For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3-space over GF(5))?
"We tried to create new realities overnight,
careful sets of words that resemble advertising slogans
in memorability and repeatability."
… Our Most Important Product …
"Omega is as real as we need it to be."
— Burt Lancaster in Sam Peckinpah's last film
Continued from Tuesday, November 18, 2014
International poster for the 2010 film "Black Swan"—
"Death is a black swan." — Graciela Chichilnisky
Wallace Stevens in "An Ordinary Evening in New Haven"
(1950) on "The Ruler of Reality" —
"Again, 'He has thought it out, he thinks it out,
As he has been and is and, with the Queen
Of Fact, lies at his ease beside the sea.'"
One such scene, from 1953 —
Another perspective, from "The Osterman Weekend" (1983) —
Corrections to the NY Times obituary of Alexander Grothendieck
are shown below. For the original Sunday, Nov. 16, NY Times
print obituary (with its online date, Nov. 14), see a copy taken
from a weblog.
For another poetic remark in memory of Grothendieck,
see a Log24 post from November 13, the day of his death.
See the easy grace link in this morning's previous post.
See also Michael Dubruiel and Log24 posts of Feb. 3
and Feb. 9, 2009.
Welcome to the Garden Club, Pilgrim
"A journalist with a literary bent, Mr. Eder wrote with an easy grace
and a practiced eye for detail. In 1974, he assessed a cultural
malaise in England during an economic downturn.
'In the West Country town of Hereford,' he began, 'the president
of a women’s club told a year-end meeting that the January bingo
game would be canceled to save electricity. Then she proposed a
New Year’s resolution. "Let us all work to get England back on her
dear old feet," she said and bumped down pinkly into her chair,
overwhelmed by applause.'" — Bruce Weber, NY Times
See also Bingo in this journal.
"When Three Into One Equals More" — New York Times headline
See also Trinity in this journal. From that search:
… The actor is
A metaphysician in the dark….
— Wallace Stevens,
"Of Modern Poetry"
Continued from Tuesday, Nov. 18
The above conclusion of a NY Times obituary
about a Monday death may also serve as the
missing conclusion of Monday's "A Search for
Missing Pieces" —
Related material starring Einstein and
Thomas Mann: "A Riddle for Davos."
Harold Rosenberg, "Art and Words,"
The New Yorker , March 29, 1969. From page 110:
"An advanced painting of this century inevitably gives rise
in the spectator to a conﬂict between his eye and his mind;
as Thomas Hess has pointed out, the fable of the emperor's
new clothes is echoed at the birth of every modemist art
movement. If work in a new mode is to be accepted, the
eye/mind conﬂict must be resolved in favor of the mind;
that is, of the language absorbed into the work. Of itself,
the eye is incapable of breaking into the intellectual system
that today distinguishes between objects that are art and
those that are not. Given its primitive function of
discriminating among things in shopping centers and on
highways, the eye will recognize a Noland as a fabric
design, a Judd as a stack of metal bins— until the eye's
outrageous philistinism has been subdued by the drone of
formulas concerning breakthroughs in color, space, and
even optical perception (this, too, unseen by the eye, of
course). It is scarcely an exaggeration to say that paintings
are today apprehended with the ears. Miss Barbara Rose,
once a promoter of striped canvases and aluminum boxes,
confesses that words are essential to the art she favored
when she writes, 'Although the logic of minimal art gained
critical respect, if not admiration, its reductiveness allowed
for a relatively limited art experience.' Recent art criticism
has reversed earlier procedures: instead of deriving principles
from what it sees, it teaches the eye to 'see' principles; the
writings of one of America's inﬂuential critics often pivot on
the drama of how he failed to respond to a painting or
sculpture the ﬁrst few times he saw it but, returning to the
work, penetrated the concept that made it signiﬁcant and
was then able to appreciate it. To qualify as a member of the
art public, an individual must be tuned to the appropriate
verbal reverberations of objects in art galleries, and his
receptive mechanism must be constantly adjusted to oscillate
to new vocabularies."
New vocabulary illustrated:
Graphic Design and a Symplectic Polarity —
Background: The diamond theorem
and a zero system .
Tom Wolfe in The Painted Word (1975):
"It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde understood, they
were not only theories but … hot news,
straight from the studios, from the scene."
"Parable of American Painting," 1954 — From The Tradition of the New , by Harold Rosenberg
"In this essay Rosenberg set out to explain what he believed to be definitively American about Abstract Expressionism. He did so by drawing on the American Revolutionary War for his metaphors, likening the new Americans to the coonskin trappers whose knowledge of their terrain enabled them to pick off the British soldiers (Redcoats), who followed the dictates of their military training. The professionally- trained soldiers were defeated because, as Rosenberg states, 'They were such extreme European professionals … they did not even see the American trees.' 'Redcoatism' was, Rosenberg argued, a symptom of the old European world's stubborn rejection of the new. It did at one time also '[dominate] the history of American art,' he wrote, but with the advent of Abstract Expressionism, times had changed. And just as the Coonskins were victorious because they stood apart from the professional military, so the new American art was triumphant because, as Rosenberg saw it, it marked a profound break with the traditions of European art."
Lectures at Bennington, 1971
Art adapted today from the Google search screen. Discuss.
Prequel from 1961 (click image for context):
Detail that may be interpreted as the Chinese
3×3 "Holy Field" and a Chinese temple bell—
"Ting-a-ling." — Kurt Vonnegut.
"Then came a 'Robot Psychologist,' known as a Psychological Matrix Rotator,
developed for the Department of Defense. It is still used to literally 'see' that
the right man gets the right Army job."
— Ronald Reagan, 1961 GE Sales Meeting
"Always with a little humor." — Yen Lo
In memory of Dr. Irving Peress,
who reportedly died on Thursday,
November 13, 2014.
A detail from the image search below —
A Google image search today for
“portal del aguila de oro” “bella vista” —
Click for a larger (4.2 MB) version.
See also yesterday’s post.
“Bit by bit, putting it together.
Piece by piece, working out the vision night and day.
All it takes is time and perseverance
With a little luck along the way.”
— Stephen Sondheim
See also, in this journal, 528, 1963, 522, and 3273.
Alexander Grothendieck, Récoltes et Semailles , 184.108.40.206. e, p. 1181:
Pour mettre la joie à son comble, j’ajoute que le dénommé Saavedra
semble avoir disparu de la circulation sans plus laisser aucune trace….
Du coup, l’histoire prend des allures de sombre intrigue policière.
Google Translate version:
To the joy at its height, I would add that the so-called Saavedra
seems to have disappeared from circulation without leaving any trace….
Suddenly, the story looks like a dark detective story.
Or horror film —
A Math Enigma —
Grothendieck reportedly died on Thursday, November 13, 2014.
From this journal a year earlier:
A German Enigma —
After clicking on the link “a grave intellectual whole” from the Nov. 13, 2013, post
see also Benedict Cumberbatch’s less serious approach to Bletchley Park:
“I want to see this film; this film’s been up my ass for the last five years.”
“What happens when you mix the brilliant wit of Noel Coward
with the intricate plotting of Agatha Christie? Set during a
weekend in an English country manor in 1932, Death by Design
is a delightful and mysterious ‘mash-up’ of two of the greatest
English writers of all time. Edward Bennett, a playwright, and
his wife Sorel Bennett, an actress, flee London and head to
Cookham after a disastrous opening night. But various guests
— Samuel French (theatrical publisher) on a play that
opened in Houston on September 9, 2011.
A paper from 1976 on symplectic torsors and finite geometry:
A pdf is now available:
FINITE GEOMETRIES IN THE THEORY OF THETA CHARACTERISTICS
Autor(en): Rivano, Neantro Saavedra
Zeitschrift: L’Enseignement Mathématique
Band (Jahr): 22 (1976)
Heft 1-2: L’ENSEIGNEMENT MATHÉMATIQUE
PDF erstellt am: 14.11.2014
Persistenter Link: http://dx.doi.org/10.5169/seals-48185
(Received by the journal on February 20, 1976.)
Saavedra-Rivano was a student of Grothendieck, who reportedly died yesterday.
“Alexandre Grothendieck est mort jeudi matin
à l’hôpital de Saint-Girons (Ariège), à l’âge de 86 ans.”
Update of 6: 16 PM ET: A memorial of sorts, from May 27 this year:
Yesterday's post and recent Hollywood news suggest
a meditation on a Progressive Matrix —
Oct. 12-14, 2005:
Click to enlarge.
Structurally related images —
A sample Raven's Progressive Matrices test item
(such items share the 3×3 structure of the hash symbol above):
Structural background —
The title is a phrase from yesterday’s post.
An example of harrowing cuteness:
Charlize Theron in “Young Adult” (2011) —
Related material for older adults: Ravenna and Nietzsche.
Remember him to Herald Square.
“We live entirely, especially if we are writers, by the imposition
of a narrative line upon disparate images….” — Joan Didion
Can the above narrative line be imposed in any sensible way
upon the above disparate images?
|“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.” — Joan Didion
See Didion and the I Ching and posts tagged Plato in China .
“There exists a considerable literature
devoted to the Lo shu , much of it infected
with the kind of crypto-mystic twaddle
met with in Feng Shui.”
— Lee C. F. Sallows, Geometric Magic Squares ,
Dover Publications, 2013, page 121
Cf. Raiders of the Lost Theorem, Oct. 13, 2014.
See also tonight’s previous post and
“Feng Shui” in this journal.
There is such a thing as an MBTI Tesseract.
See a thread at http://www.typologycentral.com/forums/
from August 17 and 18, 2010.
See also this journal on those dates: The Kermode Game.
“This setting of the Ave Verum Corpus text was composed
to celebrate the feast of Corpus Christi .” — Wikipedia
“Ave Verum Corpus .”— Madison in the BBC America TV series
“Intruders,” Season 1, Episode 3: “Time Has Come Today.”
See also the Eucharistic meditation of Feb. 13, 2006, linked to in yesterday’s post
on Guy Fawkes Day. (That British holiday originally commemorated the Catholic
Gunpowder Plot of 1605.)
From the first of previous Log24 posts tagged “Dark Fields”—
“A link in memory of Donald G. Higman,
dead on Feb. 13, 2006,
the day after Lincoln’s birthday:
On the Graphs of Hoffman-Singleton and Higman-Sims.
His truth is marching on.”
See also Foundation Square (October 25, 2014).
A link to “Nine Tailors” in this journal may serve as
a memorial to the late David M. Abshire, who
reportedly died at 88 on Halloween.
See also tonight’s previous post and a remark by
Mira Sorvino in her version of The Great Gatsby .
Courtesy of Mira Sorvino.
Enter Madison :
From “Intruders,” BBC America, Season 1, Episode 2, at 1:07 of 43:31.
“You sure know how to show a girl a good time.“
* The title is a reference to a Wisconsin-related Halloween post.
A weblog reports Chris Rock’s remarks
on Saturday Night Live this past weekend:
“It’s America, we commercialize everything.
Look at what we did to Christmas.
Christmas. Christmas is Jesus’ birthday.
It’s Jesus’ birthday. Now, I don’t know Jesus
but from what I’ve read, Jesus is the least
materialistic person to ever roam the earth.
No bling on Jesus.
Jesus kept a low profile and we turned his
birthday into the most materialistic day of the
year. Matter of fact, we have the Jesus birthday
season. It’s a whole season of materialism.
Then, at the end of the Jesus birthday season
we have the nerve to have an economist come
on TV and tell you how horrible the Jesus birthday
season was this year. Oh, we had a horrible Jesus’
birthday this year. Hopefully, business will pick up
by his Crucifixion.”
Related music and image:
“Show us the way to the next little girl …”
Natalie Wood in “Miracle on 34th Street” (1947)
Related non-materialistic meditations:
The Rhetoric of Abstract Concepts and Gods and Giants.
From a post of June 3, 2013:
New Yorker editor David Remnick at Princeton today
(from a copy of his prepared remarks):
“Finally, speaking of fabric design….”
I prefer Tom and Harold:
Tom Wolfe in The Painted Word —
“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”
Harold Rosenberg in The New Yorker (click to enlarge)—
From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 59-79 (1992):
“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”
— Steven H. Cullinane,
diamond theorem illustration
“Macy’s Herald Square occupies a singular place
in American retailing.” — NY Times today, in print
on page BU1 of the New York edition with the headline:
Makeover on 34th Street .
A Singular Time:
See Remember Me to Herald Square, at noon on
August 21, 2014, and related earlier Log24 posts.
Also on Aug. 21, 2014: from a blog post, ‘Tiles,’ by
Theo Wright, a British textile designer —
The 24 tile patterns displayed by Wright may be viewed
in their proper mathematical context at …
… there is a path. — Malcolm Lowry
“We are not saints.” — Alcoholics Anonymous , Chapter 5
The New York Times on AA’s co-founder Bill Wilson in 1934:
Click for the rest of the story.
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.
On Devil’s Night
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 —
“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.”
this way and that in the great bed, under
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”
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…
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.
See that phrase in this journal.
Robin Williams and the Stages of Math
i) shock & denial
vii) Joan Rivers
Continued from Day at the Museum, last Sunday, October 19, 2014.
This post was suggested by…
- A piece in the Bookends section of the New York Times
Sunday Book Review (page BR31 last Sunday, Oct. 19):
Daniel Mendelsohn on rereading The Catcher in the Rye .
- A detail in Day at the Museum— The New York Times ‘s
appraisal of Joan Rivers: “A Comic Without a Shut-Off Switch.”
- A Sept. 7 Log24 post, Sunday School, in memory of Joan Rivers.
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.”