Blame it on Toby
|
— Defense Secretary THE WEST WING WRITTEN BY Original Airdate: |
|
TOBY: Well… How about when we, instead of blowing Iraq back to the seventh century for harbouring terrorists and trying to develop nuclear weapons, we just imposed economic sanctions….
Misunderstanding
in the Theory of Design
"Whether or not we can follow the theorist in his demonstrations, there is one misunderstanding we must avoid at all cost. We must not confuse the analyses of geometrical symmetries with the mathematics of combination and permutation….
The earliest (and perhaps the rarest) treatise on the theory of design drives home this insight with marvellous precision."
— E. H. Gombrich, 1979, in
The Sense of Order
This is perhaps the stupidest remark I have ever read. The "treatise on the theory of design" that Gombrich refers to is
For some background, see
Truchet & Types:
Tiling Systems and Ornaments, and
Certain of the Truchet/Douat patterns have rather intriguing mathematical properties, sketched in my website Diamond Theory. These properties become clear if and only if we we do what Gombrich declares that we must not do: "confuse the analyses of geometrical symmetries with the mathematics of combination and permutation." (The verb "confuse" should, of course, be replaced by the verb "combine.")
Axis of Flim-Flam
Recommended reading from Axis of Logic —
On outgoing weapons-of-mass-destruction hunter David Kay:
“… instead of drawing the logical conclusion that he’s been duped and played for a fool, he chose instead to launch the latest salvo in the Bush administration’s undeclared war on the rank and file US intelligence community.”
The Subject Par Excellence
The previous entry connected the mad Marxist Althusser with Mount Sinai; this connection is not as whimsical as it may seem.
From Althusser's "Ideology and Ideological State Apparatuses" (La Pensée, 1970):
" 'And the Lord spoke to Moses and said to him, "I am that I am".'
God thus defines himself as the Subject par excellence, he who is through himself and for himself ('I am that I am'), and he who interpellates [Althusser’s jargon for “hails”] his subject, the individual subjected to him by his very interpellation, i.e. the individual named Moses."
This is from page 179 of Lenin and Philosophy and Other Essays.
The connection of the Althusser disciple Sprinker with the Trinity in Taking Lucifer Seriously is also not as whimsical as it may seem.
See Althusser's note (p. 180, op. cit.) stating that
"The dogma of the Trinity is precisely the theory of the duplication of the Subject (the Father) into a subject (the Son) and of their mirror-connexion (the Holy Spirit)."
Language Game
More on "selving," a word coined by the Jesuit poet Gerard Manley Hopkins. (See Saturday's Taking Lucifer Seriously.)
"… through the calibrated truths of temporal discipline such as timetabling, serialization, and the imposition of clock-time, the subject is accorded a moment to speak in."
Framing
Intelligibility, Identity, and Selfhood:
A Reconsideration of
Spatio-Temporal Models.
The "moment to speak in" of today's previous entry, 11:29 AM, is a reference to the date 11/29 of last year's entry
That entry contains, in turn, a reference to the journal Subaltern Studies. According to a review of Reading Subaltern Studies,
"… the Subaltern Studies collective drew upon the Althusser who questioned the primacy of the subject…."
Munt also has something to say on "the primacy of the subject" —
"Poststructuralism, following particularly Michel Foucault, Jacques Derrida and Jacques Lacan, has ensured that 'the subject' is a cardinal category of contemporary thought; in any number of disciplines, it is one of the first concepts we teach to our undergraduates. But are we best served by continuing to insist on the intellectual primacy of the 'subject,' formulated as it has been within the negative paradigm of subjectivity as subjection?"
How about objectivity as objection?
I, for one, object strongly to "the Althusser who questioned the primacy of the subject."
This Althusser, a French Marxist philosopher by whom the late Michael Sprinker (Taking Lucifer Seriously) was strongly influenced, murdered his wife in 1980 and died ten years later in a lunatic asylum.
For details, see
|
|
For details of Althusser's philosophy, see the oeuvre of Michael Sprinker. For another notable French tribute to Marxism, click on the picture at left. |
Taking Lucifer Seriously:
Michael Sprinker
versus
The Society of Jesus
As the previous entry indicates, I do not take Christian poetry too seriously. The Prince of Darkness is another matter. I encountered him this morning in a book on the Christian poet Hopkins by the late Michael Sprinker.
“You were never on the debating team when you were in high school, were you, ace? When you’re in a debate, you don’t try to convince the other side; they’re never going to agree with you. You try to convince the judges and the audience.”
— Michael Sprinker, quoted in The Minnesota Review, 2003
“For Hopkins, poetry was the act of producing the self, one version of that selving which he associated not only with Christ but with Lucifer.”
— Michael Sprinker, “A Counterpoint of Dissonance” — The Aesthetics and Poetry of Gerard Manley Hopkins, Johns Hopkins University Press, 1980, p. 95
A counterbalance to Sprinker on Hopkins and Lucifer is Hopkins, the Self, and God, by Walter J. Ong, S. J. (University of Toronto Press, 1986). From p. 119:
“The interior dynamism of the Three Persons in One God was not for Hopkins some sort of formula for theological juggling acts but was rather the centre of his personal devotional life and thus of his own ‘selving.’ …. He writes to Bridges 24 October 1883…
‘For if the Trinity… is to be explained by grammar and by tropes… where wd. be the mystery? the true mystery, the incomprehensible
one.’ “
For the dynamics of the Trinity, see the Jan. 22 entry, Perichoresis, or Coinherence. Another word for coinherence is “indwelling,” as expressed in what might be called the
Song of Lucifer:
Me into you,
You into me,
Me into you…
For a Christian version of this “indwelling,” see
Coinherence,
Interpenetration,
Mutual Indwelling
See also last year’s entries of 9/09.
Perichoresis, or Coinherence
Edward Gibbon, Decline and Fall of the Roman Empire, Chapter XXI —
Gibbon, discussing the theology of the Trinity, defines perichoresis as
“… the internal connection and spiritual penetration which indissolubly unites the divine persons59 ….
59 … The
or ‘circumincessio,’ is perhaps the deepest and darkest corner of the whole theological abyss.”
“Whoever fights monsters should see to it that in the process he does not become a monster. And when you look long into an abyss, the abyss also looks into you.”
Perichoresis does NOT mean “dancing around” ….
From a mailing list message:
If [a correspondent] will but open a lexicon, she will see that perichoresis (with a long o, omega) has nothing to do with “the Greek word for dance,” which is spelt with a short o (omicron). As a technical term in trinitarian theology, perichoresis means “interpenetration.”
Interpenetration in Arthur Machen
Interpenetration in T. S. Eliot:
“Between two worlds
become much like each other….”
On the Novels of Charles Williams
Coinherence in Charles Williams
The Per Speculum link is to a discussion of coinherence and the four last films of Kieslowski —
La Double Vie de Veronique (1991),
Trois Couleurs: Bleu (1993),
Trois Couleurs: Blanc (1993), and
Trois Couleurs: Rouge (1994).
See, too, previous log24 entries related to Kieslowski’s work and to coinherence:
Moulin Bleu (12/16/03),
Quarter to Three (12/20/03), and
White, Geometric, and Eternal (12/20/03).
Diamonds
This is the first anniversary of the death of Irene Diamond, patron of the arts, for whom the New York City Ballet’s Diamond Project is named. (See last year’s entries for January 20-23.)
Since tomorrow is the 100th anniversary of the birth of George Balanchine (according to the Gregorian, or “new style,” calendar), it seems appropriate to recall his ballet Diamonds, though it has no apparent connection with Irene.
Diamonds is the conclusion of a three-part work titled Jewels. (The first two parts are Emeralds, with music by Fauré, and Rubies, with music by Stravinsky.)
” ‘And then for the finale, Diamonds, I move to Tchaikovsky-always Tchaikovsky for dancing.’
Balanchine chose to use Tchaikovsky’s Symphony No. 3 in D major, gracefully cutting the first movement of the piece (by some accounts because it was too long, and by others because he felt it just wasn’t suitable for dancing).”
In other words, Balanchine “cut” Diamonds. For another use of this metaphor, see The Diamond Project. The following remark on the first movement seems appropriate on this, the anniversary of Irene Diamond’s death.
“The introduction to the first movement of the symphony is marked Moderato assai, Tempo di marcia funebre, the funeral march proceeding with increased pace….”
The following link to a part of Irene’s year-long funeral march seems appropriate:
Longtime Juilliard Benefactor Dies.
Whether her good deeds made her, like Christ and Gerard Manley Hopkins, an immortal Diamond, I do not know. Let us hope so.
Time and Chance,
Part II:
|
Click on |
“Gimme a T for Texas,
T for Tennessee.”
(See previous entry.)
The death of Max D. Barnes (previous entry) and the opening of the first Tennessee lottery suggested the following meditations.
Wikipedia on Jimmie Rodgers, known as the father of country music:
“Fundamentally, Rodgers was a white blues singer….”
A song by the father of country music:
T for Texas, T for Tennessee,
T for Texas, T for Tennessee,
T for Thelma, that gal
made a wreck out of me.
Gonna buy me a shotgun,
long as I am tall,
Buy me a shotgun,
long as I am tall,
Gonna shoot po’ Thelma,
just to see her jump and fall.
From Wikipedia:
“In modern Western popular music, call and response is most commonly found in the blues and in blues-derived music like jazz and rock’n’roll.”
If Rodgers’s song is the call, what, one wonders, would be the appropriate response?
In Memory of Max D. Barnes:
Time and Chance
Barnes, a songwriter,
died on 1/11/04.
Related material:
Fearful Meditation (8/1/03),
Time is a Weapon (9/26/03), and
In Summary (1/11/04).
Screenshot
A search on “vult decipi” at about
3:40 AM today yielded the following, from
http://www.sacklunch.net/Latin/P/
populusvultdecipidecipiatur.html
The ad for “Geometry of Latin Squares,”
my own. is in direct competition with
“Jesus Loves You.”
Good luck, Latin squares.
A Living Church
"Plato has told you a truth; but Plato is dead. Shakespeare has startled you with an image; but Shakespeare will not startle you with any more. But imagine what it would be to live with such men still living. To know that Plato might break out with an original lecture to-morrow, or that at any moment Shakespeare might shatter everything with a single song. The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast. He is always expecting to see some truth that he has never seen before."
— G. K. Chesterton, Orthodoxy
C. P. Snow on G. H. Hardy in the foreword to A Mathematician's Apology:
"… he had another favourite entertainment. 'Mark that man we met last night,' he said, and someone had to be marked out of 100 in each of the categories Hardy had long since invented and defined. STARK, BLEAK ('a stark man is not necessarily bleak: but all bleak men without exception want to be considered stark')…."
S. H. Cullinane on religion and Hollywood:
"If the incomparable Max Bialystock were to remake 'Up Close and Personal,' he might retitle it 'Distant and Impersonal.' A Google search on this phrase suggests
a plot outline for Mel Brooks & Co."
In memory of
producer Ray Stark,
an excerpt from that plot outline:
The Oxford University Press summary of
God:
Myths of the Male Divine,
by David Leeming and Jake Page
"They [Leeming and Page] describe the rise of a male sky God as 'the equal to, the true mate, of Goddess, who was still associated with Earth.' In the Iron Age, the sky God became more aggressive, separating from the Goddess and taking his place as the King God, as Zeus, Odin, and Horus. Ultimately he emerged as the creator, a more distant and impersonal force. Here Leeming and Page also illuminate an important trend–a sense that the divine is beyond gender, that it permeates all things (as seen in the Chinese Tao and En Sof of the Kabbalah). They see a movement in the biography of God toward a reunion with the Goddess."
As for the Goddess, see
(December 17, 2002).
Stark, a saint among Hollywood producers, died yesterday, January 17. If, as Chesterton might surmise, he then met Plato and Shakespeare in Heaven, the former might discuss with him the eternal Platonic form of the number 17, while the latter might offer the following links on Stark's new heavenly laptop:
This concludes the tribute to Stark. For a tribute to Bleak, click here.
Go Leonards
Yesterday's entry may be viewed as honoring Saint Leonard Eugene Dickson, who died on January 17, 1954. Dickson was the author of the three-volume classic
History of the Theory of Numbers.
Yesterday's entry was also prompted by a property of the number 17, and therefore may serve to illustrate a recurring theme… "The eternal in the temporal," an apt phrase uttered by Father Egan on page 373 of Robert Stone's religious classic,
Click on the above link for an appreciation of the Stone novel by Reynolds Price, one of the few Christians whose opinion I respect.
See also some remarks by Price from the feast day, Nov. 6, of the official Saint Leonard.
For a different Saint Leonard, see the entry of Oct. 14, 2003, which contains remarks by Leonard Bernstein on Mahler.
For a musical event that may be regarded as the fruition of Bernstein's remarks, see
Vatican invites rabbis, Muslim clerics
for concert featuring
Pittsburgh Symphony Orchestra
By Dennis B. Roddy,
Pittsburgh Post-Gazette,
Sunday, January 18, 2004
Forum
The annual World Social Forum started Jan. 16 in Bombay (“Mumbai”), India.
Background Essays:
From the right,
From the left,
Related Material:
From the right,
Marxist, Socialist, & Communist
Hate of America.
From the left,
Language Game
Ludwig Wittgenstein,
Philosophical Investigations:
373. Grammar tells what kind of object anything is. (Theology as grammar.)
Related material:
See this date last year, and
(May 2, 2003).
See also the phrase “May 2, 373.”
Games
On this date —
Alfred Tarski was born
in 1902 in Warsaw, and
Kurt Friedrich Gödel died
in 1978 in Princeton.
From last year’s entry on this date:
What is Truth?
“What is called ‘losing’ in chess
may constitute winning
in another game.”
— Ludwig Wittgenstein,
Remarks on the
Foundations of Mathematics
(revised edition, MIT Press, 1978)
At Last, Some Veritas
From the Harvard Crimson, 1/12/04:
College Faces Mental Health Crisis
“An overwhelming majority of Harvard undergraduates struggle with mental health problems, a recent Crimson poll found.”
Related material:
“The people who intermediate between lunatics and the world used to be called alienists; the go-betweens for mathematicians are called teachers. Many a student may rightly have wondered if the terms shouldn’t be reversed.”
— Book review in the current Harvard Magazine; among the authors reviewed is Harvard mathematician and administrator Benedict H. Gross.
“Dean of the College Benedict H. Gross ’71 has said improving mental health services is one of his top priorities in his first year on the job.”
— Harvard Crimson 1/12/04
“He takes us to the central activity of mathematics—which is imagining….”
— Harvard Magazine on Harvard mathematician and author Barry Mazur.
For related material on Mazur, see
“The teenagers aren’t all bad. I love ’em if nobody else does. There ain’t nothing wrong with young people. Jus’ quit lyin’ to ’em.”
Deeply Deep
“Remember your epiphanies on green oval leaves, deeply deep, copies to be sent if you died to all the great libraries of the world, including Alexandria?”
— James Joyce, Ulysses, “Proteus”
James Joyce may or may not have been a saint. Today is, accordingly, either his feast day or his secular day of remembrance.
With Joyce in mind, I surfed the Heckler & Coch weblog archives this afternoon and found a link to a page that credits
“Jørn Barger, an amateur
James Joyce scholar….”
with the first use of the term “weblog” in its current sense.
Seeking more on Barger and Joyce, I found that Barger has gone into seclusion and that his Joyce website is no longer online.
Google has a cache of his Joyce portal, however, and the portal and its sub-pages are also available at the Internet Archive Wayback Machine:
http://web.archive.org/web/2003*/
http://www.robotwisdom.com/jaj/*
These pages from Barger’s labor of love, though neither green nor oval, may serve as this year’s Joyce memorial.
In Summary
To sum up the last two entries:
“I returned and saw under the sun
that the race is not to the swift,
nor the battle to the strong,
nor bread to the wise,
nor riches to men of understanding,
nor favor to men of skill;
but time and chance
happeneth to them all”— Ecclesiastes 9:11.
Two-Dimensional Time
The following is from the Prime Quotes page at the website of Matthew R. Watkins…
“I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time, which like space may be in essence poly-dimensional and that this and other such sort of truths would become self-evident to a being whose mode of perception is according to superficially as opposed to our own limitation to linearly extended time.”
J.J. Sylvester, from “On certain inequalities relating to prime numbers”, Nature 38 (1888) 259-262, and reproduced in Collected Mathematical Papers, Volume 4, page 600 (Chelsea, New York, 1973)
Translated into contemporary English, Sylvester is saying more-or-less this:“I have sometimes thought that if we were able to perceive time in some multi-dimensional way, more like a surface than like a line, then perhaps the distribution of prime numbers would be entirely self-evident, and would not seem at all mysterious to us.”
Many thanks to Heckler & Coch (5/19/03) for pointing out the Sylvester quotation.
For related thoughts on this topic, see
The Lottery
|
New York Midday: 720 Evening: 510 |
Pennsylvania Midday: 616 Evening: 201 |
What these numbers mean to me:
720: See the recent entries
720 in the Book, and
Report to the Joint Mathematics Meetings.
616 and 201:
The dates, 6/16 and 2/01,
of Bloomsday and St. Bridget's Day.
510: A more difficult association…
Perhaps "Love at the Five and Dime"
(8/3/03 and 1/4/04).
Perhaps Fred Astaire's birthday, 5/10.
More interesting…
A search for relevant material in my own archives, using the phrase "may 10" cullinane journal, leads to the very interesting weblog Heckler & Coch, which contains the following brief entries (from May 19, 2003):
"May you live in interesting times
While widely reported as being an ancient Chinese curse, this phrase is likely to be of recent and western origin.Geometry of the I Ching
The Cullinane sequence of the 64 hexagrams"
"… there are many associations of ideas which do not correspond to any actual connection of cause and effect in the world of phenomena…."
— John Fiske, "The Primeval Ghost-World," quoted in the Heckler & Coch weblog
"The association is the idea"
— Ian Lee on the communion of saints and the association of ideas (in The Third Word War, 1978)
String Theory
Phil Sweetland of the New York Times on Gospel singer St. Jake Hess, who died on Sunday, January 4, 2004 — also the feast day of saints
“Mr. Hess was the string that tied together many of Christian music’s most famous quartets and ensembles, and he was an idol and later a colleague of [Elvis] Presley….
Mr. Hess sang at his funeral in 1977, as he had at the funeral of Hank Williams in 1953.”
“Go to other people’s funerals,
otherwise, they won’t go to yours.”
— Proverb attributed to St. Yogi Berra
“… to apprehend
The point of intersection of the timeless
With time, is an occupation for the saint…. “
— T. S. Eliot, Four Quartets
“You win again.”
— Keith Richards,
tribute to Hank Williams
Report to the
Joint Mathematics Meetings
“What was the lecture about,
Cosmo wanted to know.
‘It’s about solving equations
of the fifth degree,
which are supposed to be insoluble.'”
— Chapter 2 of
The Shadow Guests,
by Joan Aiken
For more material on insolubility
of fifth-degree equations
and on this winter’s
Joint Mathematics Meetings
(Phoenix, Jan. 7-10), see
the January 6 entry
720 in the Book.
For more material on Joan Aiken,
who died on January 4,
see the previous entry.
The number 720 is the order of
the symmetric group of degree 6.
For material related to
exceptional outer automorphisms
of this group and to
a song about Arizona, see
“Shinin’ like a diamond
she had tombstones
in her eyes.”
HURRY UP PLEASE
IT’S TIME
— T. S. Eliot,
The Waste Land, II
“A Game of Chess”
|
“Make the white Queen run so fast
|
Jan. 9 obituary of Brian Gibson —
“In 2002 he was executive producer of the film ‘Frida,’ about the artist Frida Kahlo….”
Captured for the Queen
Joan Aiken
Photo by Alex Gotfryd,
circa 1972
Jan. 9 obituary of Joan Aiken —
“Joan Aiken was born in Rye, England, a daughter of the American poet Conrad Aiken….”
“Malcolm Lowry’s Under the Volcano must be, for anyone who loves the English language, a sheer joy.”
“He was never inclined to small talk.”
— Jan. 9 obituary of Steven Edward Dorfman, writer of questions (i.e., answers) for the game show “Jeopardy!”
“What’s the Hellfire Club?”
— Joan Aiken, beginning of the final chapter of The Shadow Guests
Note that Dorfman, Gibson, and Aiken
all died on Sunday, Jan. 4, 2004.
For some related material, see
Natasha’s Dance
| “… at the still point, there the dance is….”
“… to apprehend — T. S. Eliot, Four Quartets |
It seems, according to Eliot’s criterion, that the late author John Gregory Dunne may be a saint.
Pursuing further information on the modular group, a topic on which I did a web page Dec. 30, 2003, the date of Dunne’s death, I came across a review of Apostol’s work on that subject (i.e., the modular group, not Dunne’s death, although there is a connection). The review:
“A clean, elegant,
absolutely lovely text…”
Searching further at Amazon for a newer edition of the Apostol text, I entered the search phrase “Apostol modular functions” and got a list that included the following as number four:
Natasha’s Dance:
A Cultural History of Russia,
which, by coincidence, includes all three words of the search.
For a connection — purely subjective and coincidental, of course — with Dunne’s death, see The Dark Lady (Jan. 1, 2004), which concerns another Natasha… the actress Natalie Wood, the subject of an essay (“Star!“) by Dunne in the current issue of the New York Review of Books.
The Review’s archives offer another essay, on science and religion, that includes the following relevant questions:
“Have the gates of death
been opened unto thee?
Or hast thou seen the doors
of the shadow of death?”
From my December 31 entry:
In memory of
John Gregory Dunne,
who died on
Dec. 30, 2003:
For further details, click
on the black monolith.
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
|
|
“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
|
Related recreational reading:
|
Labyrinth |
|
Whirligig
“Thus the whirligig of time
brings in his revenges.”
Twelfth Night. Act v. Sc. 1.
Twelfth night is the night of January 5-6.
Tonight is twelfth night in Australia;
12:25 AM Jan. 5 in New York City is
4:25 PM Jan. 5 in Melbourne.
Room 1010
Continuing the hotel theme of the previous entry….
John Gregory Dunne has a letter in the New York Review of Books of December 20 (St. Emil's Day in the previous entry), 1990. In this letter, he reveals that he and his wife had at one time worked on a Grand Hotel screenplay based in Las Vegas.
For related material in memory of Dunne, see In Lieu of Rosebud, which contains entries for 10/10-10/12, 2002.
Mein Irisch Kind,
Wo weilest du?
2:17
“… both a new world
And the old made explicit, understood
In the completion of its partial ecstasy,
The resolution of its partial horror.”
— T. S. Eliot, Four Quartets
Speaking of horror, today’s noon entry has a link to a page that references Stephen King’s The Shining.
On a 1970’s edition of
Stephen King’s The Shining:
“The page where Danny actually enters room 217 for the first time (King builds to this moment for a long time, it’s one of the more frightening passages in the book), is precisely on page 217. Scared the crap out of me the first time I read it.”
In honor of St. Thomas Stearns Eliot, whose feast day is today, of St. Emil Artin (see entries for St. Emil’s day, 12/20/03), and of Room 217, a check of last year’s 2/17 entries leads to St. Andrea’s weblog, which today, recalling the “white and geometric” prewar Berlin of the 12/20/03 entries, has Andrea looking, with Euclid, on beauty bare.
See also my entry “The Boys from Uruguay” and the later entry “Lichtung!” on the Deutsche Schule Montevideo in Uruguay.
“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
|
|
“720 in |
Musical Note: A Star is Born
Natalie Wood played a six-year-old
in “Miracle on 34th Street,”
six factorial equals 720,
and Wood was born on 7/20, 1938.
“How I love music.”
— John O’Hara, Hope of Heaven, 1938
For related metaphors, see
Immortal Diamond,
The Diamond Archetype, and
the first log24.net entry…
for July 20, 2002.
What, and Give Up Show Biz?
"Dying is easy. Comedy is hard."
— Saying attributed to Edmund Gwenn, star of "Miracle on 34th Street," and also attributed to "Noel Coward, David Garrick, William Holden, Edmund Kean, Marcel Marceau, Groucho Marx, and Oscar Wilde."
See also yesterday's entry on the Dark Lady. For more on Santa and the Dark Lady, see my archive for Aug.-Sept. 2002.
"Drink up, sweet. You gotta go some. How I love music. Frère Jacques, Cuernavaca, ach du lieber August. All languages. A walking Berlitz. Berlitz sounds like you with that champagne, my sweet, or how you're gonna sound."
— Hope of Heaven, by John O'Hara,
"another acidic writer to whom he
[John Gregory Dunne]
was often compared"
(Adam Bernstein, Washington Post)
For some context for the Hope of Heaven quotation, see Immortal Diamond: O'Hara, Hopkins, and Joyce, or click on the adding machine in yesterday's entry.
For more on miracles and the afterlife, see my archive for September 2002.
The Dark Lady
“… though she has been seen by many men, she is known to only a handful of them. You’ll see her — if you see her at all — just after you’ve taken your last breath. Then, before you exhale for the final time, she’ll appear, silent and sad-eyed, and beckon to you.
She is the Dark Lady, and this is her story.”
“… she played (very effectively) the Deborah Kerr part in a six-hour miniseries of From Here to Eternity….”
— John Gregory Dunne on Natalie Wood
in the New York Review of Books
dated Jan. 15, 2004
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