See Ballet Blanc in this journal.
For a darker perspective, click on the image below.
See also Cartier in The Hexagon of Opposition.
Happy birthday to Kirk Douglas.
See Ballet Blanc in this journal.
For a darker perspective, click on the image below.
See also Cartier in The Hexagon of Opposition.
Happy birthday to Kirk Douglas.
The Snow White dance from last Nov. 14
features an ad that was originally embedded
in an American Mathematical Society Notices
review describing three books of vulgarized
mathematics. These books all use "great
equations" as a framing device.
This literary strategy leads to a more abstract
snow dance. See the ballet blanc in this journal
on Balanchine's birthday (old style) in 2003.
That dance involves equation (C) below.
Recall that in a unit ring ,
"0" denotes the additive identity,
"1" the multiplicative identity, and "-1" the
additive inverse of the multiplicative identity.
Three classic equations:
(A) 1 + 1 = 2 (Characteristic 0, ordinary arithmetic)
(B) 1 + 1 = 0 (Characteristic 2 arithmetic, in which 2 = 0)
(C) 1 + 1 = -1 (Characteristic 3 arithmetic, in which 2 = -1)
Cases (B) and (C), in which the characteristic is prime,
occur in Galois geometry.
For a more elaborate snow dance, see Master Class.
See a Haaretz story commemorating the Feb. 14,
1917, birthday of a crystallographer.
Related material in this journal —
At the Still Point (June 15, 2013):
The illustration is for those who, like Andy Magid and
Steven Strogatz in the March 2014 AMS Notices,
enjoy the vulgarization of mathematics.
Backstory: Group Actions (November 14, 2012).
This journal on July 5, 2007 —
“It is not clear why MySpace China will be successful."
— The Chinese magazine Caijing in 2007, quoted in
Asia Sentinel on July 12, 2011
This journal on that same date, July 12, 2011 —
See also the eightfold cube and kindergarten blocks
at finitegeometry.org/sc.
Friedrich Froebel, Froebel's Chief Writings on Education ,
Part II, "The Kindergarten," Ch. III, "The Third Play":
"The little ones, who always long for novelty and change,
love this simple plaything in its unvarying form and in its
constant number, even as they love their fairy tales with
the ever-recurring dwarfs…."
This journal, Group Actions, Nov. 14, 2012:
"Those who insist on vulgarizing their mathematics
may regard linear and affine group actions on the eight
cubes as the dance of Snow White (representing (0,0,0))
and the Seven Dwarfs—
The December 2012 Notices of the American
Mathematical Society has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—
(Only the top part of the ad is shown; for further details
see an ICERM page.)
(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)
The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—
The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—
© 2005 The Institute for Figuring
Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)
The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.
These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.
See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.
Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—
.
Hermann Hesse's 1943 The Glass Bead Game (Picador paperback, Dec. 6, 2002, pp. 139-140)–
"For the present, the Master showed him a bulky memorandum, a proposal he had received from an organist– one of the innumerable proposals which the directorate of the Game regularly had to examine. Usually these were suggestions for the admission of new material to the Archives. One man, for example, had made a meticulous study of the history of the madrigal and discovered in the development of the style a curved that he had expressed both musically and mathematically, so that it could be included in the vocabulary of the Game. Another had examined the rhythmic structure of Julius Caesar's Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns. Or again some fanatic had once more unearthed some new cabala hidden in the musical notation of the fifteenth century. Then there were the tempestuous letters from abstruse experimenters who could arrive at the most astounding conclusions from, say, a comparison of the horoscopes of Goethe and Spinoza; such letters often included pretty and seemingly enlightening geometric drawings in several colors."
From Siri Hustvedt, author of Mysteries of the Rectangle: Essays on Painting (Princeton Architectural Press, 2005)– What I Loved: A Novel (Picador paperback, March 1, 2004, page 168)–
A description of the work of Bill Wechsler, a fictional artist:
"Bill worked long hours on a series of autonomous pieces about numbers. Like O's Journey, the works took place inside glass cubes, but these were twice as large– about two feet square. He drew his inspiration from sources as varied as the Cabbala, physics, baseball box scores, and stock market reports. He painted, cut, sculpted, distorted, and broke the numerical signs in each work until they became unrecognizable. He included figures, objects, books, windows, and always the written word for the number. It was rambunctious art, thick with allusion– to voids, blanks, holes, to monotheism and the individual, the the dialectic and yin-yang, to the Trinity, the three fates, and three wishes, to the golden rectangle, to seven heavens, the seven lower orders of the sephiroth, the nine Muses, the nine circles of Hell, the nine worlds of Norse mythology, but also to popular references like A Better Marriage in Five Easy Lessons and Thinner Thighs in Seven Days. Twelve-step programs were referred to in both cube one and cube two. A miniature copy of a book called The Six Mistakes Parents Make Most Often lay at the bottom of cube six. Puns appeared, usually well disguised– one, won; two, too, and Tuesday; four, for, forth; ate, eight. Bill was partial to rhymes as well, both in images and words. In cube nine, the geometric figure for a line had been painted on one glass wall. In cube three, a tiny man wearing the black-and-white prison garb of cartoons and dragging a leg iron has
— End of page 168 —
opened the door to his cell. The hidden rhyme is "free." Looking closely through the walls of the cube, one can see the parallel rhyme in another language: the German word drei is scratched into one glass wall. Lying at the bottom of the same box is a tiny black-and-white photograph cut from a book that shows the entrance to Auschwitz: ARBEIT MACHT FREI. With every number, the arbitrary dance of associations worked togethere to create a tiny mental landscape that ranged in tone from wish-fulfillment dream to nightmare. Although dense, the effect of the cubes wasn't visually disorienting. Each object, painting, drawing, bit of text, or sculpted figure found its rightful place under the glass according to the necessary, if mad, logic of numerical, pictorial, and verbal connection– and the colors of each were startling. Every number had been given a thematic hue. Bill had been interested in Goethe's color wheel and in Alfred Jensen's use of it in his thick, hallucinatory paintings of numbers. He had assigned each number a color. Like Goethe, he included black and white, although he didn't bother with the poet's meanings. Zero and one were white. Two was blue. Three was red, four was yellow, and he mixed colors: pale blue for five, purples in six, oranges in seven, greens in eight, and blacks and grays in nine. Although other colors and omnipresent newsprint always intruded on the basic scheme, the myriad shades of a single color dominated each cube.
The number pieces were the work of a man at the top of his form. An organic extension of everything Bill had done before, these knots of symbols had an explosive effect. The longer I looked at them, the more the miniature constructions seemed on the brink of bursting from internal pressure. They were tightly orchestrated semantic bombs through which Bill laid bare the arbitrary roots of meaning itself– that peculiar social contract generated by little squiggles, dashes, lines, and loops on a page."
From 2002:
Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest. |
ZZ
Figures from the
Poem by Eugen Jost:
Mit Zeichen und Zahlen
Numbers and Names,
With numbers and names English translation A related poem:
Alphabets
From time to time
But if a savage
— Hermann Hesse (1943), |
“Mahlburg likens his approach to an analogous one for deciding whether a dance party has an even or odd number of attendees. Instead of counting all the participants, a quicker method is to see whether everyone has a partner—in effect making groups that are divisible by 2.
In Mahlburg’s work, the partition numbers play the role of the dance participants, and the crank splits them not into couples but into groups of a size divisible by the prime number in question. The total number of partitions is, therefore, also divisible by that prime.
Mahlburg’s work ‘has effectively written the final chapter on Ramanujan congruences,’ Ono says.
‘Each step in the story is a work of art,’ Dyson says, ‘and the story as a whole is a sequence of episodes of rare beauty, a drama built out of nothing but numbers and imagination.'”
— Erica Klarreich in Science News Online, week of June 18, 2005
This would seem to meet the criteria set by Fritz Leiber for “a story that works.” (See previous entry.) Whether the muse of dance (played in “Xanadu” by a granddaughter of physicist Max Born– see recent entries) has a role in the Dyson story is debatable.
Born Dec. 11, 1882, Breslau, Germany. Died Jan. 5, 1970, Göttingen, |
Max Born |
Those who prefer less abstract stories may enjoy a mythic tale by Robert Graves, Watch the North Wind Rise, or a Christian tale by George MacDonald, At the Back of the North Wind.
Related material:
“The valley spirit never dies. It’s named the mystic woman.”
For an image of a particular
incarnation of the mystic woman
(whether as muse, as goddess,
or as the White Witch of Narnia,
I do not know) see Julie Taymor.“Down in the valley,
valley so low,
hang your head over,
hear the wind blow.”“Which is the sound of the land
Full of the same wind
That is blowing in
the same bare placeFor the listener,
who listens in the snow,
And, nothing himself, beholds
Nothing that is not there
and the nothing that is.”
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