The sermon, “God’s Architecture,” at Nassau Presbyterian
Church in Princeton on Sunday, Feb. 23, 2014. (This is the
“sermon” link in last Sunday’s 11 AM ET Log24 post.)
“I wonder what God sees when God looks at our church.
Bear with me here because I’d like to do a little architectural
redesign. I look up at our sanctuary ceiling and I see buttons.
In those large round lights, I see buttons. I wonder what would
happen if we unbutton the ceiling, Then I wonder if we were to
unzip the ceiling, pull back the rooftop, and God were to look in
from above – What does God see? What pattern, what design,
what shape takes place?” — Rev. Lauren J. McFeaters
Related material — All About Eve:
A. The Adam and Eve sketch from the March 8 “Saturday Night Live”
B. “Katniss, get away from that tree!“ —
C. Deconstructing God in last evening’s online New York Times .
“Heavensbee!” in the above video, as well as Cartier’s Groundhog Day
and Say It With Flowers.
Humans’ architecture, as described (for instance) by architecture
theorist Anne Tyng, who reportedly died at 91 on Dec. 27, 2011.
See as well Past Tense and a post from the date of Tyng’s death.
For Women’s History Month —
Conclusion of “The Storyteller,” a story
by Cynthia Zarin about author Madeleine L’Engle—
See also the exercise on the Miracle Octad Generator (MOG) at the end of
the previous post, and remarks on the MOG by Emily Jennings (non -fiction)
on All Saints’ Day, 2012 (the date the L’Engle quote was posted here).
From “Quartic Curves and Their Bitangents,” by
Daniel Plaumann, Bernd Sturmfels, and Cynthia Vinzant,
arXiv:1008.4104v2 [math.AG] 10 Jan 2011 —
The table mentioned (from 1855) is…
Exercise: Discuss the relationship, if any, to
the Miracle Octad Generator of R. T. Curtis.
From Facebook, a photo from the Feast of St. Francis, 2013:
Neantro Saavedra-Rivano, author of the 1976 paper ”Finite
Geometries in the Theory of Theta Characteristics,” in Brasilia—
On the same date, art from Inception and from Diamonds Studio
in Brazil —
On Theta Characteristics
— From Zentralblatt-math.org. 8 PM ET update: See also a related search.
Some may prefer a more politically correct— and simpler— sermon.
Background for the simpler sermon: Quilt Geometry.
For the Princeton Class of 1905 —
Joyce Carol Oates Meets Emily Dickinson.
“It is an afternoon in autumn, near dusk.
The western sky is a spider’s web of translucent gold.
I am being brought by carriage—two horses—
muted thunder of their hooves—
along narrow country roads between hilly fields
touched with the sun’s slanted rays,
to the village of Princeton, New Jersey.
The urgent pace of the horses has a dreamlike air,
like the rocking motion of the carriage;
and whoever is driving the horses
his face I cannot see, only his back—
stiff, straight, in a tight-fitting dark coat.”
“Because I could not stop for Death—
He kindly stopped for me—
The Carriage held but just Ourselves—
“Charting the Real Four-Qubit Pauli Group
via Ovoids of a Hyperbolic Quadric of PG(7,2),”
by Metod Saniga, Péter Lévay and Petr Pracna,
arXiv:1202.2973v2 [math-ph] 26 Jun 2012 —
P. 4— “It was found that Q +(5,2) (the Klein quadric)
has, up to isomorphism, a unique one — also known,
after its discoverer, as a Conwell heptad .
The set of 28 points lying off Q +(5,2) comprises
eight such heptads, any two having exactly one
point in common.”
P. 11— ”This split reminds us of a similar split of
63 points of PG(5,2) into 35/28 points lying on/off
a Klein quadric Q +(5,2).”
 G. M. Conwell, Ann. Math. 11 (1910) 60–76
A similar split occurs in yesterday’s Kummer Varieties post.
See the 63 = 28 + 35 vectors of R8 discussed there.
For more about Conwell heptads, see The Klein Correspondence,
Penrose Space-Time, and a Finite Model.
For my own remarks on the date of the above arXiv paper
by Saniga et. al., click on the image below —
A field in China —
The following link was suggested by today’s previous post
and by the ABC TV series “Resurrection“ scheduled to start
at 9 PM ET Sunday, March 9, 2014 —
The Dream of the Expanded Field continues…
From Klein’s 1893 Lectures on Mathematics —
“The varieties introduced by Wirtinger may be called Kummer varieties….”
— E. Spanier, 1956
From this journal on March 10, 2013 —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —
Two such considerations —
Update of 10 PM ET March 7, 2014 —
The following slides by one of the “Kummer Threefold” authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7, W (E7):
The Cayley reference is to “Algorithm for the characteristics of the
triple ϑ-functions,” Journal für die Reine und Angewandte
Mathematik 87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley’s other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .
See “But is it Art?,” linked to here and here —
See posts tagged The Well.
Related material: Artist Joseph Kosuth, who pictured
the dictionary definition of ”nothing” shown in the index of
today’s LA Times obituaries, and a Chinese film director,
one of those portrayed in that index.
Also mentioned on the obituaries index page —
See as well The Church of the Holy Hubcap.
* Film title, translation of Chinese: 老井; pinyin: lǎo jǐng.
See also Rosenhain and Göpel in the Wikipedia
article Kummer surface, and in this journal.
Related material: user @hyperelliptic on Twitter.
In Beijing, it is now 3 AM on March 6,
the dies natalis of St. Pearl Buck.
(Click to enlarge.)
Review of Joseph Campbell's The Inner Reaches of Outer Space—
The reviewer compares Campbell to "one of those guys who
builds his own church out of hub caps."
A simple hub cap — see ninefold in this journal.
"Alles wird viel einfacher, wenn man zuerst von der
Unendlichkeit der Theilbarkeit abstrahirt und bloss
Discrete Grössen betrachtet."
— Carl Friedrich Gauss in 1825
(Quoted here in the July 16, 2013, post Child Buyers.)
Marissa Mayer Says Cheese.
Related material: The previous post and Cheesy Consolation Prize .
"… in Speedtalk it was… difficult not to be logical."
— Robert A. Heinlein in Gulf
Related material: ABC TV at 9 PM ET
on Sunday, March 9, 2014… 3/09.
See also page 309 in the previous post, Outside the Box.
Shades of Plan 9.
Blackboard Jungle , 1955
"We are going to keep doing this
until we get it right." — June 15, 2007
"Her wallet's filled with pictures,
she gets 'em one by one" — Chuck Berry
See too a more advanced geometry lesson
that also uses the diagram pictured above.
From The Telegraph today—
And no fact of Alain Resnais’s life seemed to strike a stranger note than his assertion that the films which first inspired his ambition to become a film director were those in which Fred Astaire and Ginger Rogers danced. Or was it Dick Powell and Ruby Keeler? He could never be sure. “I wondered if I could find the equivalent of that exhilaration,” he recalled.
If he never did it was perhaps because of his highly cultivated attitude to serious cinema. His character and temperament were more attuned to the theory of film and a kind of intellectual square dance* which was far harder to bring to the screen with “exhilaration” than the art of Astaire and Rogers.
*See today's 11 AM ET Sermon.
"Heaven, I'm in Heaven!"
… With a trip to yesteryear suggested by
the Feb. 28 New York Times article
"Casting Shadows on a Fanciful World"
("Wes Anderson Evokes Nostalgia in
'The Grand Budapest Hotel' ").
Raiders of the Lost (Continued)
"Socrates: They say that the soul of man is immortal…."
From August 16, 2012—
In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not , at this point
in the dialogue, the diamond in Jowett's picture.
An 1892 figure by Jowett illustrating Plato's Meno—
A more correct version, from hermes-press.com —
Socrates: He only guesses that because the square is double, the line is double.
Socrates: Observe him while he recalls the steps in regular order. (To the Boy.) Tell me, boy, do you assert that a double space comes from a double line? Remember that I am not speaking of an oblong, but of a figure equal every way, and twice the size of this-that is to say of eight feet; and I want to know whether you still say that a double square comes from double line?
Socrates: But does not this line (AB) become doubled if we add another such line here (BJ is added)?
Socrates: And four such lines [AJ, JK, KL, LA] will make a space containing eight feet?
Socrates: Let us draw such a figure: (adding DL, LK, and JK). Would you not say that this is the figure of eight feet?
Socrates: And are there not these four squares in the figure, each of which is equal to the figure of four feet? (Socrates draws in CM and CN)
Socrates: And is not that four times four?
Socrates: And four times is not double?
[Boy] No, indeed.
Socrates: But how much?
[Boy] Four times as much.
Socrates: Therefore the double line, boy, has given a space, not twice, but four times as much.
Socrates: Four times four are sixteen— are they not?
As noted in the 2012 post, the diagram of greater interest is
Jowett's incorrect version rather than the more correct version
shown above. This is because the 1892 version inadvertently
illustrates a tesseract:
A 4×4 square version, by Coxeter in 1950, of a tesseract—
This square version we may call the Galois tesseract.
From New World Encyclopedia —
See also Tetragrammaton in this journal.
For further context, see Solomon's Cube and Oct. 16, 2013.
"Once a verbal structure is read, and reread
often enough to be possessed, it 'freezes.'
It turns into a unity in which all parts exist at
once, without regard to the specific movement
of the narrative. We may compare it to the study
of a music score, where we can turn to any
part without regard to sequential performance."
— Northrop Frye in The Great Code
Gardner reportedly died at 65 on February 19.
A post linked to here on that date suggests some
From Northrop Frye's The Great Code: The Bible and Literature , Ch. 3: Metaphor I —
"In the preceding chapter we considered words in sequence, where they form narratives and provide the basis for a literary theory of myth. Reading words in sequence, however, is the first of two critical operations. Once a verbal structure is read, and reread often enough to be possessed, it 'freezes.' It turns into a unity in which all parts exist at once, without regard to the specific movement of the narrative. We may compare it to the study of a music score, where we can turn to any part without regard to sequential performance. The term 'structure,' which we have used so often, is a metaphor from architecture, and may be misleading when we are speaking of narrative, which is not a simultaneous structure but a movement in time. The term 'structure' comes into its proper context in the second stage, which is where all discussion of 'spatial form' and kindred critical topics take their origin."
"The Great Code does not end with a triumphant conclusion or the apocalypse that readers may feel is owed them or even with a clear summary of Frye’s position, but instead trails off with a series of verbal winks and nudges. This is not so great a fault as it would be in another book, because long before this it has been obvious that the forward motion of Frye’s exposition was illusory, and that in fact the book was devoted to a constant re-examination of the same basic data from various closely related perspectives: in short, the method of the kaleidoscope. Each shake of the machine produces a new symmetry, each symmetry as beautiful as the last, and none of them in any sense exclusive of the others. And there is always room for one more shake."
— Charles Wheeler, "Professor Frye and the Bible," South Atlantic Quarterly 82 (Spring 1983), pp. 154-164, reprinted in a collection of reviews of the book.
For code in a different sense, but related to the first passage above,
see Diamond Theory Roullete, a webpage by Radamés Ajna.
For "the method of the kaleidoscope" mentioned in the second
passage above, see both the Ajna page and a webpage of my own,
"There is such a thing as a tesseract."
— Saying from Crosswicks
See also March 5, 2011.
Adapted from the above passage —
"So did L'Engle understand four-dimensional geometry?"
No and Yes.
This morning's previous post, on sacred space,
linked to "Positively White Cube Revisited,"
an article by one Simon Sheikh.
Sheikh writes well, but he seems to be a disciple
of the damned Marxist lunatic Louis Althusser.
As Pynchon put it in Gravity's Rainbow ,
"For every kind of vampire, there is a kind of cross."
In this case, a video starring Sheikh on the exhibition "All That Fits"
suggests, by its filming date (May 27, 2011), a Maltese cross.
"The stuff that dreams are made of." — Bogart
(See also Oct. 25, 2012.)
"An image comes to mind of a white, ideal space
that, more than any single picture, may be the
archetypal image of 20th-century art."
— Brian O'Doherty, "Inside the White Cube"
Cube spaces exist also in mathematics.
in Stevens' "The Man with the Blue Guitar"
Studia Anglica Posnaniensia:
An International Review of English Studies
Jan 1, 2004
See also Blue Guitar
and Cubist Language Game
as well as Dali Cube.
The online New York TImes this morning —
Paco de Lucia, Renowned Flamenco Guitarist, Dies at 66
By REUTERS FEB. 26, 2014, 8:30 A.M. E.S.T.
MADRID — Paco de Lucia, the influential Spanish guitarist who vastly expanded the international audience for flamenco and merged it with other musical styles, died suddenly on Wednesday** of a heart attack in Mexico.
The 66-year-old virtuoso, as happy playing seemingly impossible syncopated flamenco rhythms as he was improvising jazz or classical guitar, helped to legitimize flamenco in Spain itself at a time when it was shunned by the mainstream.
Related material linked to here at midnight Monday-Tuesday —
Unrelated material, suggested merely by the upload dates of
two guitar videos* — See Oct. 25, 2008, and Oct. 26, 2011.
* El Toro – Malagueña (guitarist: Canabarro) and Light and Shade
(guitarist: de Lucia).
** Update of 12:26 PM ET — Other reports now say de Lucia died
not today, Wednesday, Feb. 26, but rather on Tuesday, Feb. 25.
The title of a review of Charles Taylor's book A Secular Age
was quoted here at noon last Saturday —
"The Place of the Sacred
in the Absence of God."
My comment from last Saturday —
"The place of the sacred is not, perhaps,
Davos, but a more abstract location."
A sequel —
"Religious Experience and the Modern Self,"
by Ross Douthat in The New York Times
today at 4:25 PM ET —
"The argument comes from the Canadian
philosopher Charles Taylor and his
doorstop-thick magnum opus A Secular Age …."
Helprin Doors and Doorstop Thick.
From Log24 on Jan. 13, 2014 —
"We have a clip." — Kalle (Kristen Wiig on SNL)
I do not follow the Public Library of Science (PLOS), although,
as shown above, I do follow some of the followers.
This post was suggested by Amy Hubbard's recent reference
to a PLOS article on beliefs in Hell.
Pop culture seems more informative. Readers of the PLOS article
should also know about the Dakota, John Lennon, and Rosemary's
Baby, as well as Woody Allen, The Ninth Gate , and Plan 9 from
"What a lovely singing voice you must have."
— Bill Murray in Ghostbusters (1984)
Contestant One: Ruth Margraff
Contestant Two: Sandra Sangiao
See Stadium Devildare, Church Notes, and Ruth Margraff*.
Ruthless : A Brief Drama —
"There is no ____ , there is only Zuul."
— Adapted from Ghostbusters
* In a webpage dated July 25, 2007.
See also this journal on that date.
Katy Perry in her new "Dark Horse" video—
"So you wanna play with magic.
Boy you should know what you're fallin' for.
Baby do you dare to do this?"
Bill Murray in Ghostbusters —
"Is this a trick question?"
Or: Plan 9, Continued
For the late Harold Ramis .
See also 2/02 and 2/13.
Serge Lang, Collected Papers, Vol. 4 , p. 179—
"I find it appropriate to quote here a historical
comment made by Halberstam…."
This is Heini Halberstam, who reportedly died
on January 25, 2014.
I find it appropriate to quote here an unhistorical
comment made by a fictional character —
“The test?” I faltered, staring at the thing.
“Yes, to determine whether you can live
in the fourth dimension or only die in it.”
— From Fritz Leiber's classic story
The Leiber quote was suggested by the posts
in this journal on the day of Halberstam's death.
Some narrative notes in memory of a
Bowling Green State University math professor
who reportedly died at 72 on Feb. 13—
That date in this journal and Green Fields.
See also Nine is a Vine.
Those who prefer mathematics to narrative may
also prefer to read, instead of the notes above,
some material on the dead professor's specialty,
Diophantine equations. Recommended:
Mordell on Lang and Lang on Mordell as well as
Lang's article titled
"Mordell's Review, Siegel's Letter to Mordell,
Diophantine Geometry, and 20th Century Mathematics."
Some background —
The title was suggested by a 1921 article
by Hermann Weyl and by a review* of
a more recent publication —
The above Harvard Gazette piece on Davos is
from St. Ursula's Day, 2010. See also this journal
on that date.
See as well a Log24 search for Davos.
A more interesting piece by Peter E. Gordon
(author of the above Davos book) is his review
of Charles Taylor's A Secular Age .
The review is titled
"The Place of the Sacred
in the Absence of God."
(The place of the sacred is not, perhaps, Davos,
but a more abstract location.)
* Grundlagenkrise was a tag for a Jan. 13, 2011,
review in The New Republic of Gordon's
book on Cassirer and Heidegger at Davos.
Despite the blocking of Doodles on my Google Search
screen, some messages get through.
Today, for instance —
"Your idea just might change the world.
Enter Google Science Fair 2014"
Clicking the link yields a page with the following image—
Clearly there is a problem here analogous to
the square-triangle coordinatization problem,
but with the 4×6 rectangle of the R. T. Curtis
Miracle Octad Generator playing the role of
I once studied this 24-triangle-hexagon
coordinatization problem, but was unable to
obtain any results of interest. Perhaps
someone else will have better luck.
* For a rather different use of this word,
see Hermann Weyl in the Stanford
Encyclopedia of Philosophy.
One way of interpreting the symbol
at the end of yesterday's post is via
the phrase "necessary possibility."
See that phrase in (for instance) a post
of July 24, 2013, The Broken Tablet .
The Tablet post may be viewed in light
of a Tom Wolfe passage quoted here on
the preceding day, July 23, 2013—
On that day (July 23) another weblog had
a post titled
Wallace Stevens: Night's Hymn of the Rock.
Some related narrative —
I prefer the following narrative —
Part I: Stevens's verse from "The Rock" (1954) —
"That in which space itself is contained"
Part II: Mystery Box III: Inside, Outside (2014)
A review of this date in 2005 —
"We symbolize logical necessity
with the box ()
and logical possibility
with the diamond ()."
— Keith Allen Korcz
And what do we
symbolize by ?
See also a related brick wall.
Some context: Small World (July 12, 2004)
and Moss on the Wall (Sept. 10, 2013).
The New York Times online this evening
has two passages of interest.
From an obituary by Helen T. Verongos of
fiction writer Mavis Gallant—
"Ms. Gallant also endowed children with
special powers that vanish as they grow up.
In 'The Doctor,' she wrote: 'Unconsciously,
everyone under the age of 10 knows
everything. Under-ten can come into a room
and sense at once everything felt, kept
silent, held back in the way of love, hate and
desire, though he may not have the right
words for such sentiments. It is part of the
clairvoyant immunity to hypocrisy we are born
with and that vanishes just before puberty.' "
From a review by William Grimes of a memoir
by non-fiction writer Joachim Fest—
"Not I shrinks the Wagnerian scale of
German history in the 1930s and 1940s to
chamber music dimensions. It is intensely
personal, cleareyed and absolutely riveting,
partly because the author, thrust into an
outsider’s position, developed a keen
appreciation of Germany’s contradictions
Related material in this journal—
Octobers for Fest (Sept. 13, 2006).
For Oslo artist Josefine Lyche, who sometimes
seems to think my work resembles that of the
deranged Anthony Hopkins in the film of David
Auburn's play "Proof."
See another artist's images of Hopkins-like work
I just discovered online —
"The Proof," by David Colosi.
Edward Frenkel on Eichler's reciprocity law
(Love and Math , Kindle edition of 2013-10-01,
page 88, location 1812)—
"It seems nearly unbelievable that there
would be a rule generating these numbers.
And yet, German mathematician Martin
Eichler discovered one in 1954.11 "
"11. I follow the presentation of this result
given in Richard Taylor, Modular arithmetic:
driven by inherent beauty and human
curiosity , The Letter of the Institute for
Advanced Study [IAS], Summer 2012,
pp. 6– 8. I thank Ken Ribet for useful
comments. According to André Weil’s book
Dirichlet Series and Automorphic Forms ,
Springer-Verlag, 1971 [pp. 143-144], the
cubic equation we are discussing in this
chapter was introduced by John Tate,
following Robert Fricke."
Update of Feb. 19:
Actually, the cubic equation discussed
by Frenkel and by Taylor (see below) is
Y 2 + Y = X 3 – X 2
whereas the equation given by Weil,
quoting Tate, is
Y 2 – Y = X 3 – X 2 .
Whether this is a misprint in Weil's book,
I do not know.
At any rate, the cubic equation discussed by
Frenkel and earlier by Taylor is the same as
the cubic equation discussed in greater detail
by Henri Darmon in "A Proof of the Full
Shimura-Taniyama-Weil Conjecture Is
Announced," AMS Notices , Dec.1999.
For further background, see (for instance)
John T. Tate, "The Arithmetic of Elliptic
Curves," in Inventiones Mathematicae
Volume 23 (1974), pp. 179 – 206, esp. pp.
Richard Taylor, op. cit. —
One could ask for a similar method that given any number of polynomials in any number of variables helps one to determine the number of solutions to those equations in arithmetic modulo a variable prime number p . Such results are referred to as “reciprocity laws.” In the 1920s, Emil Artin gave what was then thought to be the most general reciprocity law possible—his abelian reciprocity law. However, Artin’s reciprocity still only applied to very special equations—equations with only one variable that have “abelian Galois group.”
Stunningly, in 1954, Martin Eichler (former IAS Member) found a totally new reciprocity law, not included in Artin’s theorem. (Such reciprocity laws are often referred to as non-abelian.) More specifically, he found a reciprocality [sic ] law for the two variable equation
Y 2 + Y = X 3 – X 2.
He showed that the number of solutions to this equation in arithmetic modulo a prime number p differs from p [in the negative direction] by the coefficient of q p in the formal (infinite) product
q (1 – q 2 )(1 – q 11) 2 (1 – q 2)2
(1 – q 22 )2 (1 – q 3)2 (1 – q 33)2
(1 – q 4)2 … =
q – 2q2 – q3 + 2q4 + q5 + 2q6
– 2q7 – 2q9 – 2q10 + q11 – 2q12 + . . .
For example, you see that the coefficient of q5 is 1, so Eichler’s theorem tells us that
Y 2 + Y = X 3 − X 2
should have 5 − 1 = 4 solutions in arithmetic modulo 5. You can check this by checking the twenty-five possibilities for (X,Y) modulo 5, and indeed you will find exactly four solutions:
(X,Y) ≡ (0,0), (0,4), (1,0), (1,4) mod 5.
Within less than three years, Yutaka Taniyama and Goro Shimura (former IAS Member) proposed a daring generalization of Eichler’s reciprocity law to all cubic equations in two variables. A decade later, André Weil (former IAS Professor) added precision to this conjecture, and found strong heuristic evidence supporting the Shimura-Taniyama reciprocity law. This conjecture completely changed the development of number theory.
With this account and its context, Taylor has
perhaps atoned for his ridiculous remarks
quoted at Log24 in The Proof and the Lie.
Thanks to SackLunch.net for the above brief catechism:
Q — What must I do to be saved?
A — Search.
Music Box … Continues.
Today's print New York Times has articles on experimental and
New Age music —
In the Church of Difficult Music and
For New Age, the Next Generation.
I prefer Old Age music… for instance, that of Tony Rice —
also the subject of an article in today's print Times .
The Times image at right above is of Croagh Patrick.
… Do you ever think of yourself as
actually dead, lying in a box with a lid on it?
Nor do I, really… It's silly to be depressed by it.
I mean one thinks of it like being alive in a box,
one keeps forgetting to take into account the fact
that one is dead… which should make all the
difference… shouldn't it? I mean, you'd never know
you were in a box, would you? It would be just like
being asleep in a box.
— Tom Stoppard
See also last Sunday's sermon (Feb. 9) and
Mystery Box III: Inside, Outside (Feb. 10).
See The Oslo Version in this journal and the New Year's Day 2014 post.
The pictures of the 56 spreads in that post (shown below) are based on
the 20 Rosenhain and 15 Göpel tetrads that make up the 35 lines of
PG(3,2), the finite projective 3-space over the 2-element Galois field.
Click for a larger image.