The above image is from Geometry of the 4×4 Square.
(The link "Visible Mathematics" in today's previous post, Quartet,
led to a post linked to that page, among others.)
Note that the seventh square above, at top right of the array of 35,
is the same as the image in Quartet.
Happy Beethoven's Birthday.
Related material: Abel 2005 and, more generally, Abel.
See also Visible Mathematics.
On Beauty
“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”
The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures. Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(32), also with 81 points. Less obvious are some related projective structures. Joseph Malkevitch has discussed the standard method of constructing GF(32) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with
Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.
For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year:
The figures are:
A symbol of Apollo from
Balanchine's Birthday and
A Minature Rosetta Stone,
a symbol of pure reason from
Visible Mathematics and
Analogical Train of Thought,
a symbol of Venus from
Why Me? and
To Graves at the Winter Solstice,
and, finally, a more
down-to-earth symbol,
adapted from a snowflake in
Those who prefer their
theological art on the scary side
may enjoy the
Christian Snowflake
link in the comments on
the "Logos" entry of
Orthodox Easter (May 1), 2005.
Part I: The 24-Cell
| 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….
“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
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.
The Fugue
"True joy is a profound remembering, and true grief is the same.
Thus it was, when the dust storm that had snatched Cal up finally died, and he opened his eyes to see the Fugue spread out before him, he felt as though the few fragile moments of epiphany he'd tasted in his twenty-six years– tasted but always lost– were here redeemed and wed. He'd grasped fragments of this delight before. Heard rumour of it in the womb-dream and the dream of love; known it in lullabies. But never, until now, the whole, the thing entire.
It would be, he idly thought, a fine time to die.
And a finer time still to live, with so much laid out before him."
Weaveworld,
Book Three:
Out of the
Empty Quarter
"The wheels of its body rolled,
the visible mathematics
of its essence turning on itself…."
For the meaning
of this picture, see
Geometry of the
4×4 Square.
For graphic designs
based on this geometry,
see Theme and Variations
and Diamond Theory.
For these designs in the
context of a Bach fugue,
see Timothy A. Smith's
essay (pdf) on
Fugue No. 21 in B-Flat Major
from Book II of
The Well-Tempered Clavier
by Johann Sebastian Bach.
Smith also offers a
Shockwave movie
that uses diamond theory
to illustrate this fugue.
For Sir Alec
From Elegance:
"Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday…."
— Bernard Holland, page C12,
The New York Times,
Monday, May 20, 1996.
Holland was pondering the identity of the Juilliard String Quartet, which had just given a series of concerts celebrating its fiftieth anniversary.
"Elegant"
— Page one,
The New York Times,
Monday, August 7, 2000.
The Times was describing the work of Sir Alec Guinness, who died on 8/5/00.
An example of the Holland name problem:
Monday, August 1, 2005 — Visible Mathematics:
"Earlier, there had been mapping projects in Saudi Arabia's Rub' al-Khali, the Empty Quarter in the south and west of the country….
'"Empty" is a misnomer… the Rub' al-Khali contains many hidden riches.'"
Friday, August 5, 2005 —
Related material:
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)
— J. L. Alperin, book review,
Bulletin (New Series) of the American
Mathematical Society 10 (1984), 121
"What is it, Major Lawrence,
that attracts you personally
to the desert?"
"It's clean."
Visible Mathematics,
continued —
"The Garden of Eden is behind us
and there is no road
back to innocence;
we can only go forward."
— Anne Morrow Lindbergh,
Earth Shine, p. xii
— Werner Heisenberg,
"Die Bedeutung des Schönen
in der exakten Naturwissenschaft,"
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg's Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974
Related material:
(in Arabic, ka'b)
and
Final Arrangements, continued
If I stand starry-eyed
That's a danger in paradise
For mortals who stand beside
An angel like you.
— Robert Wright and George Forrest
"Earlier, there had been mapping projects in Saudi Arabia's Rub' al-Khali, the Empty Quarter in the south and west of the country….
'"Empty" is a misnomer… the Rub' al-Khali contains many hidden riches.'"
— Maps from the Sky,
Saudi Aramco World, March/April 1995
|
From Weaveworld
Book Three: |
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