Log24

Tuesday, July 19, 2022

The Lost Message

Filed under: General — Tags: , — m759 @ 12:10 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.

"…groups are invariably best studied through their action on some structure…."

— R. T. Curtis, “Symmetric Generation of the Higman-Sims Group” in
Journal of Algebra  171 (1995), pp. 567-586.

Related material — Other posts now tagged Groups Act.

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.

Thursday, May 28, 2015

Show

Filed under: General,Geometry — Tags: — m759 @ 9:48 am

(A sequel to the previous post, Tell )

Inscapes

An illustration (click image for further details) —

Related reading

From my JSTOR shelf —

Click the above image for a related Log24 post, Groups Acting.

Saturday, August 27, 2011

Cosmic Cube*

IMAGE- Anthony Hopkins exorcises a Rubik cube

Prequel (Click to enlarge)

IMAGE- Galois vs. Rubik: Posters for Abel Prize, Oslo, 2008

Background —

IMAGE- 'Group Theory' Wikipedia article with Rubik's cube as main illustration and argument by a cuber for the image's use

See also Rubik in this journal.

* For the title, see Groups Acting.

Thursday, May 12, 2011

But Seriously…

Filed under: General — Tags: — m759 @ 7:00 am

Yesterday's "Succor" cited the New York Lottery of Tuesday— Midday 489, Evening 886.

One interpretation of these numbers—

  • 489 as the number of a page in the Collected Poems  of Wallace Stevens
    with verses that suggested to one author the following questions:
    "How can one express one's sense of the ground of things?
    What is the structure of Being itself…?" — Thomas Jensen Hines
  • 886 as a number applied recently to a poem by Gerard Manley Hopkins
    with the notable phrase "the unchanging register of change"

Some background from Tuesday—

Tuesday, May 10, 2011

Groups Acting

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

The LA Times  on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."

Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).

A high-minded— if not Shakespearean— non-Nordic approach to groups acting—

"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. 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. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."

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

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions

Not so high-minded— The Cosmic Cube

http://www.log24.com/log/pix11A/110509-SpideySuperStories39Sm.jpg

One way of blending high and low—

The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.

The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—

http://www.log24.com/log/pix11A/110510-GaloisAleph.GIF

The above illustration is from undated software by Ed Pegg Jr.

For mathematical background, see a 1985 note, "Visualizing GL(2,p)."

For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).

For the Nordic-averse, some background on the Hebrew connection—

Sunday, March 7, 2004

Sunday March 7, 2004

Filed under: General — Tags: , — m759 @ 6:00 pm

Apartments

From Wallace Stevens,
"Notes Toward a Supreme Fiction":

It is the celestial ennui of apartments
That sends us back to the first idea, the quick
Of this invention; and yet so poisonous

Are the ravishments of truth, so fatal to
The truth itself, the first idea becomes
The hermit in a poet’s metaphors,

Who comes and goes and comes and goes all day.
May there be an ennui of the first idea?
What else, prodigious scholar, should there be?….

From Guyan Robertson,
Groups Acting on Affine Buildings
and their Boundaries
:

From Plato's Meno:

They will get it straight one day at the Sorbonne.
We shall return at twilight from the lecture         
Pleased that the irrational is rational….              

See Logos and Logic
and the previous entry.

Friday, November 29, 2002

Friday November 29, 2002

Filed under: General,Geometry — Tags: , — m759 @ 1:06 pm

A Logocentric Archetype

Today we examine the relativist, nominalist, leftist, nihilist, despairing, depressing, absurd, and abominable work of Samuel Beckett, darling of the postmodernists.

One lens through which to view Beckett is an essay by Jennifer Martin, "Beckettian Drama as Protest: A Postmodern Examination of the 'Delogocentering' of Language." Martin begins her essay with two quotations: one from the contemptible French twerp Jacques Derrida, and one from Beckett's masterpiece of stupidity, Molloy. For a logocentric deconstruction of Derrida, see my note, "The Shining of May 29," which demonstrates how Derrida attempts to convert a rather important mathematical result to his brand of nauseating and pretentious nonsense, and of course gets it wrong. For a logocentric deconstruction of Molloy, consider the following passage:

"I took advantage of being at the seaside to lay in a store of sucking-stones. They were pebbles but I call them stones…. I distributed them equally among my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones….But this solution did not satisfy me fully. For it did not escape me that, by an extraordinary hazard, the four stones circulating thus might always be the same four."

Beckett is describing, in great detail, how a damned moron might approach the extraordinarily beautiful mathematical discipline known as group theory, founded by the French anticleric and leftist Evariste Galois. Disciples of Derrida may play at mimicking the politics of Galois, but will never come close to imitating his genius. For a worthwhile discussion of permutation groups acting on a set of 16 elements, see R. D. Carmichael's masterly work, Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937, reprinted by Dover, New York, 1956.

There are at least two ways of approaching permutations on 16 elements in what Pascal calls "l'esprit géométrique." My website Diamond Theory discusses the action of the affine group in a four-dimensional finite geometry of 16 points. For a four-dimensional euclidean hypercube, or tesseract, with 16 vertices, see the highly logocentric movable illustration by Harry J. Smith. The concept of a tesseract was made famous, though seen through a glass darkly, by the Christian writer Madeleine L'Engle in her novel for children and young adults, A Wrinkle in Tme.

This tesseract may serve as an archetype for what Pascal, Simone Weil (see my earlier notes), Harry J. Smith, and Madeleine L'Engle might, borrowing their enemies' language, call their "logocentric" philosophy.

For a more literary antidote to postmodernist nihilism, see Archetypal Theory and Criticism, by Glen R. Gill.

For a discussion of the full range of meaning of the word "logos," which has rational as well as religious connotations, click here.

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