Thursday, July 31, 2008
Symmetry in Review
“Put bluntly, who is kidding whom?”
— Anthony Judge, draft of
“Potential Psychosocial Significance
of Monstrous Moonshine:
An Exceptional Form of Symmetry
as a Rosetta Stone for
Cognitive Frameworks,”
dated September 6, 2007.
Good question.
Also from
September 6, 2007 —
the date of
Madeleine L’Engle‘s death —
Related material:
1. The performance of a work by
Richard Strauss,
“Death and Transfiguration,”
(Tod und Verklärung, Opus 24)
by the Chautauqua Symphony
at Chautauqua Institution on
July 24, 2008
2. Headline of a music review
in today’s New York Times:
Welcoming a Fresh Season of
Transformation and Death
3. The picture of the R. T. Curtis
Miracle Octad Generator
on the cover of the book
Twelve Sporadic Groups:
4. Freeman Dyson’s hope, quoted by
Gorenstein in 1986, Ronan in 2006,
and Judge in 2007, that the Monster
group is “built in some way into
the structure of the universe.”
5. Symmetry from Plato to
the Four-Color Conjecture
6. Geometry of the 4×4 Square
7. Yesterday’s entry,
“Theories of Everything“
Coda:
“There is such a thing
as a tesseract.“
— Madeleine L’Engle
For a profile of
L’Engle, click on
the Easter eggs.
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Wednesday, July 30, 2008
Like
Garrett Lisi’s, it is based on an unusual and highly symmetric mathematical structure. Lisi’s approach is related to the exceptional simple Lie group E
8.* Dharwadker uses
a structure long associated with the sporadic simple Mathieu group M
24.
| GRAND UNIFICATION
OF THE STANDARD MODEL WITH QUANTUM GRAVITY
by Ashay Dharwadker
Abstract
“We show that the mathematical proof of the four colour theorem [1] directly implies the existence of the standard model, together with quantum gravity, in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem, at the most fundamental level. We preserve all the established working theories of physics: Quantum Mechanics, Special and General Relativity, Quantum Electrodynamics (QED), the Electroweak model and Quantum Chromodynamics (QCD). We build upon these theories, unifying all of them with Einstein’s law of gravity. Quantum gravity is a direct and unavoidable consequence of the theory. The main construction of the Steiner system in the proof of the four colour theorem already defines the gravitational fields of all the particles of the standard model. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with t’Hooft’s table [8]. We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles.” |
* See, for instance, “The Scientific Promise of Perfect Symmetry” in The New York Times of March 20, 2007.
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Tuesday, July 29, 2008
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Monday, July 28, 2008
Continued
“There is a body on
the cross in my church.”
— Mary Karr, quoted
here on July 10, 2007
From Jan. 20, 2004,
opening day of the first
Tennessee lottery–
Song of the Father
“Gonna buy me a shotgun,
long as I am tall,
Buy me a shotgun,
long as I am tall,
Gonna shoot po’ Thelma,
just to see her jump and fall.”
— Jimmie Rodgers, known as
“the father of country music.”
Comments Off on Monday July 28, 2008
Sunday, July 27, 2008
For Brother Taylor:
Bobbie Gentry is 64 today.
“It was the third of June,
another sleepy, dusty Delta day….”
Third of June, 2007
Third of June, 2008
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Saturday, July 26, 2008
From Josephine Klein, Jacob’s Ladder: Essays on Experiences of the Ineffable in the Context of Contemporary Psychotherapy, London, Karnac Books, 2003–
Page 14 —
Gerard Manley Hopkins
“Quiddity and haeccity were contentious topics in medieval discussions about the nature of reality, and the poet Gerard Manley Hopkins would have encountered these concepts during his Jesuit training. W. H. Gardner, who edited much of Hopkins’s work, writes that
in 1872, while studying medieval philosophy… Hopkins came across the writing of Duns Scotus, and in that subtle thinker’s Principles of Individuation and Theory of Knowledge he discovered what seemed to be a philosophical corroboration of his own private theory of inscape and instress. [Gardner, Gerard Manley Hopkins: Poems and Prose, Penguin, 1953, p. xxiii]
In this useful introduction to his selection of Hopkins’s work, Gardner writes that Hopkins was always looking for the law or principle that gave an object ‘its delicate and surprising uniqueness.’ This was for Hopkins ‘a fundamental beauty which is the active principle of all true being, the source of all true knowledge and delight.’ Clive Bell called it ‘significant form’; Hopkins called it ‘inscape’– ‘the rich and revealing oneness of the natural object’ (pp. xx-xxiv). In this chapter, I call it quiddity.”
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Friday, July 25, 2008
56 Triangles
"This wonderful picture was drawn by Greg Egan with the help of ideas from Mike Stay and Gerard Westendorp. It's probably the best way for a nonmathematician to appreciate the symmetry of Klein's quartic. It's a 3-holed torus, but drawn in a way that emphasizes the tetrahedral symmetry lurking in this surface! You can see there are 56 triangles: 2 for each of the tetrahedron's 4 corners, and 8 for each of its 6 edges."
Exercise:
Click on image for further details.
Note that if eight points are arranged
in a cube (like the centers of the
eight subcubes in the figure above),
there are 56 triangles formed by
the 8 points taken 3 at a time.
Baez's discussion says that the Klein quartic's 56 triangles can be partitioned into 7 eight-triangle Egan "cubes" that correspond to the 7 points of the Fano plane in such a way that automorphisms of the Klein quartic correspond to automorphisms of the Fano plane. Show that the 56 triangles within the eightfold cube can also be partitioned into 7 eight-triangle sets that correspond to the 7 points of the Fano plane in such a way that (affine) transformations of the eightfold cube induce (projective) automorphisms of the Fano plane.
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Thursday, July 24, 2008
Tried out the new knol.google.com site
with a copy of The Diamond Theorem.
Comments Off on Thursday July 24, 2008
Monday, July 21, 2008
Knight Moves:
The Relativity Theory
of Kindergarten Blocks
(Continued from
January 16, 2008)
Something:
From Friedrich Froebel,
who invented kindergarten:
Click on image for details.
An Unusually
Complicated Theory:
From Christmas 2005:
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
Click on image for details.
Comments Off on Monday July 21, 2008
Saturday, July 19, 2008
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week's New Yorker:
"
A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."
Hermann Weyl on the hard core of objectivity:
"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 237)
Steven H. Cullinane on the symmetries of a 4×4 array of points:
|
A Structure-Endowed Entity
"A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way."
— Hermann Weyl in Symmetry
Let us apply Weyl's lesson to the following "structure-endowed entity."
What is the order of the resulting group of automorphisms?
|
The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
called a "colorful tale"–
Comments Off on Saturday July 19, 2008
Friday, July 18, 2008
Hard Core
David Corfield quotes Weyl in a weblog entry, "Hierarchy and Emergence," at the n-Category Cafe this morning:
"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science [Princeton, 1949], p. 237)
For the same quotation in a combinatorial context, see the foreword by A. W. Tucker, "Combinatorial Problems," to a special issue of the IBM Journal of Research and Development, November 1960 (1-page pdf).
See also yesterday's Log24 entry.
Comments Off on Friday July 18, 2008
Thursday, July 17, 2008
CHANGE
FEW CAN BELIEVE IN
|
Continued from June 18.
Jungian Symbols
of the Self —
Compare and contrast:
Jung's four-diamond figure from
Aion — a symbol of the self —
Jung's Map of the Soul,
by Murray Stein:
"… Jung thinks of the self as undergoing continual transformation during the course of a lifetime…. At the end of his late work
Aion, Jung presents a diagram to illustrate the dynamic movements of the self…."
Comments Off on Thursday July 17, 2008
Tuesday, July 15, 2008
My comment on a discussion of elliptic curves and modular forms at Secret Blogging Seminar, about 10 PM tonight:
How does this affect popularized discussions of the Taniyama-Shimura conjecture– for instance, Ivars Peterson’s, in “Curving Beyond Fermat,” November 1999– which claim, for instance, that “Elliptic curves and modular forms are mathematically so different that mathematicians initially [in the 1950’s, the early days of the conjecture] couldn’t believe that the two are related.”?
Update of about 10:45 PM tonight:
A reply by the author of the discussion, Scott Carnahan:
I don’t think anyone doubted that there is a connection between elliptic curves and modular forms on the level I described above. However, the Taniyama-Shimura conjecture refers to a more advanced idea about a deeper connection.
Carnahan then gives a one-paragraph summary, definitely not popularized, of the deeper connection.
Comments Off on Tuesday July 15, 2008
Sunday, July 13, 2008
The Drunkard’s Walk
is the title of a recent
book by Leonard Mlodinow:
Cover of British edition
“Leonard Mlodinow has had, to speak informally, a pretty random career….
A far more sober instance of randomness, however, underpins his new book, The Drunkard’s Walk. And it’s not hard to see it as a sort of Rosebud, explaining why the author finds unpredictability so compelling.”
Another sort of Rosebud–
C. P. Snow on G. H. Hardy:
“… A Mathematician’s Apology is, if read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits: yes, the crystalline clarity and candour are still there: yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again.”
Perhaps in the afterlife Hardy, an expert on the theory of numbers, does again enjoy such powers. If so, his comments on the following would be of interest:
New York Lottery today:
Mid-day 006
(the first
perfect number)
Evening 568
(an apparently random number)
Hardy, when taken to church as a child, passed the time by factorizing
hymn numbers. This suggests we note that 568 equals 8 times 71. A check of
Wikipedia on the prime number 71 reveals that it is related to 568 in another way: 568 is is the sum of the primes less than 71–
2 + 3 + 5 + 7 + 11 +
13 + 17 + 19 + 23 +
29 + 31 + 37 + 41 +
43 + 47 + 53 + 59 +
61 + 67 = 568
Clearly it is false that the sum of the primes less than a prime
p is, in general, a multiple of
p, since (2 + 3 + 5) is not a multiple of 7. The sum of primes less than an integer
x is, however, of some interest.
See The On-Line Encyclopedia
of Integer Sequences,
A046731, Sum of primes < 10^n, as well as
A006880, Number of primes < 10^n.
According to an amateur* mathematician named Cino Hilliard, “a very important relationship exists” between the sum of primes less than x and the prime counting function Pi(x) which is the number of primes less than x—
(Sum of primes less than
x) ~ Pi(
x^2).
Whether this apparent relationship is, in fact, “very important,” or merely a straightforward consequence of other number-theoretical facts, is not obvious (to those of us not expert in number theory) from Google searches. Perhaps Hardy can clear this question up for those who will, by luck or grace, meet him in the next world.
* For some background, see a profile and user group messages here and here and here.
Comments Off on Sunday July 13, 2008
Comments Off on Sunday July 13, 2008
Christ's High Table
C. P. Snow in A Mathematician's Apology :
FOREWORD
"It was a perfectly ordinary night at Christ's high table, except that Hardy was dining as a guest. He had just returned to Cambridge as Sadleirian professor, and I had heard something of him from young Cambridge mathematicians. They were delighted to have him back: he was a real mathematician, they said, not like those Diracs and Bohrs the physicists were always talking about: he was the purest of the pure. He was also unorthodox, eccentric, radical, ready to talk about anything. This was 1931, and the phrase was not yet in English use, but in later days they would have said that in some indefinable way he had star quality."
Perhaps now also at Christ's high table– Scarlett O'Hara's Younger Sister , Evelyn Keyes, who died on July 4, 2008:
"… the memory of Evelyn Keyes looking at herself on the screen, exclaiming: 'There's star quality! Look at those tits!'"
Evelyn Keyes in 99 River Street
Comments Off on Sunday July 13, 2008
Footprint
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
-- Edna St. Vincent Millay
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Saturday, July 12, 2008
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Friday, July 11, 2008
Comments Off on Friday July 11, 2008
"I say high, you say low,
you say why,
and I say I don't know.
Oh, no.
You say goodbye
and I say hello."
— Hello Goodbye *
Thanks to NBC Nightly News tonight for a story on the following:
Manhattanhenge is an evening when "the Sun sets in exact alignment with the Manhattan grid, fully illuminating every single cross-street…."
Full Sun on grid:
Friday, July 11–
8:24 PM EDT
Related material from the late
Tom Disch on St. Sarah's Day:
Saturday, May 24th, 2008
9:15 pm
What I Can See from Here
I face east toward the western wall
Of a tall many-windowed building
Some distance off. I don't see the sunset
Directly, only as it is reflected
From the facade of that building.
Those familiar with Manhattan know
How the evening sun appears to slide
Into the slot of any east/west street,
And so its beams are channeled
Along those canyon streets to strike
Large objects like that wall
And scrawl their anti-shadows there,
A Tau of twilight luminescence
At close of day. I've seen this
For some forty years and only tonight
Did I realize what I had been looking at:
The way god tries to say good-bye.
— Tom Disch
|
* Walter Everett, in The Beatles as Musicians , has a note on the song "Hello Goodbye"–
"189. The extra-long coda… was referred to as the 'Maori finale' from the start…."
(Updated Feb. 27, 2013, to replace an incorrect reference in the footnote
to a book by Stanley Cavell instead of the correct book, by Walter Everett.)
Comments Off on Friday July 11, 2008
AND MORE LOGOS:
"Serious numbers will
always be heard."
— Paul Simon
|
The HSBC Logo Designer —
Henry Steiner
 He is an internationally recognized corporate identity consultant. Based in Hong Kong, his work for clients such as HongkongBank, IBM and Unilever is a major influence in Pacific Rim design.
Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors' Club.
His Cross-Cultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995).
— Yaneff.com
|
Related material
from the past —
Fly from Fly Bottle:
Charles Taylor,
"Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477) —
"… the object sets up
a kind of frame or space or field
within which there can be epiphany."
|
Related material
from today —
Escape from a
cartoon graveyard:
Comments Off on Friday July 11, 2008
LOGOS
"Religions are hardy."
— TIME magazine,
issue dated July 14
"I confess I do not believe in time."
— Vladimir Nabokov
"I can hardly do better than
go back to the Greeks."
— G. H. Hardy
Figure 1:
The Greeks
Figure 2:
The Irrational
Comments Off on Friday July 11, 2008
Thursday, July 10, 2008
Something
From the current
issue of TIME:
“Religions are hardy. ‘Many a time
we have gotten all ready for the
funeral’ of one faith or another,
‘and found it postponed again,
on account of
the weather or something.'”
— Mark Twain
Twain was raised
as a Presbyterian
(the Calvinist tradition).
This year’s Twain award
for humor went to
George Carlin,
raised in
the Catholic tradition.
On learning he had won
the Twain award,
Carlin said,
“Thank you, Mr. Twain.
Have your people
call my people.”
Today’s Birthdays:
Born July 10, 1509 —
Comments Off on Thursday July 10, 2008
Wednesday, July 9, 2008
Ah! Bright Wings
A poem by the late Thomas Disch:
Sundays at the Colosseum
I think you always had to be a little juiced
to enjoy the show. Or Jewish!
I never attended
without a flask of red, and would salute
the dying singers–
martyrs they called themselves–
when the lions drew first blood.
The songs
went on until either terror or death
had silenced the last of them. I doubt
we would have gone so religiously
if it weren't for the singing.
Sometimes we'd even sing along.
Circuses aren't the same these days.
Pity.
— From Disch's weblog on Friday,
May 23, 2008, at 8:26 AM
Related material on a novel by Disch:
"On Wings of Song, published in 1979, tells the story of a repressive Amesville, Iowa, in the 21st century. The main character, Daniel Weinreb, tries to master the art of song and flight, 'driven by the knowledge that some have attained flight, their spirits separated from their physical bodies and propelled on the waves of their own singing voices– literally born on wings of song.'"
— Jocelyn Y. Stewart in a Los Angeles Times obituary of July 8, 2008
Comments Off on Wednesday July 9, 2008
Comments Off on Wednesday July 9, 2008
God, Time, Epiphany
8:28:32 AM
Anthony Hopkins, from
All Hallows' Eve
last year:
"For me time is God,
God is time. It's an equation,
like an Einstein equation."
James Joyce, from
June 26 (the day after
AntiChristmas) this year:
"… he glanced up at the clock
of the Ballast Office and smiled:
— It has not epiphanised yet,
he said."
Ezra Pound (from a page
linked to yesterday morning):
"It seems quite natural to me
that an artist should have
just as much pleasure in an
arrangement of planes
or in a pattern of figures,
as in painting portraits…."
From Epiphany 2008:
An arrangement of planes:
From May 10, 2008:
A pattern of figures:
See also Richard Wilhelm on
Hexagram 32 of the I Ching:
"Duration is a state whose movement is not worn down by hindrances. It is not a state of rest, for mere standstill is regression. Duration is rather the self-contained and therefore self-renewing movement of an organized, firmly integrated whole, taking place in accordance with immutable laws and beginning anew at every ending. The end is reached by an inward movement, by inhalation, systole, contraction, and this movement turns into a new beginning, in which the movement is directed outward, in exhalation, diastole, expansion."
— The Middle-English
Harrowing of Hell…
by Hulme, 1907, page 64
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Tuesday, July 8, 2008
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