"Princeton's Baccalaureate service is an end-of-the-year ceremony focused on members of the senior class. It includes prayers and readings from various religious and philosophical traditions."
One such tradition— the TV series "Lost."
Another— the Pennsylvania Lottery—
For some context,
see May 6, 2010.
See also this journal's post
"The Omen" on the date 6/6/6.
Lottery hermeneutics for yesterday's numbers—
PA— Midday 711, Evening 039.
NY— Midday 440, Evening 704.
Simple interpretive methods— numbers as dates and as hexagram numbers— yield 7/11, hexagram 39, and 7/04.
The reader may supply his own interpretations of 7/11 and 7/04; for hexagram 39, see Wilhelm's commentary—
"The hexagram pictures a dangerous abyss lying before us
and a steep, inaccessible mountain rising behind us."
— and the cover of Cold Mountain—
Adapted from cover of
German edition of Cold Mountain
This suggests revisiting The Edge of Eternity (July 5, 2005).
The hermeneutics of the NY midday 440 is more difficult. A Google search suggests that a Log24 post for Epiphany 2004, "720 in the Book," might yield a clue to the 440 riddle.
By all means, let us 440.
Continued from May 8
(Feast of Saint Robert Heinlein)
“Wells and trees were dedicated to saints. But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden. Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy.”
— Charles Williams, Witchcraft, Faber and Faber, London, 1941
Why "Saint" Robert? See his accurate depiction of evil– the Eater of Souls in Glory Road.
For more on Williams's "other capacities," see Heinlein's story "Lost Legacy."
A related story– Fritz Leiber's "The Mind Spider." An excerpt:
The conference—it was much more a hyper-intimate
gabfest—proceeded.
"My static box bugged out for a few ticks this morning,"
Evelyn remarked in the course of talking over the
trivia of the past twenty-four hours.
The static boxes were an invention of Grandfather
Horn. They generated a tiny cloud of meaningless brain
waves. Without such individual thought-screens, there was
too much danger of complete loss of individual personality
—once Grandfather Horn had "become" his infant daughter
as well as himself for several hours and the unfledged
mind had come close to being permanently lost in its own
subconscious. The static boxes provided a mental wall be-
– hind which a mind could safely grow and function, similar
to the wall by which ordinary minds are apparently
always enclosed.
In spite of the boxes, the Horns shared thoughts and
emotions to an amazing degree. Their mental togetherness
was as real and as mysterious—and as incredible—as
thought itself . . . and thought is the original angel-cloud
dancing on the head of a pin. Their present conference
was as warm and intimate and tart as any actual family
gathering in one actual room around one actual table.
Five minds, joined together in the vast mental darkness
that shrouds all minds. Five minds hugged together for
comfort and safety in the infinite mental loneliness that
pervades the cosmos.
Evelyn continued, "Your boxes were all working, of
course, so I couldn't get your thoughts—just the blurs of
your boxes like little old dark grey stars. But this time
if gave me a funny uncomfortable feeling, like a spider
Crawling down my—Grayl! Don't feel so wildly! What
Is it?”
Then… just as Grayl started to think her answer…
something crept from the vast mental darkness and infinite
cosmic loneliness surrounding the five minds of the
Horns.
Grayl was the first to notice. Her panicky thought had
ttie curling too-keen edge of hysteria. "There are six of
us now! There should only be five, but there are six.
Count! Count, I tell you! Six!"
To Mort it seemed that a gigantic spider was racing
across the web of their thoughts….
See also this journal on May 30– "720 in the Book"– and on May 31– "Memorial for Galois."
("Obnoxious nerds"— a phrase Martin Gardner recently applied to Galois— will note that 720
Transcendence through spelling:
Richard Gere and Flora Cross
as father and daughter
in "Bee Season."
Words Made Flesh: Code, Culture, Imagination—
… letters create things by the virtue of an algorithm…
Spelling is a sign, Elly. When you win the national bee, we'll know that you are ready to follow in Abulafia's footsteps. Once you're able to let the letters guide you through any word you are given, you will be ready to receive shefa."
In the quiet of the room, the sound of Eliza and her father breathing is everything.
"Do you mean," Eliza whispers, "that I'll be able to talk to God?"
Diamond Theory notes
of Feb. 4, 1986,
of April 26, 1986, and
of May 26, 1986,
Sacerdotal Jargon
(Log24, Dec. 5, 2002),
and 720 in the Book
(Log24, Epiphany 2004).
Readings for
Yom Kippur
The film Pi is, in part, about an alleged secret name of God that can be uttered only on Yom Kippur. This is my personal version of such a name– not an utterance, but instead a picture:
6:49:32 PM
Sept. 24, 2004
The Details:
Synthemes and Spreads (pdf)
(Appendix A of
"Classification of
Partial Spreads in PG(4,2),"
by Leonard H. Soicher et al.)
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)
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.”
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:
|
Labyrinth |
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