Log24

Saturday, June 8, 2024

Paradigms of Geometry

Filed under: General — Tags: , — m759 @ 7:46 am

The "Change Arises" phrase in the previous post suggests a sort of
architectural plan and elevation  for  the structure of the I Ching

Friday, June 7, 2024

The Seventh Square (Revisited)

Filed under: General — Tags: , — m759 @ 9:54 pm

A date in Wednesday's "Change Arises" post suggests a review

From posts tagged "Dec 16-18 2013" —

Wednesday, June 5, 2024

“Change Arises”* — Wednesday’s String Theory

Filed under: General — Tags: , — m759 @ 2:53 pm

* http://m759.net/wordpress/?s="Change+Arises"

Friday, May 6, 2022

Interality and the Bead Game

Filed under: General — Tags: , , , — m759 @ 3:00 pm

WIkipedia on the URL suffix ".io" —

"In computer science, "IO" or "I/O" is commonly used
as an abbreviation for input/output, which makes the
.io domain desirable for services that want to be
associated with technology. .io domains are often used
for open source projects, application programming
interfaces ("APIs"), startup companiesbrowser games,
and other online services."

An association with the Bead Game from a post of April 7, 2018

IMAGE- 'Solomon's Cube'

Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

The recent use by a startup company of the URL "interality.io" suggests
a fourth  reading for the 1955 list of Lévi-Strauss — in and out
i.e., inner and outer group automorphisms —  from a 2011 post
on the birthday of T. S. Eliot :

A transformation:

Inner and outer group automorphisms

Click on the picture for details.

Friday, April 29, 2022

Code Bleu

Filed under: General — Tags: , , , — m759 @ 2:17 pm

From The New York Times  on May 5, 2011 —

"… What Paris says to me is love story, awash with painters,
shots of the Seine, Champagne. Thank God I have a
can’t-miss notion to sell you. I call it ‘Midnight in Paris.’ ”

“Romantic title,” I had to admit. “Is there a script?”

“Actually, there’s nothing on paper yet, but I can spitball
the main points,” he said, slipping on his tap shoes.

“Maybe some other time,” I said, mindful of Cubbage’s
unbroken string of theatrical Hiroshimas.

— Woody Allen

The above passage is in memory of a French film director
who, like the reporter in yesterday's post Primary Colors,
reportedly died on April 21, 2022.

See also Aitchison at Hiroshima and Easter for Aitchison.

Thursday, April 28, 2022

Primary Colors

Filed under: General — Tags: , , , — m759 @ 5:05 pm

Saturday, January 29, 2022

On the Diamond-Theorem Group* of Order 322,560

Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.

This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M24.

* See a Log24 post of June 1, 2013.

Friday, January 8, 2021

Groups Act

Filed under: General — Tags: , — m759 @ 2:08 pm

"Somehow, a message had been lost on me. Groups act .
The elements of a group do not have to just sit there,
abstract and implacable; they can do  things, they can
'produce changes.' In particular, groups arise
naturally as the symmetries of a set with structure."

— Thomas W. Tucker, review of Lyndon's Groups and Geometry
in The American Mathematical Monthly , Vol. 94, No. 4
(April 1987), pp. 392-394.

"The concept of group actions is very useful in the study of
isomorphisms of combinatorial structures."

— Olli Pottonen, "Classification of Steiner Quadruple Systems"
(Master's thesis, Helsinki, 2005, p. 48).

“In a sense, we would see that change arises from
the structure of the object.”

— Nima Arkani-Hamed, quoted in "A  Jewel at the Heart of
Quantum Physics
," by Natalie Wolchover, Quanta Magazine ,
Sept. 17, 2013.

See as well "Change Arises" in this  journal.

Friday, December 25, 2020

Change Arises: Mathematical Examples

Filed under: General — Tags: , , , — m759 @ 12:59 am

From old posts tagged Change Arises

From Christmas 2005:

 

The Eightfold Cube: The Beauty of Klein's Simple Group
Click on image for details.

For the eightfold cube
as it relates to Klein's
simple group, see
"A Reflection Group
of Order 168
."

For an rather more
complicated theory of
Klein's simple group, see

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve'

Click on image for details.

The phrase "change arises" is from Arkani-Hamed in 2013, describing
calculations in physics related to properties of the positive Grassmannian

 

A related recent illustration from Quanta Magazine —

The above illustration of seven cells is not unrelated to
the eightfold-cube model of the seven projective points in
the Fano plane.

Thursday, December 24, 2020

Change Arises

Filed under: General — Tags: , , — m759 @ 1:44 am

See posts so tagged.

"Change arises from the structure of the object." — Arkani-Hamed

Related material from 1936 —

Related material from 1905, with the "object" a 4×4 array —

Related material from 1976, with the "object"
a 4×6 array — See Curtis.

Related material from 2018, with the "object"
a cuboctahedron — See Aitchison.

Wednesday, December 20, 2017

January 2018 AMS Notices

Filed under: General,Geometry — Tags: — m759 @ 3:03 pm

Update of 9:29 PM ET Dec. 20, 2017 —

See in particular, in the above Notices , the article

"Algebraic Structures on Polytopes," by Federico Ardila,
within the 2018 Joint Mathematics Meeting Lecture Sampler.

Related reading:

arXiv:1711.09102v1 [hep-th] 24 Nov 2017,

"Scattering Forms and the Positive Geometry of
Kinematics, Color and the Worldsheet," by
Nima Arkani-Hamed, Yuntao Bai, Song He, Gongwang Yan
(Submitted to the arXiv on 24 Nov. 2017).

Wednesday, April 2, 2014

Change Arises

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

IMAGE- Search for the source of the quotation 'Change arises from the structure of the object'

For a different view of change arising, click on the tag above.

Tuesday, April 1, 2014

Kindergarten Geometry

Filed under: General,Geometry — Tags: , , , — m759 @ 11:22 pm

(Continued)

A screenshot of the new page on the eightfold cube at Froebel Decade:

IMAGE- The eightfold cube at Froebel Decade

Click screenshot to enlarge.

Saturday, November 30, 2013

Waiting for Ogdoad

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

Continued from October 30 (Devil’s Night), 2013.

“In a sense, we would see that change
arises from the structure of the object.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the
Classic of Change ,” the I Ching .

The physicist quoted above was discussing a rather
complicated object. His words apply to a much simpler
object, an embodiment of the eight trigrams underlying
the I Ching  as the corners of a cube.

The Eightfold Cube and its Inner Structure

See also

(Click for clearer image.)

The Cullinane image above illustrates the seven points of
the Fano plane as seven of the eight I Ching  trigrams and as
seven natural ways of slicing the cube.

For a different approach to the mathematics of cube slices,
related to Gauss’s composition law for binary quadratic forms,
see the Bhargava cube  in a post of April 9, 2012.

Saturday, June 1, 2013

Permanence

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

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

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

The diamond theorem  group, published without acknowledgment
of its source by the Mathematical Association of America in 2011—

IMAGE- The diamond-theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

Sunday, April 1, 2012

The Palpatine Dimension

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

A physics quote relayed at Peter Woit's weblog today—

"The relation between 4D N=4 SYM and the 6D (2, 0) theory
is just like that between Darth Vader and the Emperor.
You see Darth Vader and you think 'Isn’t he just great?
How can anyone be greater than that? No way.'
Then you meet the Emperor."

— Arkani-Hamed

Some related material from this  weblog—

(See Big Apple and Columbia Film Theory)

http://www.log24.com/log/pix12/120108-Space_Time_Penrose_Hawking.jpg

The Meno Embedding:

Plato's Diamond embedded in The Matrix

Some related material from the Web—

IMAGE- The Penrose diamond and the Klein quadric

See also uses of the word triality  in mathematics. For instance…

A discussion of triality by Edward Witten

Triality is in some sense the last of the exceptional isomorphisms,
and the role of triality for n = 6  thus makes it plausible that n = 6
is the maximum dimension for superconformal symmetry,
though I will not give a proof here.

— "Conformal Field Theory in Four and Six Dimensions"

and a discussion by Peter J. Cameron

There are exactly two non-isomorphic ways
to partition the 4-subsets of a 9-set
into nine copies of AG( 3,2).
Both admit 2-transitive groups.

— "The Klein Quadric and Triality"

Exercise: Is Witten's triality related to Cameron's?
(For some historical background, see the triality  link from above
and Cameron's Klein Correspondence and Triality.)

Cameron applies his  triality to the pure geometry of a 9-set.
For a 9-set viewed in the context of physics, see A Beginning

From MIT Commencement Day, 2011—

A symbol related to Apollo, to nine, and to "nothing"

A minimalist favicon—

IMAGE- Generic 3x3 square as favicon

This miniature 3×3 square— http://log24.com/log/pix11A/110518-3x3favicon.ico — may, if one likes,
be viewed as the "nothing" present at the Creation. 
See Feb. 19, 2011, and Jim Holt on physics.

Happy April 1.

Monday, July 21, 2008

Monday July 21, 2008


Knight Moves:

The Relativity Theory
of Kindergarten Blocks

(Continued from
January 16, 2008)

"Hmm, next paper… maybe
'An Unusually Complicated
Theory of Something.'"

Garrett Lisi at
Physics Forums, July 16

Something:

From Friedrich Froebel,
who invented kindergarten:

Froebel's Third Gift: A cube made up of eight subcubes

Click on image for details.

An Unusually
Complicated Theory:

From Christmas 2005:

The Eightfold Cube: The Beauty of Klein's Simple Group

Click on image for details.

For the eightfold cube
as it relates to Klein's
simple group, see
"A Reflection Group
of Order 168
."

For an even more
complicated theory of
Klein's simple group, see

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve'

Click on image for details.

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