Wednesday, April 15, 2020

Oslo Prophet (after Varignon)

Filed under: General — Tags: , — m759 @ 12:06 PM

See also Invariance, a Log24 post from yesterday morning —

Note the resemblance to Plato’s Diamond.

Friday, January 11, 2019

Permutations at Oslo

Filed under: General — Tags: , , — m759 @ 8:45 PM

Webpage at Oslo of Josefine Lyche, 'Plato's Diamond'

See also yesterday’s  Archimedes at Hiroshima  and the
above 24 graphic permutations on  All Souls’ Day 2010.

For some backstory, see Narrative Line (November 10, 2014).

Monday, January 12, 2015

Points Omega*

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

The previous post displayed a set of
24 unit-square “points” within a rectangular array.
These are the points of the
Miracle Octad Generator  of R. T. Curtis.

The array was labeled  Ω
because that is the usual designation for
a set acted upon by a group:

* The title is an allusion to Point Omega , a novel by
Don DeLillo published on Groundhog Day 2010.
See “Point Omega” in this journal.

Friday, May 21, 2010

The Oslo Version

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

From an art exhibition in Oslo last year–

Image-- Josefine Lyche's combination of Polster's phrase with Cullinane's images in her gallery show, Oslo, 2009-- 'The Smallest Perfect Universe -- Points and Hyperplanes'

The artist's description above is not in correct left-to-right order.
Actually the hyperplanes above are at left, the points at right.

Compare to "Picturing the Smallest Projective 3-Space,"
a note of mine from April 26, 1986—

Image-- Points and hyperplanes in the finite 3-space PG(3,2), April 1986, by Cullinane

Click for the original full version.

Compare also to Burkard Polster's original use of
the phrase "the smallest perfect universe."

Polster's tetrahedral model of points and hyperplanes
is quite different from my own square version above.

See also Cullinane on Polster.

Here are links to the gallery press release
and the artist's own photos.

Monday, June 1, 2020

The Gefter Boundary

Filed under: General — Tags: , — m759 @ 5:09 PM

“The message was clear: having a finite frame of reference
creates the illusion of a world, but even the reference frame itself
is an illusion. Observers create reality, but observers aren’t real.
There is nothing ontologically distinct about an observer, because
you can always find a frame in which that observer disappears:
the frame of the frame itself, the boundary of the boundary.”

— Amanda Gefter in 2014, quoted here on Mayday 2020.

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

See as well the previous post.

A Graveyard Smash: Galois Geometry Meets Nordic Aliens

Filed under: General — Tags: — m759 @ 4:02 PM

See also Vril Chick.

Tuesday, April 14, 2020


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

Note the resemblance to Plato’s Diamond.

Click the Pritchard passage above for an interactive version.

Monday, February 2, 2015

Spielraum as Ω

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

From "Origins of the Logical Theory of Probability: von Kries, Wittgenstein, Waismann," by Michael Heidelberger —

"Von Kries calls a range of objective possibilities of a hypothesis or event (under given laws) its Spielraum   (literally: play space), which can mean ‘room to move’, ‘leeway’, ‘latitude of choice’, ‘degree of freedom’ or ‘free play’ and ‘clearance’ – or even ‘scope’. John Maynard Keynes translated it as ‘field’, but the term ‘range’ has generally been adopted in English. Von Kries now holds that if numerical probability were to make any sense at all it must be through this concept of the Spielraum  . Von Kries’s theory is therefore called a ‘Spielraum  theory’ or ‘range theory of probability’."

— International Studies in the Philosophy of Science , Volume 15, Issue 2, 2001, pp. 177-188

See also the tag Points Omega
(Scroll down to January 11-12, 2015.)

Related material:

"Now, for example, in how far are
the six sides of a symmetric die
'equally possible' upon throwing?"

— From "The Natural-Range Conception
     of Probability," by Dr. Jacob Rosenthal,
     page 73 in Time, Chance, and
     Reduction: Philosophical Aspects of
Mechanics , ed. by 
     Gerhard Ernst and Andreas Hüttemann, 
     Cambridge U. Press, 2010, pp. 71-90

Sunday, January 11, 2015

Real Beyond Artifice

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

A professor at Harvard has written about
“the urge to seize and display something
real beyond artifice.”

He reportedly died on January 3, 2015.

An image from this journal on that date:

Another Gitterkrieg  image:

 The 24-set   Ω  of  R. T. Curtis

Click on the images for related material.

Thursday, June 23, 2011

Script Magic…

Filed under: General — m759 @ 3:33 AM

In a Jewish Cathedral

From The New York Times Magazine  of Sunday, April 6, 1986—

"David Rayfiel's Script Magic" by Alex Ward

WHEN THE CALL came last year to revise ''The Morning After,'' Rayfiel was working on a screenplay about the 1911 Triangle Shirtwaist Company fire for Barbra Streisand and Jane Fonda. He has now resumed work— as the principal writer, not the reviser— on that script. But chances are good that he will have further interruptions. Pollack will probably call and say, as he usually does, ''David, I need access to your brain.'' And Rayfiel will probably say, as he usually does, ''That's O.K., I'm not using it.'' He will revise another script, and be reluctant about taking credit for it.

''I guess it's like the medieval stonecutters who worked on the cathedrals,'' he says. ''There's all that elaborate work. The saints were carved by one guy, the cherubs by someone else. They didn't care about getting credit, they knew what they'd done. I'm like that. I'm the guy who does the cherubs.''

Related material:

Proginoskes in this journal and Abracadabra from the midnight of June 18-19.

See also Rayfiel's obituary in today"s Times .

For some quite different work, also  from April 1986, see—

Oslo: Points and Hyperplanes.

Tuesday, April 26, 2011

25 Years Ago Today

Filed under: General,Geometry — m759 @ 11:02 PM

Picturing the smallest projective 3-space

       Click to enlarge.

The above points and hyperplanes underlie the symmetries discussed
in the diamond theorem. See The Oslo Version  and related remarks
for a different use in art.

Tuesday, June 15, 2010

Imago, Imago, Imago

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

Recommended— an online book—

Flight from Eden: The Origins of Modern Literary Criticism and Theory,
by Steven Cassedy, U. of California Press, 1990.

See in particular

Valéry and the Discourse On His Method.

Pages 156-157—

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. “Every act of understanding is based on a group,” he says (C, 1:331). “My specialty—reducing everything to the study of a system closed on itself and finite” (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one “group” undergoes a “transformation” and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: “The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind’s momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. “Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind” (O, 1:36). “Psychology is a theory of transformations, we just need to isolate the invariants and the groups” (C, 1:915). “Man is a system that transforms itself” (C, 2:896).


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

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

Compare Jung’s image in Aion  of the Self as a four-diamond figure:


and Cullinane’s purely geometric four-diamond figure:


For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.

What remains fixed (globally, not pointwise) under these transformations is the system  of points and hyperplanes from the diamond theorem. This system was depicted by artist Josefine Lyche in her installation “Theme and Variations” in Oslo in 2009.  Lyche titled this part of her installation “The Smallest Perfect Universe,” a phrase used earlier by Burkard Polster to describe the projective 3-space PG(3,2) that contains these points (at right below) and hyperplanes (at left below).

Image-- Josefine Lyche's combination of Polster's phrase with<br /> Cullinane's images in her gallery show, Oslo, 2009-- 'The Smallest<br /> Perfect Universe -- Points and Hyperplanes'

Although the system of points (at right above) and hyperplanes (at left above) exemplifies Valéry’s notion of invariant, it seems unlikely to be the sort of thing he had in mind as an image of the Self.

Sunday, May 23, 2010

For Your Consideration —

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

Cannes Festival Readies for Awards Night

Uncertified Copy

Image-- Uncertified copy of 1986 figures by Cullinane in a 2009 art exhibit in Oslo

The pictures in the detail are copies of
figures created by S. H. Cullinane in 1986.
They illustrate his model of hyperplanes
and points in the finite projective space
known as PG(3,2) that underlies
Cullinane's diamond theorem.

The title of the pictures in the detail
is that of a film by Burkard Polster
that portrays a rival model of PG(3,2).

The artist credits neither Cullinane nor Polster.

Wednesday, June 8, 2005

Wednesday June 8, 2005

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

Kernel of Eternity

Today is the feast day of Saint Gerard Manley Hopkins, “immortal diamond.”

“At that instant he saw, in one blaze of light, an image of unutterable conviction, the reason why the artist works and lives and has his being–the reward he seeks–the only reward he really cares about, without which there is nothing. It is to snare the spirits of mankind in nets of magic, to make his life prevail through his creation, to wreak the vision of his life, the rude and painful substance of his own experience, into the congruence of blazing and enchanted images that are themselves the core of life, the essential pattern whence all other things proceed, the kernel of eternity.”

— Thomas Wolfe, Of Time and the River

“… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)”

— Peter J. Cameron,
The Geometry of the Mathieu Groups (pdf)

“… donc Dieu existe, réponse!

— attributed, some say falsely, to Leonhard Euler

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