. . . is now at loshu.space. (Update on 10 Dec. — See also loshu.group.)
See as well GL(2,3) in this journal.
Thursday, December 9, 2021
Thursday, August 25, 2005
Thursday August 25, 2005
Analogical
Train of Thought
From S. H. Cullinane,
Visualizing GL(2,p),
March 26, 1985–
Train of Thought
Part I: The 24-Cell
From S. H. Cullinane,
Visualizing GL(2,p),
March 26, 1985–
| From John Baez, “This Week’s Finds in Mathematical Physics (Week 198),” September 6, 2003: Noam Elkies writes to John Baez:
The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – Gian-Carlo Rota |
Like footprints erased in the sand….
Part II: Discrete Space
“Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.”
“A very short space of time through very short times of space….
Am I walking into eternity along Sandymount strand?”
— James Joyce, Ulysses, Proteus chapter
A very short space of time through very short times of space….
“It is demonstrated that space-time should possess a discrete structure on Planck scales.”
— Peter Szekeres, abstract of Discrete Space-Time
“A theory…. predicts that space and time are indeed made of discrete pieces.”
— Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004
“… a fundamental discreteness of spacetime seems to be a prediction of the theory….”
— Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity
“Theories of discrete space-time structure are being studied from a variety of perspectives.”
— Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London
Disclaimer:
The above speculations by physicists
are offered as curiosities.
I have no idea whether
any of them are correct.
Related material:
Stephen Wolfram offers a brief
History of Discrete Space.
For a discussion of space as discrete
by a non-physicist, see John Bigelow‘s
Space and Timaeus.
Part III: Quaternions
in a Discrete Space
in a Discrete Space
Apart from any considerations of
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
(Aug. 4, 2005):
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
(Aug. 4, 2005):
Comments Off on Thursday August 25, 2005
Saturday, August 6, 2005
Saturday August 6, 2005
For André Weil on
the seventh anniversary
of his death:
the seventh anniversary
of his death:
A Miniature
Rosetta Stone
In a 1940 letter to his sister Simone, André Weil discussed a sort of “Rosetta stone,” or trilingual text of three analogous parts: classical analysis on the complex field, algebraic geometry over finite fields, and the theory of number fields.
John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:
“… a 24-element group called the ‘binary tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and the vertices of a regular polytope in 4 dimensions called the ’24-cell.’ The most important fact is that these are all the same thing!”
For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,
The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from the general linear group GL(2,3). For the relationship of this group to the quaternions, see the Log24 entry for August 4 (the birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).
The 3×3 square shown above may, as my August 4 entry indicates, be used to picture the quaternions and, more generally, the 48-element group GL(2,3). It may therefore be regarded as the structure underlying the miniature Rosetta stone described by Baez.
“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”
— J. L. Alperin, book review,
Bulletin (New Series) of the American
Mathematical Society 10 (1984), 121
Comments Off on Saturday August 6, 2005
Thursday, August 4, 2005
Thursday August 4, 2005
Visible Mathematics, continued
Today's mathematical birthdays:
Saunders Mac Lane, John Venn,
and Sir William Rowan Hamilton.
It is well known that the quaternion group is a subgroup of GL(2,3), the general linear group on the 2-space over GF(3), the 3-element Galois field.
The figures below illustrate this fact.
Related material: Visualizing GL(2,p)
"The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled."
— J. L. Alperin, book review,
Bulletin (New Series) of the American
Mathematical Society 10 (1984), 121
Comments Off on Thursday August 4, 2005
Thursday, May 27, 2004
Thursday May 27, 2004
Ineluctable
On the poetry of Geoffrey Hill:
"… why read him? Because of the things he writes about—war and peace and sacrifice, and the search for meaning and the truths of the heart, and for that haunting sense that, in spite of war and terror and the indifferences that make up our daily hells, there really is some grander reality, some ineluctable presence we keep touching. There remains in Hill the daunting possibility that it may actually all cohere in the end, or at least enough of it to keep us searching for more.
There is a hard edge to Hill, a strong Calvinist streak in him, and an intelligence that reminds one of Milton….."
— Paul Mariani, review in America of Geoffrey Hill's The Orchards of Syon
"Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one."
"A very short space of time through very short times of space…. Am I walking into eternity along Sandymount strand?"
— James Joyce, Ulysses, Proteus chapter
"Time has been unfolded into space."
"Pattern and symmetry are closely related."
— James O. Coplien on Symmetry Breaking
"… as the critic S. L. Goldberg puts it, 'the chapter explores the Protean transformations of matter in time . . . apprehensible only in the condition of flux . . . as object . . . and Stephen himself, as subject. In the one aspect Stephen is seeking the principles of change and the underlying substance of sensory experience; in the other, he is seeking his self among its temporal manifestations'….
— Goldberg, S.L. 'Homer and the Nightmare of History.' Modern Critical Views: James Joyce. Ed. Harold Bloom. New York: Chelsea House, 1986. 21-38."
— from the Choate site of David M. Loeb
In summary:
 Joyce |
See also Time Fold.
(By the way, Jorn Barger seems
to have emerged from seclusion.)
Comments Off on Thursday May 27, 2004
Sunday, October 5, 2003
Sunday October 5, 2003
At Mount Sinai:
Art Theory for Yom Kippur
From the New York Times of Sunday, October 5, 2003 (the day that Yom Kippur begins at sunset):
“Rabbi Ephraim Oshry, whose interpretations of religious law helped sustain Lithuanian Jews during Nazi occupation…. died on Sept. 28 at Mount Sinai Hospital in Manhattan. He was 89.”
For a fictional portrait of Lithuanian Jews during Nazi occupation, see the E. L. Doctorow novel City of God.
For meditations on the spiritual in art, see the Rosalind Krauss essay “Grids.” As a memorial to Rabbi Oshry, here is a grid-based version of the Hebrew letter aleph:
|
|
|
Click on the aleph for details.
“In the garden of Adding,
Live Even and Odd….”
— The Midrash Jazz Quartet in
City of God, by E. L. Doctorow
Here are two meditations
on Even and Odd for Yom Kippur:
Meditation I
From Rosalind Krauss, “Grids”:
“If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.
Or, to take a more up-to-date example, we could think about Ad Reinhardt who, despite his repeated insistence that ‘Art is art,’ ended up by painting a series of black nine-square grids in which the motif that inescapably emerges is a Greek cross. There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora’s box of spiritual reference that is opened once one uses it.”
Meditation II
Here, for reference, is a Greek cross
within a nine-square grid:
Related religious meditation for
Doctorow’s “Garden of Adding”…
Comments Off on Sunday October 5, 2003
Thursday, January 9, 2003
Thursday January 9, 2003
Balanchine's Birthday
Today seems an appropriate day to celebrate Apollo and the nine Muses.
From a website on Balanchine's and Stravinsky's ballet, "Apollon Musagete":
In his Poetics of Music (1942) Stravinsky says: "Summing up: What is important for the lucid ordering of the work– for its crystallization– is that all the Dionysian elements which set the imagination of the artist in motion and make the life-sap rise must be properly subjugated before they intoxicate us, and must finally be made to submit to the law: Apollo demands it." Stravinsky conceived Apollo as a ballet blanc– a "white ballet" with classical choreography and monochromatic attire. Envisioning the work in his mind's eye, he found that "the absence of many-colored hues and of all superfluities produced a wonderful freshness." Upon first hearing Apollo, Diaghilev found it "music somehow not of this world, but from somewhere else above." The ballet closes with an Apotheosis in which Apollo leads the Muses towards Parnassus. Here, the gravely beautiful music with which the work began is truly recapitulated "on high"– ceaselessly recycled, frozen in time.
— Joseph Horowitz
Another website invoking Apollo:
The icon that I use… is the nine-fold square…. The nine-fold square has centre, periphery, axes and diagonals. But all are present only in their bare essentials. It is also a sequence of eight triads. Four pass through the centre and four do not. This is the garden of Apollo, the field of Reason….
In accordance with these remarks, here is the underlying structure for a ballet blanc:
This structure may seem too simple to support movements of interest, but consider the following (click to enlarge):
As Sir Arthur Quiller-Couch, paraphrasing Horace, remarks in his Whitsun, 1939, preface to the new edition of the Oxford Book of English Verse, "tamen usque recurret Apollo."
The alert reader will note that in the above diagrams, only eight of the positions move.
Which muse remains at the center?
Consider the remark of T. S. Eliot, "At the still point, there the dance is," and the fact that on the day Eliot turned 60, Olivia Newton-John was born. How, indeed, in the words of another "sixty-year-old smiling public man," can we know the dancer from the dance?