Friday, June 9, 2017

Overarching Theme

Filed under: Uncategorized — m759 @ 8:23 PM

“Communications disorders were the overarching theme of my mother’s career.”

— Anne Louise Oaklander, daughter of a famed autism expert, Isabelle Rapin,
     who reportedly died at 89 on May 24.

See also a post on Mark Zuckerberg's recent Harvard commencement address.

Some background — Overarching in this journal.

Friday, May 26, 2017

Taormina Test

Filed under: Uncategorized — 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 —

Thursday, February 23, 2017

Bullshit Studies

Filed under: Uncategorized — m759 @ 1:21 PM


" The origin of new ways of doing things may often be
a disciplinary crisis. The definition of such a crisis
provided by Barry Mazur in Mykonos (2005) applies
equally well to literary creation. '[A crisis occurs] when
some established overarching framework, theoretical
vocabulary or procedure of thought is perceived as
inadequate in an essential way, or not meaning
what we think it means.' "

— Circles Disturbed :
The Interplay of Mathematics and Narrative

Edited by Apostolos Doxiadis & Barry Mazur
Princeton University Press, 2012. See
Chapter 14, Section 5.1, by Uri Margolin.

See also "overarching" in this journal.

Saturday, February 18, 2017

Not Strange Enough?

Filed under: Uncategorized — m759 @ 7:59 PM

Peter Woit today discusses a book by one Zeeya Merali:

Some earlier remarks by Merali:

Zeeya Merali in Nature  on 28 August 2013

"… a small band of researchers who think that
the usual ideas are not yet strange enough.
If nothing else, they say, neither of the two great
pillars of modern physics — general relativity,
which describes gravity as a curvature of space
and time, and quantum mechanics, which governs
the atomic realm — gives any account for
the existence of space and time.

. . . .

'All our experiences tell us we shouldn't have two
dramatically different conceptions of reality —
there must be one huge overarching theory,' says
Abhay Ashtekar, a physicist at Pennsylvania State
University in University Park."

See as well Overarching and Doctor Strange in this  journal.

Monday, September 12, 2016

The Kummer Lattice

Filed under: Uncategorized — 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.

Monday, May 23, 2016


Filed under: Uncategorized — 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.

Friday, February 26, 2016


Filed under: Uncategorized — m759 @ 4:00 PM


"The study of social memory allows scholars to
understand how different memories form within
a collective group, thus exploring the societal
and ideological elements of disparate groups
that form the over-arching memory of Melkisedeq."

The Melkisedeq Memoirs , by Cale Staley,
2015 master's thesis at the University of Iowa

Elements of groups that I prefer —

"Right through hell
 there is a path…."
 — Malcolm Lowry,
Under the Volcano

Friday, December 4, 2015


Filed under: Uncategorized — m759 @ 12:00 PM

"Encouraged by Proposition 5, one may hope…."

— Katrin Wendland in the previous post

Related material:  Euclid Book I, Proposition 5.

Thursday, December 3, 2015

Overarching Symmetry

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


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.

Thursday, September 10, 2015

Super Overarching Symmetry

Filed under: Uncategorized — m759 @ 12:48 PM


Santa Fe Institute logo (see previous post) —

Symmetry , by Hermann Weyl

The image “http://www.log24.com/log/pix06/060319-Weyl.jpg” cannot be displayed, because it contains errors.

Wednesday, September 9, 2015

Death Link

Filed under: Uncategorized — m759 @ 9:29 PM

Continued from a post of August 20, 2015 —


(Continued)  See posts of August 9, 2015.
See also a death on that date.

The death link above leads to an obituary for one
"John Henry Holland, Who Computerized Evolution."

A book by Holland published July 13, 2012, by The MIT Press

Signals and Boundaries

Building Blocks for Complex Adaptive Systems

By John H. Holland


Complex adaptive systems (cas), including ecosystems, governments, biological cells, and markets, are characterized by intricate hierarchical arrangements of boundaries and signals. In ecosystems, for example, niches act as semi-permeable boundaries, and smells and visual patterns serve as signals; governments have departmental hierarchies with memoranda acting as signals; and so it is with other cas. Despite a wealth of data and descriptions concerning different cas, there remain many unanswered questions about "steering" these systems. In Signals and Boundaries, John Holland argues that understanding the origin of the intricate signal/border hierarchies of these systems is the key to answering such questions. He develops an overarching framework for comparing and steering cas through the mechanisms that generate their signal/boundary hierarchies.

How nice to have an overarching framework.

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

Sunday, May 17, 2015

Moon Shadow

Filed under: Uncategorized — 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: Uncategorized — 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.

Thursday, March 12, 2015

Overarching Symmetry*

Filed under: Uncategorized — m759 @ 6:29 AM

for fans of the late C. P. Snow

* See earlier references here to that phrase.

Monday, February 9, 2015

Overarching Symmetry

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

Continued from earlier posts.

The Washington Post  online yesterday:

"Val Logsdon Fitch, the Nebraska rancher’s son who shared the Nobel Prize for detecting a breakdown in the overarching symmetry of physical laws, thus helping explain how the universe evolved after the Big Bang, died Feb. 5 in Princeton, N.J. He was 91.

His death was confirmed by Princeton University, where he had been a longtime faculty member and led the physics department for several years.

Dr. Fitch and his Princeton colleague James Cronin received the Nobel Prize in physics in 1980 for high-energy experiments conducted in 1964 that overturned fundamental assumptions about symmetries and invariances that are characteristic of the laws of physics."

— By Martin Weil

Fans of synchronicity may prefer some rather
ig -Nobel remarks quoted here  on the date
of Fitch's death:

"The Harvard College Events Board presents
Harvard Thinks Big VI, a night of big ideas
and thinking beyond traditional boundaries.
On Thursday February 5th at 8 pm in
Sanders Theatre …."

— Log24 post The Big Spielraum

Monday, June 10, 2013

Galois Coordinates

Filed under: Uncategorized — 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: Uncategorized — 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)


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 1, 2013

The Crosswicks Curse

Filed under: Uncategorized — m759 @ 9:00 PM


"There is  such a thing as a tesseract." —A novel from Crosswicks

Related material from a 1905 graduate of Princeton,
"The 3-Space PG(3,2) and Its Group," is now available
at Internet Archive (1 download thus far).

The 3-space paper is relevant because of the
connection of the group it describes to the
"super, overarching" group of the tesseract.

Tuesday, April 30, 2013


Filed under: Uncategorized — m759 @ 9:29 AM

Found this morning in a search:

logline  is a one-sentence summary of your script.
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.

Saturday, April 27, 2013

Mark and Remark

Filed under: Uncategorized — m759 @ 11:00 AM

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

Not just novels.


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


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

Friday, April 26, 2013


Filed under: Uncategorized — 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: Uncategorized — 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.

Powered by WordPress