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Ratan Naval Tata was born on Dec. 28, 1937, in Bombay, now Mumbai, during the British Raj. His family belonged to the Parsi religion, a small Zoroastrian community that originated in Persia, fled persecution by the Muslim majority there centuries ago and found refuge in India. Mr. Tata became a leader of that community. — New York Times obituary on 9 October 2024 |
See also theta functions in this journal.
For those who prefer narratives to mathematics . . .
Tiger at the Fire Temple
"This volume is the first of three in a series surveying
the theory of theta functions. Based on lectures given by
the author at the Tata Institute of Fundamental Research
in Bombay, these volumes constitute a systematic exposition
of theta functions, beginning with their historical roots as
analytic functions in one variable (Volume I), touching on
some of the beautiful ways they can be used to describe
moduli spaces (Volume II), and culminating in a methodical
comparison of theta functions in analysis, algebraic geometry,
and representation theory (Volume III)."
Α, ϴ, Ω
Related line:
Also from a Culture Desk of sorts:
Related art — Background colors for the letters in the NPR logo —
Anthony Oettinger, who was quoted in the previous post,
once warned me to beware of those promoting "creativity."
Related material:
Another outstanding mentor, Randy R. Ross, taught me physics
at Jamestown (NY) Community College.
At the blackboard, after adding a pair of fangs to the crossbar
in a capital Theta , Ross once quipped: "Beware of big Theta!."
Related material: Theta functions and finite geometry.
See as well . . .
Saturday evening's post Diamond Globe suggests a review of …
Iain Aitchison on symmetric generation of M24 —
* A Greek version for the late John SImon:
«Ἀνερρίφθω κύβος».
The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter
revived "Beautiful Mathematics" as a title:
This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below.
In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —
". . . a special case of a much deeper connection that Ian Macdonald
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)
The adjective "modular" might aptly be applied to . . .
The adjective "affine" might aptly be applied to . . .
The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.
Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but
did not discuss the 4×4 square as an affine space.
For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —
— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —
For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."
For Macdonald's own use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms,"
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.
This post was suggested by the names* (if not the very abstruse
concepts ) in the Aug. 20, 2013, preprint "A Panoramic Overview
of Inter-universal Teichmuller Theory," by S. Mochizuki.
* Specifically, Jacobi and Kummer (along with theta functions).
I do not know of any direct connection between these names'
relevance to the writings of Mochizuki and their relevance
(via Hudson, 1905) to my own much more elementary studies of
the geometry of the 4×4 square.
A post suggested by remarks of J. D. Salinger in
The New Yorker of November 19, 1955 —
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Wikipedia: Taiji (philosophy) Etymology The word 太極 comes from I Ching : "易有太極，是生兩儀，兩儀生四象，四象生八卦，八卦定吉凶，吉凶生大業。" Taiji (太極) is a compound of tai 太 "great; grand; supreme; extreme; very; too" (a superlative variant of da 大 "big; large; great; very") and ji 極 "pole; roof ridge; highest/utmost point; extreme; earth's pole; reach the end; attain; exhaust". In analogy with the figurative meanings of English pole, Chinese ji 極 "ridgepole" can mean "geographical pole; direction" (e.g., siji 四極 "four corners of the earth; world's end"), "magnetic pole" (Beiji 北極 "North Pole" or yinji 陰極 "negative pole; cathode"), or "celestial pole" (baji 八極 "farthest points of the universe; remotest place"). Combining the two words, 太極 means "the source, the beginning of the world". Common English translations of the cosmological Taiji are the "Supreme Ultimate" (Le Blanc 1985, Zhang and Ryden 2002) or "Great Ultimate" (Chen 1989, Robinet 2008); but other versions are the "Supreme Pole" (Needham and Ronan 1978), "Great Absolute", or "Supreme Polarity" (Adler 1999). |
See also Polarity in this journal.
* A phrase adapted, via Salinger,
from a poem by Sappho—
Ἴψοι δὴ τὸ μέλαθρον,
Υ᾽μήναον
ἀέρρετε τέκτονεσ ἄνδρεσ,
Υ᾽μήναον
γάμβροσ ἔρχεται ἶσοσ Ά᾽ρευϊ,
[Υ᾽μήναον]
ανδροσ μεγάλο πόλυ μείζων
[Υ᾽μήναον]
Πολυμερῶς καὶ πολυτρόπως πάλαι ὁ Θεὸς λαλήσας . . . .
( Long Day's Journey into Nighttown continues. )
A search in this journal for material related to the previous post
on theta characteristics yields…
"The Solomon Key is the working title of an unreleased
novel in progress by American author Dan Brown.
The Solomon Key will be the third book involving the
character of the Harvard professor Robert Langdon,
of which the first two were Angels & Demons (2000) and
The Da Vinci Code (2003)." — Wikipedia
"One has O+(6) ≅ S8, the symmetric group of order 8! …."
— "Siegel Modular Forms and Finite Symplectic Groups,"
by Francesco Dalla Piazza and Bert van Geemen,
May 5, 2008, preprint.
"It was only in retrospect
that the silliness
became profound."
— Review of
Faust in Copenhagen
"The page numbers
are generally reliable."
For further backstory, click the above link "May 5, 2008,"
which now leads to all posts tagged on080505.
See searches for "theta characteristics" in Google and in this journal.
A definition of particular interest for finite geometry —
The Grushevsky-Manni paper above was submitted to the arXiv
on 9 Dec. 2012. For some synchronistically related remarks
suitable for Michaelmas, see this journal on that date.
An article in the new April issue of Notices of the American
Mathmatical Society suggests a search for connections
between the Calkin-Wilf tree and the modular group.
The search yields, for instance (in chronological order) …
"Cutting sequences for geodesic flow on the modular surface
and continued fractions," David J. Grahinet, Jeffrey C. Lagaria,
arXiv, 2 April 2001
"Orderings of the rationals and dynamical systems,"
Claudio Bonanno, Stefano Isola, arXiv, 14 May 2008.
"Periods of negative-regular continued fractions. Rational numbers."
Sergey Khrushchev and Michael Tyaglov, slides PDF, 11 Sept. 2012
"The Minkowski ?(x) function, a class of singular measures,
theta-constants, and mean-modular forms," Giedrius Alkauskas,
arXiv, 20 Sept. 2012
"Forests of complex numbers,"
Melvyn B. Nathanson, arXiv, 1 Dec. 2014
Update of March 21, 2015:
For many more related papers, search by combining the
phrase "modular group" with phrases denoting forking structures
other than Calkin-Wilf, such as "cubic tree," "Stern-Brocot tree,"
and "Farey tree" (or "Farey sequence" or "Farey series" or
"Farey graph" ).
A paper from 1976 on symplectic torsors and finite geometry:
FINITE GEOMETRIES IN THE THEORY OF THETA CHARACTERISTICS
Autor(en): Rivano, Neantro Saavedra
Objekttyp: Article
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.
Part I:
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.)
An excerpt:
“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 .
Part II:
“Heavensbee!” in the above video, as well as Cartier’s Groundhog Day
and Say It With Flowers.
Part III:
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.
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.
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 .
This post was suggested by the recent Log24
posts Film Politics, The Wind Rises, and
Figure and Ground, as well as the related
Wikipedia article The Wind Has Risen.
Cover design by Helen Yentus.
Μή, φίλα ψυχά, βίον ἀθάνατον σπεῦδε,
τὰν δ' ἔμπρακτον ἄντλει μαχανάν.
— Pindar, Pythian III , epigraph to
Le Cimetière Marin by Paul Valéry (1920)
O mon âme, n’aspire à la vie immortelle,
mais épuise le champ du possible.
— Pindar, 3e Pythique , epigraph to
The Myth of Sisyphus by Albert Camus (1942)
O my soul, do not aspire to immortal life,
but exhaust the limits of the possible.
— Pindar, Pythian iii , as translated
from the French (or Greek) by Justin O'Brien
in the Knopf Myth of Sisyphus , 1955
A sequel to the quotation here March 8 (Pinter Play)
of Joan Aiken's novel The Shadow Guests—
Supposing that one's shadow guests are
Rosenhain and Göpel (see March 18)…
Hans Freudenthal at Encyclopedia.com on Charles Hermite:
"In 1855 Hermite took advantage of Göpel’s and Rosenhain’s work
when he created his transformation theory (see below)."
"One of his invariant theory subjects was the fifth-degree equation,
to which he later applied elliptic functions.
Armed with the theory of invariants, Hermite returned to
Abelian functions. Meanwhile, the badly needed theta functions
of two arguments had been found, and Hermite could apply what
he had learned about quadratic forms to understanding the
transformation of the system of the four periods. Later, Hermite’s
1855 results became basic for the transformation theory of Abelian
functions as well as for Camille Jordan’s theory of 'Abelian' groups.
They also led to Herrnite’s own theory of the fifth-degree equation
and of the modular equations of elliptic functions. It was Hermite’s
merit to use ω rather than Jacobi’s q = eπi ω as an argument and to
prepare the present form of the theory of modular functions.
He again dealt with the number theory applications of his theory,
particularly with class number relations or quadratic forms.
His solution of the fifth-degree equation by elliptic functions
(analogous to that of third-degree equations by trigonometric functions)
was the basic problem of this period."
See also Hermite in The Catholic Encyclopedia.
we are just like a couple of tots…
— Sinatra
Born 1973 in Bergen. Lives and works in Oslo.
Education
2000 – 2004 National Academy of Fine Arts, Oslo
1998 – 2000 Strykejernet Art School, Oslo, NO
1995 – 1998 Philosophy, University of Bergen
University of Bergen—
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It might therefore seem that the idea of digital and analogical systems as rival fundaments to human experience is a new suggestion and, like digital technology, very modern. In fact, however, the idea is as old as philosophy itself (and may be much older). In his Sophist, Plato sets out the following ‘battle’ over the question of ‘true reality’: What we shall see is something like a battle of gods and giants going on between them over their quarrel about reality [γιγαντομαχία περì της ουσίας] ….One party is trying to drag everything down to earth out of heaven and the unseen, literally grasping rocks and trees in their hands, for they lay hold upon every stock and stone and strenuously affirm that real existence belongs only to that which can be handled and offers resistance to the touch. They define reality as the same thing as body, and as soon as one of the opposite party asserts that anything without a body is real, they are utterly contemptuous and will not listen to another word. (…) Their adversaries are very wary in defending their position somewhere in the heights of the unseen, maintaining with all their force that true reality [την αληθινήν ουσίαν] consists in certain intelligible and bodiless forms. In the clash of argument they shatter and pulverize those bodies which their opponents wield, and what those others allege to be true reality they call, not real being, but a sort of moving process of becoming. On this issue an interminable battle is always going on between the two camps [εν μέσω δε περι ταυτα απλετος αμφοτέρων μάχη τις (…) αει συνέστηκεν]. (…) It seems that only one course is open to the philosopher who values knowledge and truth above all else. He must refuse to accept from the champions of the forms the doctrine that all reality is changeless [and exclusively immaterial], and he must turn a deaf ear to the other party who represent reality as everywhere changing [and as only material]. Like a child begging for 'both', he must declare that reality or the sum of things is both at once [το όν τε και το παν συναμφότερα] (Sophist 246a-249d). The gods and the giants in Plato’s battle present two varieties of the analog position. Each believes that ‘true reality’ is singular, that "real existence belongs only to" one side or other of competing possibilities. For them, difference and complexity are secondary and, as secondary, deficient in respect to truth, reality and being (την αληθινήν ουσίαν, το όν τε και το παν). Difference and complexity are therefore matters of "interminable battle" whose intended end for each is, and must be (given their shared analogical logic), only to eradicate the other. The philosophical child, by contrast, holds to ‘both’ and therefore represents the digital position where the differentiated two yet belong originally together. Here difference, complexity and systematicity are primary and exemplary. It is an unfailing mark of the greatest thinkers of the tradition, like Plato, that they recognize the digital possibility and therefore recognize the principal difference of it from analog possibilities.
— Cameron McEwen, "The Digital Wittgenstein," |
* See that phrase in this journal.
In memory of director Ulu Grosbard (continued from yesterday)
From http://scripturetext.com/matthew/13-44.htm —
Again the kingdom of heaven is like unto treasure hid in a field
the which when a man hath found he hideth and for joy thereof
goeth and selleth all that he hath and buyeth that field
ΚΑΤΑ ΜΑΤΘΑΙΟΝ 13:44 Greek NT: Byzantine/Majority Text (2000)
παλιν ομοια εστιν η βασιλεια των ουρανων θησαυρω κεκρυμμενω εν τω αγρω
LEXICON —
παλιν adverb
palin pal'-in: (adverbially) anew, i.e. (of place) back, (of time) once more, or (conjunctionally) furthermore or on the other hand — again.
ομοια adjective – nominative singular feminine
homoios hom'-oy-os: similar (in appearance or character) — like, + manner.
εστιν verb – present indicative – third person singular
esti es-tee': he (she or it) is; also (with neuter plural) they are
η definite article – nominative singular feminine
ho ho: the definite article; the (sometimes to be supplied, at others omitted, in English idiom) — the, this, that, one, he, she, it, etc.
βασιλεια noun – nominative singular feminine
basileia bas-il-i'-ah: royalty, i.e. (abstractly) rule, or (concretely) a realm — kingdom, + reign.
των definite article – genitive plural masculine
ho ho: the definite article; the (sometimes to be supplied, at others omitted, in English idiom) — the, this, that, one, he, she, it, etc.
ουρανων noun – genitive plural masculine
ouranos oo-ran-os': the sky; by extension, heaven (as the abode of God); by implication, happiness, power, eternity; specially, the Gospel (Christianity) — air, heaven(-ly), sky.
θησαυρω noun – dative singular masculine
thesauros thay-sow-ros': a deposit, i.e. wealth — treasure.
κεκρυμμενω verb – perfect passive participle – dative singular masculine
krupto kroop'-to: to conceal (properly, by covering) — hide (self), keep secret, secret(-ly).
εν preposition
en en: in, at, (up-)on, by, etc.
τω definite article – dative singular masculine
ho ho: the definite article; the (sometimes to be supplied, at others omitted, in English idiom) — the, this, that, one, he, she, it, etc.
αγρω noun – dative singular masculine
agros ag-ros': a field (as a drive for cattle); genitive case, the country; specially, a farm, i.e. hamlet — country, farm, piece of ground, land.
For Kathrin Bringmann, who has been mentioned as a possible candidate for a Fields Medal.
The four Fields medal winners were announced today at the International Congress of Mathematicians in Hyderabad, India. Bringmann was not among them.
Bringmann was, however, the winner of the 2009 SASTRA Ramanujan Prize—
See The Hindu of September 30, 2009 and this journal on that date—
For more about Bringmann's work, see an article on what has been called Ramanujan's "final problem."
For another problem with a claim to this title, see "Mathematician Untangles Legendary Problem" and search in this journal for Dyson + crank.
First to Illuminate
“From the History of a Simple Group” (pdf), by Jeremy Gray:
“The American mathematician A. B. Coble [1908; 1913]* seems to have been the first to illuminate the 27 lines and 28 bitangents with the elementary theory of geometries over finite fields.
The combinatorial aspects of all this are pleasant, but the mathematics is certainly not easy.”
* [Coble 1908] A. Coble, “A configuration in finite geometry isomorphic with that of the 27 lines on a cubic surface,” Johns Hopkins University Circular 7:80-88 (1908), 736-744.
[Coble 1913] A. Coble, “An application of finite geometry to the characteristic theory of the odd and even theta functions,” Trans. Amer. Math. Soc. 14 (1913), 241-276.
720 in the Book
Searching for an epiphany on this January 6 (the Feast of the Epiphany), I started with Harvard Magazine, the current issue of January-February 2004.
An article titled On Mathematical Imagination concludes by looking forward to
“a New Instauration that will bring mathematics, at last, into its rightful place in our lives: a source of elation….”
Seeking the source of the phrase “new instauration,” I found it was due to Francis Bacon, who “conceived his New Instauration as the fulfilment of a Biblical prophecy and a rediscovery of ‘the seal of God on things,’ ” according to a web page by Nieves Mathews.
Hmm.
The Mathews essay leads to Peter Pesic, who, it turns out, has written a book that brings us back to the subject of mathematics:
Abel’s Proof: An Essay
on the Sources and Meaning
of Mathematical Unsolvability
by Peter Pesic,
MIT Press, 2003
From a review:
“… the book is about the idea that polynomial equations in general cannot be solved exactly in radicals….
Pesic concludes his account after Abel and Galois… and notes briefly (p. 146) that following Abel, Jacobi, Hermite, Kronecker, and Brioschi, in 1870 Jordan proved that elliptic modular functions suffice to solve all polynomial equations. The reader is left with little clarity on this sequel to the story….”
— Roger B. Eggleton, corrected version of a review in Gazette Aust. Math. Soc., Vol. 30, No. 4, pp. 242-244
Here, it seems, is my epiphany:
“Elliptic modular functions suffice to solve all polynomial equations.”
Incidental Remarks
on Synchronicity,
Part I
Those who seek a star
on this Feast of the Epiphany
may click here.
Most mathematicians are (or should be) familiar with the work of Abel and Galois on the insolvability by radicals of quintic and higher-degree equations.
Just how such equations can be solved is a less familiar story. I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.
The topic is of interest to me because, as my recent web page The Proof and the Lie indicates, I was deeply irritated by the way recent attempts to popularize mathematics have sown confusion about modular functions, and I therefore became interested in learning more about such functions. Modular functions are also distantly related, via the topic of “moonshine” and via the “Happy Family” of the Monster group and the Miracle Octad Generator of R. T. Curtis, to my own work on symmetries of 4×4 matrices.
Incidental Remarks
on Synchronicity,
Part II
There is no Log24 entry for
December 30, 2003,
the day John Gregory Dunne died,
but see this web page for that date.
Here is what I was able to find on the Web about Pesic’s claim:
From Wolfram Research:
From Solving the Quintic —
“Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein’s approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.”
From Siegel Theta Function —
“Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions. (Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkhäuser, 1984.)”
From Polynomial —
“… the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable. Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron. Klein’s method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or ‘Siegel functions’ must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be ‘natural’ generalizations of the elliptic functions.”
Belardinelli, G. “Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébrique générales.” Mémoral des Sci. Math. 145, 1960.
King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.
Chow, T. Y. “What is a Closed-Form Number.” Amer. Math. Monthly 106, 440-448, 1999.
From Angel Zhivkov,
Preprint series,
Institut für Mathematik,
Humboldt-Universität zu Berlin:
“… discoveries of Abel and Galois had been followed by the also remarkable theorems of Hermite and Kronecker: in 1858 they independently proved that we can solve the algebraic equations of degree five by using an elliptic modular function…. Kronecker thought that the resolution of the equation of degree five would be a special case of a more general theorem which might exist. This hypothesis was realized in [a] few cases by F. Klein… Jordan… showed that any algebraic equation is solvable by modular functions. In 1984 Umemura realized the Kronecker idea in his appendix to Mumford’s book… deducing from a formula of Thomae… a root of [an] arbitrary algebraic equation by Siegel modular forms.”
— “Resolution of Degree Less-than-or-equal-to Six Algebraic Equations by Genus Two Theta Constants“
Incidental Remarks
on Synchronicity,
Part III
From Music for Dunne’s Wake:
“Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged.”
— Carrie Fisher,
Postcards from the Edge
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|
“720 in |
“The group Sp4(F2) has order 720,”
as does S6. — Angel Zhivkov, op. cit.
Those seeking
“a rediscovery of
‘the seal of God on things,’ “
as quoted by Mathews above,
should see
The Unity of Mathematics
and the related note
Sacerdotal Jargon.
For more remarks on synchronicity
that may or may not be relevant
to Harvard Magazine and to
the annual Joint Mathematics Meetings
that start tomorrow in Phoenix, see
For the relevance of the time
of this entry, 10:10, see
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Related recreational reading:
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Labyrinth |
|
Monolith
“Music can name the unnameable
and communicate the unknowable.”
— Quotation attributed to Leonard Bernstein
“Finally we get to Kubrick’s ultimate trick…. His secret is in plain sight…. The film is the monolith. In a secret that seems to never have been seen by anyone: the monolith in the film has the same exact dimensions as the movie screen on which 2001 was projected.”
— Alchemical Kubrick 2001, by Jay Weidner
My entry of Saturday, August 30,
included the following illustration:
My entry of Monday, September 1,
concluded with the black monolith.
“There is little doubt that the black monolith
in 2001 is the Philosopher’s Stone.”
— Alchemical Kubrick 2001, by Jay Weidner
The philosopher Donald Davidson
died on Saturday, August 30.
The New York Times says that as an undergraduate, Davidson “persuaded Harvard to let him put on ‘The Birds’ by Aristophanes and played the lead, Peisthetairos, which meant memorizing 700 lines of Greek. His friend and classmate Leonard Bernstein, with whom he played four-handed piano, wrote an original score for the production.”
Perhaps they are still making music together.
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