abyss

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

Hexagram 29,
The Abyss (Water)

This post was suggested by an August 6, 2010, post by the designer
(in summer or fall, 2010) of the Stack Exchange math logo (see
the previous Log24 post, Art Space Illustrated) —

http://www.8164.org/☵☲/  .

In that post, the designer quotes the Wilhelm/Baynes I Ching  to explain
his choice of Hexagram 63, Water Over Fire, as a personal icon —

"When water in a kettle hangs over fire, the two elements
stand in relation and thus generate energy (cf. the
production of steam). But the resulting tension demands
caution. If the water boils over, the fire is extinguished
and its energy is lost. If the heat is too great, the water
evaporates into the air. These elements here brought in
to relation and thus generating energy are by nature
hostile to each other. Only the most extreme caution
can prevent damage."

See also this  journal on Walpurgisnacht (April 30), 2010 —


Hexagram 29:



Hexagram 30:

"Hates California,
it's cold and it's damp.

Image--'The Fire,' by Katherine Neville

A thought from another German-speaking philosopher

"Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache."

See also The Crimson 's abyss in today's 4:35 AM post Art Space, Continued.

Tuesday, March 28, 2017

Bit by Bit

Filed under: General,Geometry — Tags: , — m759 @ 11:45 AM

From Log24, "Cube Bricks 1984" —

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

Also on March 9, 2017 —

For those who prefer graphic  art —

Broken Symmetries  in  Diamond Space  


Filed under: General,Geometry — Tags: , — m759 @ 12:06 AM

Click here to enlarge.  Click the image for the source page.

The "this page" reference is to …

Finite Geometry of the Square and Cube.

Also from March 14, 2017 —

Related material

'Children of the Central Structure,' adapted from 'Children of the Damned'

Thursday, March 2, 2017


Filed under: General — Tags: — m759 @ 1:40 PM

Or:  "A Hologram for the King" Meets "Big"

* A reference to an alleged motto of Plato's Academy


Filed under: General,Geometry — Tags: — m759 @ 12:48 PM

"We tell ourselves stories in order to live." — Joan Didion

The New York Times Magazine  online today —

"As a former believer and now a nonbeliever, Carrère,
seeking answers, sets out, in The Kingdom , to tell
the story of the storytellers. He is trying to understand
what it takes to be able to tell a story, any story.
And what he finds, once again, is that you have to find
your role in it."

Wyatt Mason in The New York Times Magazine ,
     online March 2, 2017 

Like Tom Hanks?

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Click image for related posts.

Wednesday, October 5, 2016


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

From a Google image search yesterday

Sources (left to right, top to bottom) —

Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)

Monday, September 26, 2016

The Embedding

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

From this morning's 3:33 AM ET post

Adapted from a post of Dec. 8, 2012, "Defining the Contest" —

      From a post of Sept. 22,
  "Binary Opposition Illustrated" —

From Sunday's news


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

Eleanor Arroway and Palmer Joss in the "Occam's Razor"
scene from the 1997 film "Contact"

Sunday, September 25, 2016

Introduction to Pragmatism

Filed under: General — Tags: — m759 @ 7:29 AM

Stanford Encyclopedia of Philosophy
on the origins of Pragmatism:

"Pragmatism had been born in the discussions at
a ‘metaphysical club’ in Harvard around 1870
(see Menand…*). Peirce and James participated
in these discussions along with some other philosophers
and philosophically inclined lawyers. As we have
already noted, Peirce developed these ideas in his
publications from the 1870s."

From "How to Make Our Ideas Clear,"
by Charles Sanders Peirce in 1878 —

"The very first lesson that we have a right to demand that logic shall teach us is, how to make our ideas clear; and a most important one it is, depreciated only by minds who stand in need of it. To know what we think, to be masters of our own meaning, will make a solid foundation for great and weighty thought. It is most easily learned by those whose ideas are meagre and restricted; and far happier they than such as wallow helplessly in a rich mud of conceptions. A nation, it is true, may, in the course of generations, overcome the disadvantage of an excessive wealth of language and its natural concomitant, a vast, unfathomable deep of ideas. We may see it in history, slowly perfecting its literary forms, sloughing at length its metaphysics, and, by virtue of the untirable patience which is often a compensation, attaining great excellence in every branch of mental acquirement. The page of history is not yet unrolled which is to tell us whether such a people will or will not in the long-run prevail over one whose ideas (like the words of their language) are few, but which possesses a wonderful mastery over those which it has. For an individual, however, there can be no question that a few clear ideas are worth more than many confused ones. A young man would hardly be persuaded to sacrifice the greater part of his thoughts to save the rest; and the muddled head is the least apt to see the necessity of such a sacrifice. Him we can usually only commiserate, as a person with a congenital defect. Time will help him, but intellectual maturity with regard to clearness comes rather late, an unfortunate arrangement of Nature, inasmuch as clearness is of less use to a man settled in life, whose errors have in great measure had their effect, than it would be to one whose path lies before him. It is terrible to see how a single unclear idea, a single formula without meaning, lurking in a young man's head, will sometimes act like an obstruction of inert matter in an artery, hindering the nutrition of the brain, and condemning its victim to pine away in the fullness of his intellectual vigor and in the midst of intellectual plenty. Many a man has cherished for years as his hobby some vague shadow of an idea, too meaningless to be positively false; he has, nevertheless, passionately loved it, has made it his companion by day and by night, and has given to it his strength and his life, leaving all other occupations for its sake, and in short has lived with it and for it, until it has become, as it were, flesh of his flesh and bone of his bone; and then he has waked up some bright morning to find it gone, clean vanished away like the beautiful Melusina of the fable, and the essence of his life gone with it. I have myself known such a man; and who can tell how many histories of circle-squarers, metaphysicians, astrologers, and what not, may not be told in the old German story?"

Peirce himself may or may not have been entirely successful
in making his ideas clear.  See Where Credit Is Due  (Log24, 
June 11, 2016) and the Wikipedia article Categories (Peirce).

* Menand, L., 2001. The Metaphysical Club A Story of
Ideas in America
, New York:  Farrar, Straus and Giroux

Saturday, September 24, 2016

The Seven Seals

Filed under: General,Geometry — Tags: — m759 @ 7:23 AM

From Hermann Weyl's 1952 classic Symmetry —

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

Some Galois geometry —

See the previous post for more narrative.

Core Structure

Filed under: General,Geometry — Tags: — m759 @ 6:40 AM

For the director of "Interstellar" and "Inception"

At the core of the 4x4x4 cube is …


                                                      Cover modified.

The Eightfold Cube

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 29, 2016

Sunday School

Filed under: General — Tags: — m759 @ 8:00 AM


Tuesday, May 10, 2016

Game Theory

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

The following passage appeared in this journal
on the night of May 23-24, 2015.

IMAGE- A fictional vision of resurrection within a tesseract

The afternoon  of May 23, 2015, was significant
for devotees of mathematics and narrative.

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)

Friday, April 8, 2016

Ogdoads by Curtis

Filed under: General,Geometry — Tags: , — m759 @ 12:25 PM

As was previously noted here, the construction of the Miracle Octad Generator
of R. T. Curtis in 1974 involved his "folding" the 1×8 octads constructed in 1967
by Turyn into 2×4 form.

This resulted in a way of picturing a well-known correspondence (Conwell, 1910)
between partitions of an 8-set and lines of the projective 3-space PG(3,2).

For some background related to the "ogdoads" of the previous post, see
A Seventh Seal (Sept. 15, 2014).

Wednesday, December 16, 2015

Board Awards

Filed under: General — m759 @ 2:45 PM

“How do you  get young people excited about space?”

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

Sunday, December 6, 2015

Form and Idea

Filed under: General,Geometry — Tags: , , , — m759 @ 3:24 PM

"Those early works are succinct and uncompromising
in how they give shape to the philosophical perplexities
of form and idea…."

J. J. Charlesworth, artnet news, Dec. 16, 2014

"Form" and "idea" are somewhat synonymous, 
as opposed to "form" and "substance." A reading:

IMAGE- 'American Hustle' and Art Cube


Wednesday, August 26, 2015

“The Quality Without a Name”

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

The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander's book
The Timeless Way of Building  (Oxford University Press, 1979).

A quote from the publisher:

"Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself."

Three paragraphs from the book (pp. xiii-xiv):

19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.

20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.

21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.

Compare to, and contrast with, these illustrations of "Boolean space":

(See also similar illustrations from Berkeley and Purdue.)

Detail of the above image —

Note the "unfolding," as Christopher Alexander would have it.

These "Boolean" spaces of 1, 2, 4, 8, and 16 points
are also Galois  spaces.  See the diamond theorem —

Tuesday, June 9, 2015

Colorful Song

Filed under: General,Geometry — Tags: — m759 @ 8:40 PM

For geeks* —

Domain, Domain on the Range , "

where Domain = the Galois tesseract  and
Range = the four-element Galois field.

This post was suggested by the previous post,
by a Log24 search for Knight + Move, and by
the phrase "discouraging words" found in that search.

* A term from the 1947 film "Nightmare Alley."

Saturday, May 23, 2015


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

On the artist Hilma af Klint (1862-1944):

"She belonged to a group called 'The Five'…."

Related material — Real Life (Log24, May 20, 2015).

From that post:

IMAGE- Immersion in a fictional vision of resurrection within a tesseract

Wednesday, May 20, 2015

Real Life

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

From Amazon.com —

From the Milwaukee Journal Sentinel  Tuesday afternoon —

A 46-year-old Jesuit priest who was a Marquette University
assistant professor of theology collapsed on campus
Tuesday morning and died, President Michael Lovell
announced to the campus community in an email.

"Rev. Lúcás (Yiu Sing Luke) Chan, S.J., died after
collapsing this morning in Marquette Hall. Just last Sunday,
Father Chan offered the invocation at the Klingler College
of Arts and Sciences graduation ceremony."

Synchronicity check

From this journal on the above publication date of
Chan's book — Sept. 20, 2012 —

IMAGE- Immersion in a fictional vision of resurrection within a tesseract

From a Log24 post on the preceding day, Sept. 19, 2012 —

The Game in the Ship cannot be approached as a job,
a vocation, a career, or a recreation. To the contrary,
it is Life and Death itself at work there. In the Inner Game,
we call the Game Dhum Welur , the Mind of God."

 — The Gameplayers of Zan

Wednesday, April 22, 2015

Purely Aesthetic

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

G. H. Hardy in A Mathematician's Apology —

What ‘purely aesthetic’ qualities can we distinguish in such theorems as Euclid’s or Pythagoras’s?

I will not risk more than a few disjointed remarks. In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way. 

Related material:

Thursday, March 26, 2015

The Möbius Hypercube

Filed under: General,Geometry — Tags: , — m759 @ 12:31 AM

The incidences of points and planes in the
Möbius 8 configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.* 
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and face-planes of two mutually inscribed tetrahedra —

Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his 
Fig. 24, a schematic representation of the point-plane
incidences, as dotless, and some as hollow dots.  The figure,
"Gallucci's version of Möbius's 84," is shown below.
The hollow dots, representing the 8 points  (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.

Here a plane  (represented by a dotless intersection) contains
the four points  that are represented in the square array as lying
in the same row or same column as the plane. 

The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube. 

In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.

Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in 
Coxeter's labels above may be viewed as naming the positions 
of the 1's in (0,1) vectors (x4, x3, x2, x1) over the two-element
Galois field.  In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .

*  "Self-Dual Configurations and Regular Graphs," 
    Bulletin of the American Mathematical Society,
    Vol. 56 (1950), pp. 413-455

The subscripts' usual 1-2-3-4 order is reversed as a reminder
    that such a vector may be viewed as labeling a binary number 
    from 0  through 15, or alternately as labeling a polynomial in
    the 16-element Galois field GF(24).  See the Log24 post
     Vector Addition in a Finite Field (Jan. 5, 2013).

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, February 3, 2015

Expanding the Spielraum

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

A short poem by several authors:

"The role of
the 16 singular points
on the Kummer surface
is now played by
the 64 singular points
on the Kummer threefold."

— From Remark 2.4 on page 9 of
"The Universal Kummer Threefold,"
by Qingchun Ren, Steven V Sam,
Gus Schrader, and Bernd Sturmfels,
August 6, 2012 — June 12, 2013.

See also "Expanded Field" in this journal.

IMAGE- Concepts of Space

Illustration from "Sunday School," July 20, 2014.

Other Log24 background:  Kummer, Spielraum, Art Space.

Friday, January 23, 2015

Complex Symplectic Fantasy

Filed under: General — Tags: — m759 @ 8:08 PM

"We are not isolated free chosers,
monarchs of all we survey, but
benighted creatures sunk in a reality
whose nature we are constantly and
overwhelmingly tempted to deform
by fantasy."

—Iris Murdoch, "Against Dryness"
in Encounter , p. 20 of issue 88 
(vol. 16 no. 1, January 1961, pp. 16-20)

"We need to turn our attention away from the consoling
dream necessity of Romanticism, away from the dry
symbol, the bogus individual, the false whole, towards
the real impenetrable human person."

— Iris Murdoch, 1961

"Impenetrability!  That's what I  say!"

Humpty Dumpty, 1871

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:

Thursday, September 11, 2014

Oh, Moon of Alabama

Filed under: General — Tags: — m759 @ 11:09 AM

The mention of Gauss in today's previous post, along with
recent news, suggested this post.

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

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

Further details:  Child Buyers (July 16, 2013).

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.]

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 .

Tuesday, February 25, 2014

The Argument

Filed under: General — Tags: — m759 @ 8:30 PM

The title of a review of Charles Taylor's book A Secular Age
was quoted here at noon last Saturday —

"The Place of the Sacred
in the Absence of God

My comment from last Saturday —

"The place of the sacred is not, perhaps,
Davos, but a more abstract location."

A sequel —

"Religious Experience and the Modern Self,"
by Ross Douthat in The New York Times 
today at 4:25 PM ET —

"The argument comes from the Canadian
philosopher Charles Taylor and his
doorstop-thick magnum opus A Secular Age …." 

Related material: 

Helprin Doors and Doorstop Thick.

Tuesday, January 21, 2014

Class of 64

Filed under: General — Tags: — m759 @ 7:31 AM

NY Times  researcher from this morning’s previous post
tweeted last fall about art forgery and China.

Related material — Art Cube.

IMAGE- 'American Hustle' and Art Cube

Illustration from December 25, 2013.

Wednesday, December 25, 2013

Rotating the Facts

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

IMAGE- 'American Hustle' and Art Cube

"She never looked up while her mind rotated the facts,
trying to see them from all sides, trying to piece them
together into theory. All she could think was that she
was flunking an IQ test."

— Steve Martin, An Object of Beauty

"So you should not feel so all alone…"
— Adapted song lyric

Wednesday, December 18, 2013

Bing Bang Theory

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

Microsoft in 2009 on its new search engine name—

"We like Bing because it sounds off in our heads
when we think about that moment of discovery
and decision making— when you resolve those
important tasks."

A search on Bing today —

IMAGE- Top search result on Bing for 'diamond space' on Dec. 18, 2013

A colorful tale —

IMAGE- The Diamond 16 Puzzle, with commentary

"Bing bang, I saw the whole gang
Bobby Darin, 1958

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