Friday, May 19, 2017

From Algebra to Geometry

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

Monday, May 25, 2015

A Stitch in Time

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

The most recent version of a passage
quoted in posts tagged "May 19 Gestalt" —

"You've got to pick up every stitch." — Donovan

Tuesday, May 19, 2015

The Lindbergh Manifesto

Filed under: Uncategorized — Tags: — m759 @ 3:24 AM

"Creation is the birth of something, and
something cannot come from nothing."

— Photographer Peter Lindbergh at his website

From a biography of Lindbergh —

" it took Lindbergh awhile to find his true métier.
Born in Krefeld, Germany, in 1944….
Barely out of his teens, he became a painter who
embraced conceptual art and — for reasons he
has since forgotten — adopted the professional
name « Sultan. »   Lindbergh was a few years
short of his 30th birthday when he turned to

— "The Man Who Loves Women," by Pamela Young,
Toronto Globe & Mail , September 19, 1996

A Lindbergh work (at right below) from his conceptual-art days —

For a connection between the above work by Paul Talman and the
above "Mono Type 1" of Lindbergh, see…

Monday, May 19, 2014

Cube Space

Filed under: Uncategorized — Tags: — m759 @ 8:00 PM

A sequel to this afternoon's Rubik Quote:

"The Cube was born in 1974 as a teaching tool
to help me and my students better understand
space and 3D. The Cube challenged us to find
order in chaos."

— Professor Ernő Rubik at Chrome Cube Lab

IMAGE- Weyl on symmetry

(Click image below to enlarge.)

Sunday, May 19, 2013

Priority Claim

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

From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):

"By our construction, this vector space is the dual
of our hypercube F24 built on I \ O9. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis
in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O9."

[Cur89] reference:
 R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 
32 (1989), 345-353 (received on
July 20, 1987).

— Anne Taormina and Katrin Wendland,
    "The overarching finite symmetry group of Kummer
      surfaces in the Mathieu group 24 ,"
     arXiv.org > hep-th > arXiv:1107.3834

"First mentioned by Curtis…."

No. I claim that to the best of my knowledge, the 
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.

Update of the above paragraph on July 6, 2013—

No. The vector space structure was described by
(for instance) Peter J. Cameron in a 1976
Cambridge University Press book —
Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pages 59 and 60.

The vector space structure as it occurs in a 4×4 array
of the sort that appears in the Curtis Miracle Octad
Generator may first have been pointed out by me,
Steven H. Cullinane,
 in an AMS abstract submitted in
October 1978, some nine years before the Curtis article.

See Notes on Finite Geometry for some background.

See in particular The Galois Tesseract.

For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).

Wednesday, May 19, 2010


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

Photo caption in NY Times  today— a pianist "preforming" in 1967. (See today's previous post.)

The pianist's life story seems in part to echo that of Juliette Binoche in the film "Bleu." Binoche appeared in this journal yesterday, before I had seen the pianist in today's Times  obituaries. The Binoche appearance was related to the blue diamond in the film "Duelle " (Tuesday morning's post) and the saying of Heraclitus "immortals mortal, mortals immortal" (Tuesday afternoon's post).

This somewhat uncanny echo brings to mind Nabokov

Life Everlasting—based on a misprint!
I mused as I drove homeward: take the hint,
And stop investigating my abyss?
But all at once it dawned on me that this
Was the real point, the contrapuntal theme;
Just this: not text, but texture; not the dream
But topsy-turvical coincidence,
Not flimsy nonsense, but a web of sense.

Whether sense or nonsense, the following quotation seems relevant—

"Archetypes function as living dispositions, ideas in the Platonic sense, that preform and continually influence our thoughts and feelings and actions." –C.G. Jung in Four Archetypes: Mother, Rebirth, Spirit, Trickster, the section titled "On the Concept of the Archetype."

That section is notable for its likening of Jungian archetypes to Platonic ideas and to axial systems of crystals. See also "Cubist Tune," March 18 —


Blue tesseract cover<br /><br />
art, blue crystals in 'Bleu,' lines from 'Blue Guitar'

Blue Note à Quatre

Filed under: Uncategorized — Tags: , — m759 @ 3:00 AM

The Concert à Quatre  "was Messiaen's last work, left unfinished on his desk at his death. His widow undoubtedly followed his wishes and style in completing the orchestration." —Leslie Gerber

Related material:


See also yesterday's Stone Junction, this morning's note on Heidegger 's Geviert, and Moulin Bleu from Beethoven's birthday, 2003—

Juliette Binoche in "Bleu"

Mathematics and Gestalt

Filed under: Uncategorized — Tags: — m759 @ 2:01 AM

"We acknowledge a theorem's beauty
 when we see how the theorem 'fits'
 in its place, how it sheds light around itself,
 like a Lichtung, a clearing in the woods."

 — Gian-Carlo Rota, Indiscrete Thoughts

Here Rota is referring to a concept of Heidegger.

Some context—

"Gestalt Gestell Geviert: The Way of the Lighting,"
 by David Michael Levin in The Philosopher's Gaze

Tuesday, May 19, 2009

Tuesday May 19, 2009

Filed under: Uncategorized — Tags: , — m759 @ 7:20 PM
Exquisite Geometries

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

"Block Designs," 1995, by Andries E. Brouwer

"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24."

The Miracle Octad Generator (MOG) of R.T. Curtis (webpage)

"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a little-known 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."

William M. Kantor, 1981

The 1931 paper of Carmichael is now available online from the publisher for $10.

Saturday, May 19, 2007

Saturday May 19, 2007

Filed under: Uncategorized — Tags: , — m759 @ 9:29 AM
Point of View

"In a sense, too, Wallace Stevens has spent a lifetime writing a single poem. What gives his best work its astonishing power and vitality is the way in which a fixed point of view, maturing naturally, eventually takes in more than a constantly shifting point of view could get at.

The point of view is romantic, 'almost the color of comedy'; but 'the strength at the center is serious.'  Behind Wallace Stevens stand Wordsworth and Coleridge as well as Rimbaud and Mallarmé, and, surprisingly enough, La Fontaine and Pope. This poetic lineage is important only in so far as it proves that a master can claim the world as ancestor. Knowing where he stands, the poet can move as a free man in the company of free men."

Samuel French Morse, review 
of The Collected Poems
of Wallace Stevens, in
The New York Times
(October 3, 1954)
Related material:

The point of view
expressed in Log24 on
  today's date in 2004:

For a related gloss on Stevens's remark
"the strength at the center is serious,"
see "Serious" (also on an October 3).

Wednesday, May 19, 2004

Wednesday May 19, 2004

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


In memory of Lynn H. Loomis:

The above diagram is from a
(paper) journal note of October 21, 1999.

It pictures the relationship of my own discovery, diamond theory (at center), to the field, harmonic analysis, of Professor Loomis, a writer whose style I have long admired.

A quotation from the 1999 note:

"…it is not impossible to draw a fairly sharp dividing line between our mental disposition in the case of esthetic response and that of the responses of ordinary life.  A far more difficult question arises if we try to distinguish it from the responses made by us to certain abstract mental constructions such as those of pure mathematics…. Perhaps the distinction lies in this, that in the case of works of art the whole end and purpose is found in the exact quality of the emotional state, whereas in the case of mathematics the purpose is the constatation of the universal validity of the relations without regard to the quality of the emotion accompanying apprehension.  Still, it would be impossible to deny the close similarity of the orientation of faculties and attention in the two cases."
— Roger Fry, Transformations (1926), Doubleday Anchor paperback, 1956, p. 8

In other words, appreciating mathematics is much like appreciating art.

(Digitized diagram courtesy of Violet.)

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