See a University of Wisconsin obituary for Schneider,
a leading expert on linear algebra who reportedly died
at 87 on Tuesday, October 28, 2014.
Some background on linear algebra and “magic” squares:
tonight’s 3 AM (ET) post and a search in this
journal for Knight, Death, and the Devil.
Click image to enlarge.
Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square
The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.
(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)
This journal Tuesday, Oct. 28, 2014, at 5 PM ET:
"What is a tai chi master, and what is it that he unfolds?"
From an earlier post, Hamlet's father's ghost
on "the fretful porpentine":
Hamlet , Act 1, Scene 5 —
Ghost:
“I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes, like stars, start from their spheres,
Thy knotted and combinèd locks to part
And each particular hair to stand on end,
Like quills upon the fretful porpentine:
But this eternal blazon must not be
To ears of flesh and blood."
Galway Kinnell:
"I roll
this way and that in the great bed, under
the quilt
that mimics this country of broken farms and woods"
— "The Porcupine"
For quilt-block designs that do not mimic farms or woods,
see the cover of Diamond Theory . See also the quotations
from Wallace Stevens linked to in the last line of yesterday's
post in memory of Kinnell.
"… a bee for the remembering of happiness" — Wallace Stevens
For poet Galway Kinnell, Princeton ’48:
Kinnell was named "Tiger of the Week" in a
Princeton Alumni Weekly post of August 27, 2014.
See his obituary in today's New York Times
as well as posts here on August 27, 2014.
A print copy of next Sunday’s New York Times Book Review
arrived in today’s mail. From the front-page review:
Marcel Theroux on The Book of Strange New Things ,
a novel by Michel Faber —
“… taking a standard science fiction premise and
unfolding it with the patience and focus of a
tai chi master, until it reveals unexpected
connections, ironies and emotions.”
What is a tai chi master, and what is it that he unfolds?
Perhaps the taijitu symbol and related material will help.
The Origin of Change
“Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined
On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come.”
— Wallace Stevens,
“Notes Toward a Supreme Fiction,”
Canto IV of “It Must Change”
For Karl Pribram and Katherine Neville,
a sequel to this morning's Figural Processing —
See also Christmas 2013.
Part I:
Part II:
Click images for some context.
A post in honor of Évariste Galois (25 October 1811 – 31 May 1832)
From a book by Richard J. Trudeau titled The Non-Euclidean Revolution —
See also “non-Euclidean” in this journal.
One might argue that Galois geometry, a field ignored by Trudeau,
is also “non-Euclidean,” and (for those who like rhetoric) revolutionary.
The "Chern" of today's previous post is mathematician
Shiing-Shen Chern (b. Oct. 26, 1911, d. Dec. 3, 2004).
For an observance of the 2011 centennial of his birth,
see a website in China.
See also this journal on the centennial date —
Erlanger and Galois, a post of Oct. 26, 2011.
In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean geometry or of Galois geometry.
In Euclidean geometry, these grids illustrate a property of
the inner triangle.
In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids. This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).
The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:
See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.
For 5×5 geometry that is not so elementary, see…
Hafner's abstract:
We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.
The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)
See Langer (Harvard U. Press, Third Edition, Jan. 31, 1957, pp. 3-4-5).
See also Old Key : Pythagoras, harmony, and the 3-4-5 triangle.
Robin Williams and the Stages of Math
i) shock & denial
ii) anger
iii) bargaining
iv) depression
v) acceptance
And then…
vi) checking
vii) Joan Rivers
See also
Continued from Day at the Museum, last Sunday, October 19, 2014.
This post was suggested by…
From The Catcher in the Rye , a passage just before the
museum passage quoted by Mendelsohn:
“She was having a helluva time tightening her skate.
She didn’t have any gloves on or anything and her hands
were all red and cold. I gave her a hand with it. Boy, I
hadn’t had a skate key in my hand for years. It didn’t feel
funny, though. You could put a skate key in my hand
fifty years from now, in pitch dark, and I’d still know
what it is. She thanked me and all when I had it tightened
for her. She was a very nice, polite little kid. God, I love it
when a kid’s nice and polite when you tighten their skate
for them or something. Most kids are. They really are.
I asked her if she’d care to have a hot chocolate or something
with me, but she said no, thank you. She said she had to meet
her friend. Kids always have to meet their friend. That kills me.
Even though it was Sunday and Phoebe wouldn’t be there
with her class or anything, and even though it was so damp
and lousy out, I walked all the way through the park over to
the Museum of Natural History. I knew that was the museum
the kid with the skate key meant.”
(Continued from Nov. 15, 2011)
Ben Bradlee, legendary Washington Post editor, dies at 93
See also a post of Jan. 20, 2011, and an earlier post on Twelfth Night, 2010.
A star figure and the Galois quaternion.
The square root of the former is the latter.
Two news items on art as a tool:
Two Log24 posts related to the 3×3 grid, the underlying structure for China’s
ancient Lo Shu “magic” square:
Finally, leftist art theorist Rosalind Krauss in this journal
on AntiChristmas, 2010:
Which is the tool here, the grid or Krauss?
(Night at the Museum continues.)
"Strategies for making or acquiring tools
While the creation of new tools marked the route to developing the social sciences,
the question remained: how best to acquire or produce those tools?"
— Jamie Cohen-Cole, “Instituting the Science of Mind: Intellectual Economies
and Disciplinary Exchange at Harvard’s Center for Cognitive Studies,”
British Journal for the History of Science vol. 40, no. 4 (2007): 567-597.
Obituary of a co-founder, in 1960, of the Center for Cognitive Studies at Harvard:
"Disciplinary Exchange" —
In exchange for the free Web tools of HTML and JavaScript,
some free tools for illustrating elementary Galois geometry —
The Kaleidoscope Puzzle, The Diamond 16 Puzzle,
The 2x2x2 Cube, and The 4x4x4 Cube
"Intellectual Economies" —
In exchange for a $10 per month subscription, an excellent
"Quilt Design Tool" —
This illustrates not geometry, but rather creative capitalism.
Related material from the date of the above Harvard death: Art Wars.
The online Harvard Crimson today:
“ ‘I don’t like how they check your bags
when you leave the library
even though you have to swipe your
student ID to get in.’
But what else would I be carrying in this
Gutenberg Bible-sized backpack? ”
Nicole Kidman at the end of “Hemingway & Gellhorn” (2012)
Or: The Long, Long Trailer
See also a Log24 post from the date of the above tweet: Welcome to the Ape Stuff.
“So it’s the laughter we will remember.” Speak for yourself, Barbra.
Barron's Educational Series (click to enlarge):
The Tablet of Ahkmenrah:
"With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series
Another educational series (this journal):
Art theorist Rosalind Krauss and The Ninefold Square
On Walter Isaacson’s The Innovators :
“Yet as the book’s five hundred–plus pages unwind, Isaacson interrupts himself to present small bromides about what it means to innovate and what we might learn from these innovators, our presumed betters. “Innovation requires articulation,” he tells us, after explaining how the main strength of Grace Hopper, a trailblazing computer scientist for the US Navy, was her ability to speak in the languages of mathematicians, engineers, programmers, and soldiers alike. ‘One useful leadership talent is knowing when to push ahead against doubters and when to heed them,’ he offers later.
The book is peppered with these kinds of passages, which often intrude on the narrative, depriving us of moments of real emotional power.”
— Jacob Silverman in Bookforum , Sept/Oct/Nov 2014
From Isaacson’s book:
Related material:
In memory of T. S. Eliot…
… and in memory of Stanley Chase, producer of Colossus: The Forbin Project
and of Threepenny Opera :
Ninefold square from Colossus
(“There is another system”) —
Fourfold square introducing Brecht
in Dreigroschen Trifft Vierfarben —
Raiders of the Lost Archetype
“… an unexpected development: the discovery of a lost archetype….”
— “The Lost Theorem,” by Lee Sallows, Mathematical Intelligencer, Fall 1997
Related material:
A scene from the 1954 film:
A check of this journal on the above MetaFilter date — Jan. 24, 2012 —
yields a post tagged “in1954.” From another post with that tag:
Backstory: Posts tagged Root Circle.
Rebecca Newberger Goldstein, quoted in a webpage dated
October 7, 2014 (presumably according to Australian time):
"For the Athenians, kleos mattered more than anything,
according to Goldstein.
'Kleos is fame: it’s the deed that brings fame, it’s the poem
that sings your triumphs, it’s having your life replicated in
other minds, acquiring a kind of moreness, a kind of
secular immortality.' "
Related material:
A check of Goldstein's definition…
… and an image for Broomsday:
From Argument for the Existence of Rebecca (Feb. 6, 2010)
This post was suggested by today's previous posts, Broom Bridge Day
and Taking the Fork, as well as by Alyssa is Wonderland.
For the meaning of the title, see Serpent + Derrida and Symbology.
Wikipedia on Broom (or Broome, or Brougham) Bridge,
where on 16 October 1843 Hamilton discovered quaternions:
"The 16 October is sometimes referred to as
Broomsday (in reference to Broome Bridge)
and as a nod to the literary commemorations
on 16 June (Bloomsday in honour of James Joyce)."
See also, in this journal, The Craft.
For Dorothy Maharam Stone, mathematician, who reportedly
died on Sept. 27, 2014 (Gwyneth Paltrow’s birthday):
“When I die…. I want it to be Hollywood all the way.
I don’t want some rabbi rambling on; I want
Meryl Streep crying, in five different accents….”
How about Meryl Streep as Ethel Greenglass Rosenberg
saying Kaddish in Angels in America ?
The title is from a Log24 post, "Diabolically Complex Riddle," of Sept. 27, 2014.
(See also a search for "Diabolic" in this journal, which yields an application to
"magic" squares.)
My own contribution to an event of the Mathematical Association of America:
Rick’s Tricky Six and The Judas Seat.
The Polster tetrahedral model of a finite geometry appears, notably,
in a Mathematics Magazine article from April 2009—
Parallelograms and the structure of the 3×3 array —
Click to enlarge:
A different approach to parallelograms and arrays —
Click for original post:
(Continued from Nov. 16, 2013.)
The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.
One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:
In context (click to enlarge):
Update of later the same day —
Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —
“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”
* Defined by Sallows later in the book:
“Geometric or, less formally, geomagic is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles or pieces ) may appear in the cells instead of numbers.”
** See some geometric matrices by Cullinane in a March 2001 webpage.
Earlier actual specimens — see Diamond Theory excerpts published in
February 1977 and a brief description of the original 1976 monograph:
“51 pp. on the symmetries & algebra of
matrices with geometric-figure entries.”
— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society
The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.
From the online Encyclopaedia Britannica:
Piero della Francesca, original name
Piero di Benedetto dei Franceschi
(born c. 1416/17, Sansepolcro, Republic of Florence [Italy]—
died Oct. 12, 1492, Sansepolcro),
painter whose serene, disciplined exploration of perspective
had little influence on his contemporaries but came to be
recognized in the 20th century as a major contribution to
the Italian Renaissance. The fresco cycle “The Legend of
the True Cross” (1452–66) and the diptych portrait of
Federico da Montefeltro, duke of Urbino, and his consort
(1465) are among his best known works.
Dialogue from "Django Unchained" —
"What's a bounty?" "It's like a reward."
Today's noon post links to posts on Tony Scott
that in turn lead to…
A post from June 27, 2005—
the date of Domino Harvey's death.
A link at the end of that post leads to…
"Dr. Chandra?" "Yes?" "Will I dream?"
See also…
Vikram Chandra, Geek Sublime:
The Beauty of Code, the Code of Beauty
This post is a sequel to Pythagorean Selfie (Sept. 30, 2014)
and October Nine: Lyche at Bodø.
Today’s Instagram photos from Josefine Lyche, still at Bodø:
The figure at left appears to be diving. This suggests a review of posts on
the late film director Tony Scott.
“Chase worked for years to make a movie of
‘The Threepenny Opera.’ He finally got it done in 1989 as
‘Mack the Knife,’ with Menahem Golan directing.”
— David Colker, LA Times obituary, Oct. 9, 2014
See also, from Tuesday, Oct. 7, 2014, the date of Chase’s death,
the Log24 posts Grids and Space, Concepts of Space, and As Is.
(The phrase is from Don DeLillo and Josefine Lyche.)
See “Complex Grid.”
See as well Bill O’Reilly’s remark, “Do not be a coxcomb,”
and an artist‘s self-portrait:
Grid Designer
(Continued from Sept. 3, 2009)
George Steiner on chess:
"At the sight of a set, even the tawdriest of plastic pocket sets,
one’s fingers arch and a coldness as in a light sleep steals over
one’s spine. Not for gain, not for knowledge or reknown, but
in some autistic enchantment, pure as one of Bach’s inverted
canons or Euler’s formula for polyhedra."
— George Steiner in “A Death of Kings,” The New Yorker,
issue dated September 7, 1968, page 133
A related remark from Dudeney:
See also a different context for 16 squares and 322,560 arrangements.
(Continued from Grids and Space and posts tagged Riddle for Caltech)
Click to enlarge.
See also Apollo in this journal.
“Nine is a very powerful Nordic number.”
— Katherine Neville, who deserves some sort of prize for literature.
— Heidegger, “Hölderlin and the Essence of Poetry,”
translated by Douglas Scott, in Existence and Being ,
Regnery, 1949
"That simple operator, 'as,' turns out to carry within its philosophical grammar
a remarkable complex field* of operations…."
— Charles Altieri, Painterly Abstraction in Modernist American Poetry,
Cambridge University Press, 1989, page 343
See also Rota on Heidegger (What "As" Is, July 6, 2010), and Lead Belly
on the Rock Island Line — "You got to ride it like you find it."
* Update of Oct. 10, 2014: See also "Complex + Grid" in this journal.
Or: Plan 9 Continues
(Suggested by this afternoon’s post Concepts of Space.)
See also Card…
… and Tick Tick Hash.
Or: Phantasmagoria Meets Pandemonium
Part I: Phantasmagoria
Rebecca Goldstein on first encountering Plato —
| “I was reading Durant’s section on Plato, struggling to understand his theory of the ideal Forms that lay in inviolable perfection out beyond the phantasmagoria. (That was the first, and I think the last, time that I encountered that word.)” |
| “We tell ourselves stories in order to live….We interpret what we see, select the most workable of multiple choices. We live entirely, especially if we are writers, by the imposition of a narrative line upon disparate images, by the ‘ideas’ with which we have learned to freeze the shifting phantasmagoria which is our actual experience.” |
Part II: Pandemonium
Terry Teachout in Commentary on Oct. 1, 2014:
“When making art or writing about it, the aesthete
tries never to moralize. Nor will he look with favor
upon artists who do so, no matter whether their
particular brand of moralizing is religious or secular.
But he can and must be fully, intensely alive to the
moral force of art whose creators aspire merely to
make the world around us more beautiful, and in
so doing to pierce the veil of the visible and give us
a glimpse of the permanently true. That is his job:
to help make sense of the pandemonium amid which
we live.”
Rivka Galchen in The New York Times Sunday Book Review
issue of October 5, 2014 (online Sept. 30):
“The story describes honestly something that is,
which is very different from proposing what ought to be.”
See also Pandemonium in this journal.
From the MacTutor biography of Otto Neugebauer:
“… two projects which would be among the most important
contributions anyone has made to mathematics. He persuaded
Springer-Verlag to publish a journal reviewing all mathematical
publications, which would complement their reviewing journals
in other topics. In 1931 the first issue of Zentralblatt für Matematik
appeared, edited by Neugebauer.” [Mathematical Reviews was
the other project.]
Neugebauer appeared in Sunday morning’s post In Nomine Patris .
A review from Zentralblatt appeared in the Story Creep link from
this morning’s post Mysterious Correspondences.
Ben Brantley in The New York Times today on a Broadway opening:
“As Christopher navigates his way through an increasingly
unfamiliar landscape, both physical and emotional, the arcs
of his adventures are drawn into being.
So are the shards of sensory overload.”
Arc — See a search for Line at Infinity:
Shard — See Shard and Pythagorean Selfie:
(Continued from Beautiful Mathematics, Dec. 14, 2013)
“Seemingly unrelated structures turn out to have
mysterious correspondences.” — Jim Holt, opening
paragraph of a book review in the Dec. 5, 2013, issue
of The New York Review of Books
One such correspondence:
For bibliographic information and further details, see
the March 9, 2014, update to “Beautiful Mathematics.”
See as well posts from that same March 9 now tagged “Story Creep.”
I.e., Neugebauer (See The Source, 9 PM Saturday, Oct. 4, 2014.)
See also Alms for Oblivion (January 22, 2006).
With Sarah Silverman …
… Continued from The Story of N (October 15, 2010).
“I remember how the darkness doubled….”
"In ancient Greece, 9 was the number of
the Muses, patron goddesses of the arts.
They were the daughters of Mnemosyne ('memory'),
the source of imagination, which in turn is
the carrier of archetypal, elementary ideas to
artistic realization in the field of space-time."
— Joseph Campbell in The Inner Reaches of Outer Space
In memoriam:
See also Raiders of the Lost Well and…
Ground plan for a game of Noughts and Crosses
“Nine is a very powerful Nordic number.”
— Katherine Neville, The Magic Circle
See also Arcade Fire in this journal.
From the preface to Introduction to the Construction of Class Fields ,
by Harvey Cohn (Cambridge University Press, 1985):
“It is an elementary observation that an integral right triangle
has an even area. Suppose the hypotenuse is prime.
Q. How do we determine from the prime value of the hypotenuse
when the area is divisible by 4, 8, 16, or any higher power of 2?
A. We use class fields constructed by means of transcendental
functions, of course!
The question might have been asked by Pythagoras in about
500 BC….”
The question seems to assume something apparently not known to Pythagoras:
The area is determined uniquely by the prime hypotenuse.
Nontrivial exercise: Prove or disprove this assertion.
Background to the exercise: See Fermat’s Christmas Theorem on the Web,
and a specific remark about prime hypotenuses in a letter from Fermat to
Mersenne on Christmas Day, 1640, quoted in The Mathematical Career
of Pierre de Fermat, 1601-1665 , by Michael Sean Mahoney (Princeton
University Press, 2nd ed,, 1994), pp. 316-317.
From the title page of a book quoted here yesterday:
Another title, in memory of a former mayor of Tel Aviv who reportedly died yesterday:
“President Reuven Rivlin called Lahat ‘Tel Aviv’s Herod,’
referring to the semi-mythical Jewish king whose epic
construction projects, including the Temple Mount,
Masada and Caesarea, remain standing 2,000 years after
his death.” — The Times of Israel
The title is the usual pronunciation of MSRI,
the Mathematical Sciences Research Institute
at 17 Gauss Way, Berkeley, California.
The late Scandinavian novelist Stieg Larsson
might prefer to call this street Gardner Way.
The following post suggests the Spiders and Snakes of Fritz Leiber’s
Changewar , a mythology inspired by the hallucinations of delirium tremens .
Loren Olson, Harvard ’64, a professor of mathematics
at Norway’s Tromsø University,* died June 22, 2014.
In his memory, a search in this journal for Lie Group.
That search yields a post titled Lie Groups for Holy Week (March 30, 2010).
A quotation related to that post:
* The city of Tromsø hosted some art related to group theory in 2010.
Neither that art nor my own related remarks on group theory are very
relevant to physics (yet).
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