Log24

Friday, May 26, 2017

Taormina Test

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

Mark Zuckerberg in a commencement speech
at Harvard yesterday —

"Movies and pop culture get this all wrong.
The idea of a single eureka moment
is a dangerous lie. It makes us feel inadequate
since we haven’t had ours. It prevents people
with seeds of good ideas from getting started.
Oh, you know what else movies get wrong about
innovation? No one writes math formulas on glass.
That’s not a thing."

The Thing from Taormina —

Taormina on symmetry-surfing

Tuesday, July 9, 2013

Taormina Dualism

Filed under: General — m759 @ 7:23 PM

"At some point in Greek history, it was noticed that the capital upsilon—Y—
looked like a path branching left and right. The comparison, like so much
traditional material, was ascribed to the Pythagoreans, in accordance with
the dualism just mentioned; our earliest source for it, however, is as late as
the Roman poet Persius (Satires, 3.56)." 

— "The Garden of Forking Paths" in the weblog
   Varieties of Unreligious Experience, Nov. 21, 2006

Amy Adams at the Lancia Café in Taormina, Sicily, on June 15, 2013.
Adams was in Taormina for the Italian premiere of her Superman film.

See also this  journal on that date— June 15, 2013.

Posts related to the Garden of Forking Paths:  Witch Ball (Jan. 24, 2013),
Sermon for Harvard (Sept. 19, 2010), and Amy Adams + Craft.

Tuesday, April 24, 2018

Illustrators of the Word

Filed under: General,Geometry — m759 @ 1:30 AM

Tom Wolfe in The Painted Word  (1975) 

“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”

The above group of 322,560 permutations appears also in a 2011 book —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

— and in 2013-2015 papers by Anne Taormina and Katrin Wendland:

Monday, April 23, 2018

Super Symmetry Surfing

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

Midrash —

    

Backstory — Search this journal for Taormina.

Sunday, May 28, 2017

Freeze Frame

Filed under: General — m759 @ 11:15 PM

In memory of John Severson, the founder of Surfer  magazine —

"Freeze-frame surfer, and as a live Hendrix 'E Z Rider' blares
over the soundtrack, the surfer lifts his arms and rises like Christ
into the sky."

Rolling Stone , August 5, 1971, on the film Rainbow Bridge

Severson reportedly died on Friday, May 26, 2017.

For a rather different sort of surfing, see this  journal on that date.

Friday, May 26, 2017

Day at the Museum

Filed under: General — m759 @ 12:00 PM

From the 1994 film review linked to above —

Reality Bites – Peter Travers in Rolling Stone , Feb. 1994

"Life after college – the time between graduation and
finding a job that pays your rent without making you puke.
Panic time. By spinning something fresh out of something
familiar, Reality Bites  scores the first comedy knockout of
the new year. It also brings out the vibrant best in Winona
Ryder and Ethan Hawke as friends who resist being lovers,
makes a star of Janeane Garofalo as their tart-tongued
buddy and puts Ben Stiller on the map as a director." 

Headline Style at The New York Times

Filed under: General — m759 @ 1:00 AM

Sounds like a job for Amy Adams.

Amy Adams at the Lancia Café in Taormina, Sicily, on June 15, 2013.
Adams was in Taormina for the Italian premiere of her Superman film.

Monday, September 12, 2016

The Kummer Lattice

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

The previous post quoted Tom Wolfe on Chomsky's use of
the word "array." 

An example of particular interest is the 4×4  array
(whether of dots or of unit squares) —

      .

Some context for the 4×4 array —

The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .

Further background on the Kummer lattice:

Alice Garbagnati and Alessandra Sarti, 
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action." 
To appear in Rocky Mountain J. Math.

The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite  geometry, see the website
Finite Geometry of the Square and Cube.

Some further context

"To our knowledge, the relation of the Golay code
to the Kummer lattice is a new observation."

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 
"

As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface.  The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.

* Update of Sept. 14: "Uncoordinatized," but parametrized  by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface
.

"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least, 
as an everlasting symbol of mathematical culture."

— Werner Kleinert, Mathematical Reviews ,
     as quoted at Amazon.com

Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).

Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .

Some related work of my own (click images for related posts)—

Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)

IMAGE- Desargues's theorem in light of Galois geometry

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Anticommuting Dirac matrices as spreads of projective lines

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

Monday, May 23, 2016

Springer

Filed under: General — m759 @ 10:00 AM

In memory of the late mathematician John Nash
and of the late actor Alan Young ...

A Talking Horse — 

What the horse says: "First online: 28 August 2013."

See also OverarchingPsychonauts, and Spider Tale in this journal.

Thursday, December 3, 2015

Overarching Symmetry

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

(Continued)

From p. 34 of the preprint "Snapshots of Conformal Field Theory,"
by Katrin Wendland, arXiv, 11 April 2014

50. Gannon, T.: Much ado about Mathieu (arXiv:1211.5531 [math.RT])

85. Taormina, A., Wendland, K.: The overarching finite symmetry group
of Kummer surfaces in the Mathieu group M24. JHEP  08, 125 (2013)

86. Taormina, A., Wendland, K.: Symmetry-surfing the moduli space
of Kummer K3s (arXiv:1303.2931 [hep-th])

87. Taormina, A., Wendland, K.: A twist in the M24 moonshine story
(arXiv:1303.3221 [hep-th])

The Wendland paper was published on Jan. 7, 2015, in
Mathematical Aspects of Quantum Field Theories ,
edited by Damien Calaque and Thomas Strobl
(Springer Mathematical Physics Studies), pages 89-129.

Monday, May 25, 2015

A Stitch in Time

Filed under: General,Geometry — 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

Sunday, May 17, 2015

Moon Shadow

Filed under: General — m759 @ 7:07 AM

IMAGE- The diamond theorem and umbral moonshine

"I'm being followed by a moon shadow…."  — Song lyric

Friday, March 13, 2015

Mathieu Moonshine

Filed under: General — m759 @ 1:26 PM

(Continued from yesterday's "earlier references" link.)

Yesterday at the Simons Foundation's Quanta Magazine :

See also earlier Log24 references to Mathieu moonshine .
I do not know the origin of this succinct phrase, taken from
an undated web page of Anne Taormina.

Friday, October 10, 2014

Autistic Enchantment

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

(Continued from Sept. 3, 2009)

George Steiner on chess:

"At the sight of a set, even the tawdriest of plastic pocket sets,
one’s fingers arch and a coldness as in a light sleep steals over
one’s spine. Not for gain, not for knowledge or reknown, but
in some autistic enchantment, pure as one of Bach’s inverted
canons or Euler’s formula for polyhedra."

— George Steiner in “A Death of Kings,” The New Yorker,
issue dated September 7, 1968, page 133

A related remark from Dudeney:

See also a different context for 16 squares and 322,560 arrangements.

Saturday, September 21, 2013

Mathematics and Narrative (continued)

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

Mathematics:

A review of posts from earlier this month —

Wednesday, September 4, 2013

Moonshine

Filed under: Uncategorized — m759 @ 4:00 PM

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.)

Thursday, September 5, 2013

Moonshine II

Filed under: Uncategorized — Tags:  — m759 @ 10:31 AM

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)
."

Narrative:

Aooo.

Happy birthday to Stephen King.

Wednesday, September 4, 2013

Moonshine

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

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the 
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.) 

A Google search documents the moonshine
relating Rosenhain's and Göpel's 19th-century work
in complex analysis to M24  via the book of Hudson and
the geometry of the 4×4 square.

Tuesday, June 25, 2013

Lexicon (continued)

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

Online biography of author Cormac McCarthy—

" he left America on the liner Sylvania, intending to visit
the home of his Irish ancestors (a King Cormac McCarthy
built Blarney Castle)." 

Two Years Ago:

Blarney in The Harvard Crimson

Melissa C. Wong, illustration for "Atlas to the Text,"
by Nicholas T. Rinehart:

Thirty Years Ago:

Non-Blarney from a rural outpost—

Illustration for the generalized diamond theorem,
by Steven H. Cullinane: 

See also Barry's Lexicon .

Monday, June 10, 2013

Galois Coordinates

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

Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."

A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."

A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galois-field coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory  monograph.

But such a survey might not  find any such pre-1976
coordinatization of a 4×4 array  by the 16 elements
of the vector 4-space  over the Galois field with two
elements, GF(2).

Such coordinatizations are important because of their
close relationship to the Mathieu group 24 .

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group 24 ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.

Related material: 

Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—

*  A rather abstract  2011 paper that uses the phrase
   "Galois coordinates" may have some implications 
   for the naive form of the relativity problem
   related to square and cubical arrays.

Sunday, June 9, 2013

Sicilian Reflections

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

(Continued from Sept. 22, 2011)

See Taormina in this journal, and the following photo of "Anne Newton"—

Click photo for context.

Related material:

"Super Overarching" in this journal,
  a group of order 322,560, and

See also the MAA Spectrum  program —

— and an excerpt from the above book:

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Backstory

Sunday, May 19, 2013

Priority Claim

Filed under: General,Geometry — 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).

Saturday, May 11, 2013

Core

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

Promotional description of a new book:

"Like Gödel, Escher, Bach  before it, Surfaces and Essences  will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking."

"Like Gödel, Escher, Bach  before it…."

Or like Metamagical Themas

Rubik core:

Swarthmore Cube Project, 2008

Non- Rubik cores:

Of the odd  nxnxn cube:

Of the even  nxnxn cube:

The image “http://www.log24.com/theory/images/cube2x2x2.gif” cannot be displayed, because it contains errors.

Related material: The Eightfold Cube and

"A core component in the construction
is a 3-dimensional vector space  over F."

—  Page 29 of "A twist in the M24 moonshine story," 
      by Anne Taormina and Katrin Wendland.
      (Submitted to the arXiv on 13 Mar 2013.)

Tuesday, April 30, 2013

Logline

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

Found this morning in a search:

logline  is a one-sentence summary of your script.
www.scriptologist.com/Magazine/Tips/Logline/logline.html
It's the short blurb in TV guides that tells you what a movie
is about and helps you decide if you're interested 

The search was suggested by a screenwriting weblog post,
"Loglines: WHAT are you doing?".

What is your story about?
No, seriously, WHAT are you writing about?
Who are the characters? What happens to them?
Where does it take place? What’s the theme?
What’s the style? There are nearly a million
little questions to answer when you set out
to tell a story. But it all starts with one
super, overarching question.
What are you writing about? This is the first
big idea that we pull out of the ether, sometimes
before we even have any characters.
What is your story about?

The screenwriting post was found in an earlier search for
the highlighted phrase.

The screenwriting post was dated December 15, 2009.

What I am doing now  is checking for synchronicity.

This  weblog on December 15, 2009, had a post
titled A Christmas Carol. That post referred to my 1976
monograph titled Diamond Theory .

I guess the script I'm summarizing right now is about
the heart of that theory, a group of 322,560 permutations
that preserve the symmetry of a family of graphic designs.

For that group in action, see the Diamond 16 Puzzle.

The "super overarching" phrase was used to describe
this same group in a different context:

IMAGE- Anne Taormina on 'Mathieu Moonshine' and the 'super overarching symmetry group'

This is from "Mathieu Moonshine," a webpage by Anne Taormina.

A logline summarizing my  approach to that group:

Finite projective geometry explains
the surprising symmetry properties
of some simple graphic designs— 
found, for instance, in quilts.

The story thus summarized is perhaps not destined for movie greatness.

Sunday, April 28, 2013

C’mon Baby…

Filed under: General — Tags: — m759 @ 3:13 AM
 

Let's do the twist.

The image at left
is from a poster
for a film released
on March 28, 2003.

See this journal
on that date.

A phrase from yesterday's noon post:

Sinking the Magic 8-Ball .

A scene from the above film is related to this phrase.
Another image from the film poster:

A review of the film:

"The final 'twist' seems to negate the entire story,
like a bad shaggy-dog joke."

Such a joke:

“Words and numbers are of equal value,
  for, in the cloak of knowledge,
  one is warp and the other woof.”

— The princesses Rhyme and Reason
      in The Phantom Tollbooth

"A core component in the construction
is a 3-dimensional vector space over F."

—  Page 29 of "A twist in the M24 moonshine story,"
      by Anne Taormina and Katrin Wendland.
      (Submitted to the arXiv on 13 Mar 2013.)

The number of points in such a space is, of course, 8.

Saturday, April 27, 2013

Mark and Remark

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

“Fact and fiction weave in and out of novels like a shell game.” —R.B. Kitaj

Not just novels.

Fact: 

IMAGE- Anne Taormina on 'Mathieu Moonshine' and the 'super overarching symmetry group'

The mark preceding A in the above denotes the semidirect product.

Symbol from the box-style
I Ching  (Cullinane, 1/6/89).
This is Hexagram 55,
“Abundance [Fullness].”

The mathematical quote, from last evening’s Symmetry, is from Anne Taormina.

The I Ching  remark is not.

Another version of Abbondanza 

IMAGE- Taormina sunset from inabbondanza.com on June 22, 2009

Fiction:

Found in Translation and the giorno  June 22, 2009here.

Friday, April 26, 2013

Symmetry

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

Anne Taormina on Mathieu Moonshine —

IMAGE- Anne Taormina on 'Mathieu Moonshine' and the 'super overarching symmetry group'

This is, of course, the same group (of order 322,560) underlying the Diamond 16 Puzzle.

Sunday, March 17, 2013

Back to the Present

Filed under: General,Geometry — m759 @ 4:24 PM

The previous post discussed some tesseract
related mathematics from 1905.

Returning to the present, here is some arXiv activity
in the same area from March 11, 12, and 13, 2013.

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