Saturday, December 21, 2013

House of Secrets*

Filed under: General,Geometry — m759 @ 6:01 PM

The title is taken from a book for ages 8-12 published
on Shakespeare's birthday, April 23, 2013.

Also from that date, a note for older readers—

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

Half a dozen of the other —

For further context, see all  posts for the cruelest month of this past year.

* Secrets :  A sometimes dangerous word.

Saturday, September 30, 2017

Where Angels Fear to Tread

Filed under: General — m759 @ 8:38 AM

From the online New York Times  this morning —

"Origin  is Mr. Brown’s eighth novel. It finds his familiar protagonist,
the brilliant Harvard professor of symbology and religious iconography
Robert Langdon, embroiled once more in an intellectually challenging,
life-threatening adventure involving murderous zealots, shadowy fringe
organizations, paradigm-shifting secrets with implications for the future
of humanity, symbols within puzzles and puzzles within symbols and
a female companion who is super-smart and super-hot.

As do all of Mr. Brown’s works, the new novel does not shy away from
the big questions, but rather rushes headlong into them."

— Profile of Dan Brown by Sarah Lyall

See also yesterday's Log24 post on the Feast of St. Michael and All Angels.

Wednesday, February 22, 2017

A Dead Ringer

Filed under: General — m759 @ 1:00 PM

"According to Thelemic legend, in 1918 Aleister Crowley
came into contact with a interdimensional entity
named Lam, who by the way is a dead ringer for
the popular conception of the 'alien grey '
depicted on the cover of Whitley Strieber's Communion ."

Related material —

"Robert Langdon stood mesmerized at the glass portal,
absorbing the power of the landscape below him."


From 'Buffy the Vampire Slayer,' season 1, episode 2

"Ting-a-ling." — Kurt Vonnegut Jr.

Friday, May 22, 2015

Mathmagic Land

Filed under: General — Tags: , — m759 @ 12:00 PM

Donald Duck with Pythagorean pentagram on hand

Donald in Mathmagic Land

Manly P. Hall

Sunday, November 30, 2014

Electric Dreams

Filed under: General — Tags: — m759 @ 12:31 PM

Continued from Black Friday 

"Tell me no secrets, tell me some lies."

Don't It Make My Brown Eyes Blue

Thursday, June 26, 2014

Study This Example, Part II

Filed under: General,Geometry — Tags: — m759 @ 11:06 AM

(Continued from 10:09 AM today)

The quotation below is from a webpage on media magnate
Walter Annenberg.

Annenberg Hall at Harvard, originally constructed to honor
the Civil War dead, was renamed in 1996 for his son Roger,
Harvard Class of ’62.


“It was said that Roger was ‘moody and sullen’
spending large parts of his time reading poetry
and playing classical music piano. It had been
reported that Roger attempted suicide at the
age of eleven by slitting his wrists. He recovered
and was graduated Magna Cum Laude from
Episcopal Academy in our area. For awhile,
Roger attended Harvard, but he was removed
from the school’s rolls after Roger stopped doing
his school work and spent almost all his time
reading poetry in his room. He then was sent to
an exclusive and expensive treatment center
in Bucks County, Pennsylvania. At that facility,
Roger became more remote. It was said that he
often didn’t recognize or acknowledge his father.
On August 7, 1962, Roger Annenberg died from
an overdose of sleeping pills.”

A more appropriate Annenberg memorial, an article
in The Atlantic  magazine on June 25, notes that…

“Among those who ended up losing their battles
with mental illness through suicide are
Virginia Woolf, Ernest Hemingway, Vincent van Gogh,
John Berryman, Hart Crane, Mark Rothko, Diane Arbus,
Anne Sexton, and Arshile Gorky.”

Sunday, October 27, 2013

Reverse Chronology

Filed under: General — m759 @ 1:00 PM

Scene I:

"Pinter's particular usage of reverse chronology 
in structuring the plot is innovative…."

— Wikipedia on the play "Betrayal," a version of which
    opens tonight

Scene II:

Reverse Chronology in Wikipedia —

"As a hypothetical example, if the fairy tale Jack and the Beanstalk 
was told using reverse chronology, the opening scene would depict
Jack chopping the beanstalk down and killing the giant. The next
scene would feature Jack being discovered by the giant and climbing
down the beanstalk in fear of his life. Later, we would see Jack running
into the man with the infamous magic beans, then, at the end of the film,
being sent off by his mother to sell the cow."

Scene III:

Dialogue for Scene III — 

"Sell the damn cow, Jack."

Epilogue:  Jack + Jill.

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — m759 @ 4:30 AM

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Friday, June 15, 2012


Filed under: General — m759 @ 3:00 AM

In memory of Paul Sussman, author of archaeological 
mystery novels about Egypt—

IMAGE- Harvard Divinity School bookplate dated 1910

"… the sacred symbols of the cosmic elements
were hid away hard by the secrets of Osiris."  

Thrice-Great Hermes: Excerpts and Fragments ,
      by George Robert Stowe Mead,
     Theosophical Publishing Society, 1906   

Sussman's last novel, not yet published, was

The Labyrinth of Osiris .

Sussman, 45, reportedly died suddenly on May 31, 2012.

A perhaps relevant thought—

"A world of made
is not a world of born— pity poor flesh
and trees, poor stars and stones, but never this
fine specimen of hypermagical 

– e. e. cummings, 1944

Friday, July 15, 2011

Inside CBS News

Filed under: General — m759 @ 5:24 AM


Related material from the June 30 Washington Post

"Wizardly in his abilities and appearance—
he had a scraggly, gray beard—
Mr. Morris was the digital gatekeeper
of the American government’s computer secrets."

See also audience participation features at The New York Times


Monday, January 31, 2011

Darkness at Noon

Filed under: General — m759 @ 12:00 PM

In today's Wall Street Journal , Peter Woit reviews a new book on dark matter and dark energy.

For a more literary approach, see "dark materials" in this  journal.

Before thir eyes in sudden view appear
The secrets of the hoarie deep, a dark
Illimitable Ocean without bound,
Without dimension, where length, breadth, and highth,
And time and place are lost; where eldest Night
And Chaos, Ancestors of Nature, hold
Eternal Anarchie, amidst the noise
Of endless warrs and by confusion stand.
For hot, cold, moist, and dry, four Champions fierce
Strive here for Maistrie, and to Battel bring amidst the noise
Thir embryon Atoms....
                                ... Into this wilde Abyss,
The Womb of nature and perhaps her Grave,
Of neither Sea, nor Shore, nor Air, nor Fire,
But all these in thir pregnant causes mixt
Confus'dly, and which thus must ever fight,
Unless th' Almighty Maker them ordain
His dark materials to create more Worlds,
Into this wilde Abyss the warie fiend
Stood on the brink of Hell and look'd a while,
Pondering his Voyage....

-- John Milton, Paradise Lost , Book II

Related material:

1. The “spider” symbol of Fritz Leiber’s short story “Damnation Morning”—

2. Angels and demons here and in the Catholic Church.

3. The following diagram by one “John Opsopaus”—


Sunday, May 2, 2010

Legal Fiction–

Filed under: General — m759 @ 1:00 AM

Murphy's Last Stand

In memory of Walter F. Murphy, a leading constitutional scholar and McCormick Professor of Jurisprudence at Princeton from 1968 to 1995. Murphy was also the author of a bestselling 1979 novel, The Vicar of Christ. He died at 80 on Tuesday, April 20, 2010, quod vide.

His novel, according to this morning's New York Times, "tells the story of an American who fights valiantly in the Korean War (as Professor Murphy did), becomes chief justice of the United States, resigns to become a monk and is eventually elected the first American pope." An eventful tale.

For a good review of Murphy's novel, see "The Doomed Hero." This review, and yesterday's Log24 Law Day post, which mentions the concept of "the mighty music of the innermost heaven," suggest revisiting a Log24 post of August 28, 2009 and a hymn by Brian Wilson—


  Send "In My Room" Ringtone to Cell

There's a world where I can go
and tell my secrets to
In my room
In my room

In this world I lock out
all my worries and my fears
In my room
In my room

Do my dreaming and my scheming lie awake and pray
Do my crying and my sighing laugh at yesterday

Now it's dark and I'm alone
but I won't be afraid
In my room
In my room

Thursday, March 25, 2010

Marshall, JC, and…

Filed under: General — m759 @ 12:00 PM

Southern Comfort


"Jim’s appetite for booze and drugs was legendary. In fact, one of the secrets of his success was that he kept taking pictures after all the other photographers went to bed.

What’s most striking about his photographs is how even in the most chaotic moments he finds clarity and candor. Jim’s photographs are remarkable for the ease with which they convey something deep and real about their subjects….

What comes across is a deep empathy for the musicians he photographed, and an ability to capture their pride and sense of purpose, even when circumstances were less than ideal. I love the shot of… Johnny Cash, heavy and brooding before his 1968 performance at Folsom Prison, with the guard tower looming over his shoulder."

Unauthorized copy from Marshall's gallery


"Recently, over scotches in one of Jim’s favorite New York bars, I asked what he sees in this collection of his work. 'How the fuck should I know?' he said. 'I was there. I took some photographs. This is them….'"

In Memoriam… The classic film Asphalt Jungle and a hymn for Hunter S. Thompson that will do as well for Sterling Hayden, Marshall, and Cash.

Sunday, January 10, 2010

Leaving Las Vegas, continued

Filed under: General — m759 @ 11:00 AM

A sermon for the father of Kal-El.
See also related material in this journal
on Vegas and on Maniacs.

Wednesday, July 15, 2009

Wednesday July 15, 2009

Filed under: General — Tags: — m759 @ 4:01 AM
of obituaries page,
New York Times,
Monday morning:

Detail, obits page, NY Times Monday morning, July 13, 2009

Detail of arts page,
New York Times, Wednesday morning:

(Click ad for more on the Monday night death of Dash Snow.)

Arts page detail, morning of Wednesday, July 15, 2009

Headlines collage by Dash Snow

Hurt yet?

Update of 5:01 AM:

Lavery Hits
Literary Jackpot

From the top right of
this morning's online
New York Times front page:

Christoph Niemann on witchcraft and snow

Click on voodoo doll
for further details.

See also…

1. Monday's link to a
Wallace Stevens poem,
"Snow and Stars"

2. The conclusion of this
morning's Times obituary
for artist Dash Snow, which
gives his daughter's name…

3. David Lavery's excellent
of the classic
 Conrad Aiken story
"Silent Snow, Secret Snow."

Saturday, June 27, 2009

Saturday June 27, 2009

Filed under: General — m759 @ 2:56 PM

Dark Materials

Before thir eyes in sudden view appear
The secrets of the hoarie deep, a dark
Illimitable Ocean without bound,
Without dimension, where length, breadth, and highth,
And time and place are lost; where eldest Night
And Chaos, Ancestors of Nature, hold
Eternal Anarchie, amidst the noise
Of endless warrs and by confusion stand.
For hot, cold, moist, and dry, four Champions fierce
Strive here for Maistrie, and to Battel bring amidst the noise
Thir embryon Atoms....
                                ... Into this wilde Abyss,
The Womb of nature and perhaps her Grave,
Of neither Sea, nor Shore, nor Air, nor Fire,
But all these in thir pregnant causes mixt
Confus'dly, and which thus must ever fight,
Unless th' Almighty Maker them ordain
His dark materials to create more Worlds,
Into this wilde Abyss the warie fiend
Stood on the brink of Hell and look'd a while,
Pondering his Voyage....

-- John Milton, Paradise Lost, Book II

Sunday, April 13, 2008

Sunday April 13, 2008

Filed under: General — m759 @ 8:29 PM
National Treasure

Nicolas Cage in National Treasure: Book of Secrets

Pennsylvania Lottery today:

Mid-day 504, Evening 628.

Related material:

Today’s previous entry

and entries of

5/04 (2007), and 6/28 (2007).

Happy birthday, Thomas Jefferson:

“… God to a nation
         dealt that day’s dear chance.
 To man, that needs would worship
         block or barren stone….”

— “To what serves Mortal Beauty?,”
     by Gerard Manley Hopkins, S. J.

    (Quoted here on Aug. 29, 2006)

Thursday, August 9, 2007

Thursday August 9, 2007

Filed under: General — m759 @ 12:00 PM
“Serious numbers  
will always be heard.”

— Paul Simon

(See St. Luke’s Day, 2005.)  

Bulletin of the American Mathematical Society
Volume 31, Number 1, July 1994, Pages 1-14

Selberg’s Conjectures
and Artin L-Functions

M. Ram Murty


In its comprehensive form, an identity between an automorphic L-function and a “motivic” L-function is called a reciprocity law. The celebrated Artin reciprocity law is perhaps the fundamental example. The conjecture of Shimura-Taniyama that every elliptic curve over Q is “modular” is certainly the most intriguing reciprocity conjecture of our time. The “Himalayan peaks” that hold the secrets of these nonabelian reciprocity laws challenge humanity, and, with the visionary Langlands program, we have mapped out before us one means of ascent to those lofty peaks. The recent work of Wiles suggests that an important case (the semistable case) of the Shimura-Taniyama conjecture is on the horizon and perhaps this is another means of ascent. In either case, a long journey is predicted…. At the 1989 Amalfi meeting, Selberg [S] announced a series of conjectures which looks like another approach to the summit. Alas, neither path seems the easier climb….

[S] A. Selberg, Old and new
      conjectures and results
      about a class of Dirichlet series,
      Collected Papers, Volume II,
      Springer-Verlag, 1991, pp. 47-63.

Zentralblatt MATH Database
on the above Selberg paper:

“These are notes of lectures presented at the Amalfi Conference on Number Theory, 1989…. There are various stimulating conjectures (which are related to several other conjectures like the Sato-Tate conjecture, Langlands conjectures, Riemann conjecture…)…. Concluding remark of the author: ‘A more complete account with proofs is under preparation and will in time appear elsewhere.'”

Related material: Previous entry.

Monday, July 30, 2007

Monday July 30, 2007

Filed under: General — m759 @ 8:00 AM
 Behind Every
Great Man…

The image “http://www.log24.com/log/pix07/070730-OdileCrick.jpg” cannot be displayed, because it contains errors.

Odile Crick with her husband, Francis H.C. Crick, in Cambridge, England. Mrs. Crick, an artist, illustrated the work of her husband, whose team received a Nobel Prize for its DNA research.
Photo Credit: Courtesy Of The Salk Institute For Biological Studies

Washington Post, July 21, 2007

“Her graceful drawing of the double-helix structure of DNA with intertwined helical loops has become a symbol of the achievements of science and its aspirations to understand the secrets of life. The image represents the base pairs of nucleic acids, twisted around a center line to show the axis of the helix. Terrence J. Sejnowski, a neuroscientist at the Salk Institute for Biological Studies in La Jolla, where Francis Crick later worked, said: ‘Mrs. Crick’s drawing was an abstract representation of DNA, but it was accurate with regard to its shape and size of its spacing.

‘The models you see now have all the atoms in them,’ Sejnowski said. ‘The one in Nature was the backbone and gave the bare outline. It may be the most famous [scientific] drawing of the 20th century, in that it defines modern biology.'”

— Adam Bernstein in
The Washington Post, July 21, 2007

Saturday, June 23, 2007

Saturday June 23, 2007

Filed under: General — m759 @ 9:00 AM
Faust in Copenhagen:
A Struggle for
the Soul of Physics

By Gino Segrè

Illustrated. 310 pp.
Viking. $25.95.

 The image “http://www.log24.com/log/pix07/FaustInCopenhagen.jpg” cannot be displayed, because it contains errors.

Review in the June 24
New York Times Book Review:

“As though their knowledge of the quantum secrets came with the power of prophecy, some three dozen of Europe’s best physicists ended their 1932 meeting in Copenhagen with a parody of Goethe’s ‘Faust.’….

It was only in retrospect that the silliness became profound. The players were becoming possessors of ‘a truth with implicit powers of good and evil,’ Gino Segrè writes in ‘Faust in Copenhagen,’ his inventive new book about the era. And ‘the devil… was in the details.'” –George Johnson

Related material:

This week’s entries
on Pauli and Faust,
the entries of
  June 3 through June 6,
and the five entries
ending on April 7, 2005,
with “In the Details

Thursday, October 5, 2006

Thursday October 5, 2006

Filed under: General,Geometry — m759 @ 9:11 AM
In Touch with God

(Title of an interview with
the late Paul Halmos, mathematician)

Since Halmos died on Yom Kippur, his thoughts on God may be of interest to some.

From a 1990 interview:

“What’s the best part of being a mathematician? I’m not a religious man, but it’s almost like being in touch with God when you’re thinking about mathematics. God is keeping secrets from us, and it’s fun to try to learn some of the secrets.”

I personally prefer Annie Dillard on God:

“… if Holy the Firm is matter at its dullest, Aristotle’s materia prima, absolute zero, and since Holy the Firm is in touch with the Absolute at base, then the circle is unbroken.  And it is…. Holy the Firm is in short the philosopher’s stone.”

Some other versions of
the philosopher’s stone:

The image �http://www.log24.com/log/pix06/060101-SixOfOne.jpg� cannot be displayed, because it contains errors.

And, more simply,
April 28, 2004:

This last has the virtue of
being connected with Halmos
via his remarks during the
“In Touch with God” interview:

“…at the root of all deep mathematics there is a combinatorial insight… the really original, really deep insights are always combinatorial….”
“Combinatorics, the finite case, is where the genuine, deep insight is.”

See also the remark of Halmos that serves as an epigraph to Theme and Variations.

Finally, it should be noted that
the 4×9 black rectangle

The image “http://www.log24.com/log/pix06A/061004-Halmos100x225.jpg” cannot be displayed, because it contains errors.

has also served
at least one interpreter
as a philosopher’s stone,
and is also the original
“Halmos tombstone.”

(See previous entry.)

Wednesday, October 26, 2005

Wednesday October 26, 2005

Filed under: General — m759 @ 2:56 PM
Human Conflict
Number Five

(Album title, 10,000 Maniacs)

The image “http://www.log24.com/log/pix05B/051026-Human.jpg” cannot be displayed, because it contains errors.

This album contains
Planned Obsolescence:

is truth for life
watch religion fall obsolete
will be truth for life
technology as nature
truth for life
in fortran tongue the

with wealth and prominence
man so near perfection
it’s an absence of interim
secure no demurrer
defense against divine
defense against his true
human conflict number five
dissolved all illusion
destroyed with conclusion
and illusion never restored

any modern man can see
that religion is

prophetic vision

Secrets of the I Ching

(Album title, 10,000 Maniacs)

Time of this entry: 2:56:37

Question suggested by the
lottery in the state of Grace
(Kelly) on the night Sinatra died:

What is 256 about?

Answer: 37.

In other words…

The image “http://www.log24.com/images/IChing/hexagram37.gif” cannot be displayed, because it contains errors.

37. The Family (The Clan)

The image “http://www.log24.com/log/pix05B/051016-Mont.jpg” cannot be displayed, because it contains errors.

For details, see Log24,
11 AM Sunday, October 16:

Philadelphia Stories.

Wednesday October 26, 2005

Filed under: General — m759 @ 7:48 AM
Today’s Birthday:
Natalie Merchant

From Wikipedia:

Hope Chest:
The Fredonia Recordings 1982-1983

is a 1990 album by 10,000 Maniacs.
It compiles tracks from their early releases
Human Conflict Number Five and
Secrets of the I Ching.”

For Natalie,
a new web page summing up the
benefits of a Fredonia education:
Certified Crank

Friday, November 14, 2003

Friday November 14, 2003

Filed under: General — Tags: — m759 @ 2:56 AM

Philip K. Dick Meets Joan Didion

From the ending of
The Man in the High Castle:

Juliana said, “I wonder why the oracle would write a novel. Did you ever think of asking it that?” ….

“You may say the question aloud,” Hawthorne said. “We have no secrets here.”

Juliana said, “Oracle, why did you write The Grasshopper Lies Heavy? What are we supposed to learn?”

“You have a disconcertingly superstitious way of phrasing your question,” Hawthorne said. But he had squatted down to witness the coin throwing. “Go ahead,” he said; he handed her three Chinese brass coins with holes in the center. “I generally use these.”

She began throwing the coins; she felt calm and very much herself. Hawthorne wrote down her lines for her. When she had thrown the coins six times, he gazed down and said:

“Sun at the top. Tui at the bottom. Empty in the center.”

IMAGE- Hexagram 61

“Do you know what hexagram that is?” she said. “Without using the chart?”

“Yes,” Hawthorne said.

“It’s Chung Fu,” Juliana said. “Inner Truth. I know without using the chart, too. And I know what it means.”

From the ending of
Play It As It Lays:

I lie here in the sunlight, watch the hummingbird.  This morning I threw the coins in the swimming pool, and they gleamed and turned in the water in such a way that I was almost moved to read them.  I refrained.

One thing in my defense, not that it matters.  I know something Carter never knew, or Helene, or maybe you.  I know what “nothing” means, and keep on playing.

Why, BZ would say.

Why not, I say.

Wednesday, November 5, 2003

Wednesday November 5, 2003

Filed under: General — m759 @ 5:32 PM

Legacy Codes

“In writing The Legacy Codes, the term itself became the play’s central metaphor. In newspaper accounts of the Wen Ho Lee case, the classified legacy codes which caused the uproar were described as computer simulations of plutonium explosions. The term is also used by computer experts for any archaic codes which are still necessary to run complex computer programs. For me the term can also be interpreted as the DNA genetic code, it can be interpreted as what is passed on in families regarding culture, family secrets, genetic traits. It also can relate to how people and institutions want to be remembered in the future.”

Playwright Cherylene Lee

The Legacy Codes opens at 7 tonight in Manhattan.

Monday, September 1, 2003

Monday September 1, 2003

Filed under: General — m759 @ 3:33 PM

The Unity of Mathematics,

or “Shema, Israel”

A conference to honor the 90th birthday (Sept. 2) of Israel Gelfand is currently underway in Cambridge, Massachusetts.

The following note from 2001 gives one view of the conference’s title topic, “The Unity of Mathematics.”

Reciprocity in 2001

by Steven H. Cullinane
(May 30, 2001)

From 2001: A Space Odyssey, by Arthur C. Clarke, New American Library, 1968:

The glimmering rectangular shape that had once seemed no more than a slab of crystal still floated before him….  It encapsulated yet unfathomed secrets of space and time, but some at least he now understood and was able to command.

How obvious — how necessary — was that mathematical ratio of its sides, the quadratic sequence 1: 4: 9!  And how naive to have imagined that the series ended at this point, in only three dimensions!

— Chapter 46, “Transformation”

From a review of Himmelfarb, by Michael Krüger, New York, George Braziller, 1994:

As a diffident, unsure young man, an inexperienced ethnologist, Richard was unable to travel through the Amazonian jungles unaided. His professor at Leipzig, a Nazi Party member (a bigot and a fool), suggested he recruit an experienced guide and companion, but warned him against collaborating with any Communists or Jews, since the objectivity of research would inevitably be tainted by such contact. Unfortunately, the only potential associate Richard can find in Sao Paulo is a man called Leo Himmelfarb, both a Communist (who fought in the Spanish Civil War) and a self-exiled Jew from Galicia, but someone who knows the forests intimately and can speak several of the native dialects.

“… Leo followed the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity, which I could not even imitate.”

… E. M. Forster famously advised his readers, “Only connect.” “Reciprocity” would be Michael Kruger’s succinct philosophy, with all that the word implies.

— William Boyd, New York Times Book Review, October 30, 1994

Reciprocity and Euler

Applying the above philosophy of reciprocity to the Arthur C. Clarke sequence

1, 4, 9, ….

we obtain the rather more interesting sequence
1/1, 1/4, 1/9, …..

This leads to the following problem (adapted from the St. Andrews biography of Euler):

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series

1/1 + 1/4 + 1/9 + 1/16 + 1/25 + …

— a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that the series sums to (pi squared)/6. He generalized this series, now called zeta(2), to zeta functions of even numbers larger than two.

Related Reading

For four different proofs of Euler’s result, see the inexpensive paperback classic by Konrad Knopp, Theory and Application of Infinite Series (Dover Publications).

Related Websites

Evaluating Zeta(2), by Robin Chapman (PDF article) Fourteen proofs!

Zeta Functions for Undergraduates

The Riemann Zeta Function

Reciprocity Laws
Reciprocity Laws II

The Langlands Program

Recent Progress on the Langlands Conjectures

For more on
the theme of unity,

Monolithic Form

Wednesday, October 23, 2002

Wednesday October 23, 2002

Filed under: General — m759 @ 5:04 PM

In Memoriam

From the New York Times of Oct. 23, 2002:

Richard M. Helms Dies at 89;
Dashing Ex-Chief of the C.I.A.

      Associated Press

Richard M. Helms, a former C.I.A. director, died today.


WASHINGTON Oct. 23 — “Richard Helms, a former director of central intelligence who defiantly guarded some of the darkest secrets of the cold war, died today of multiple myeloma. He was 89.

An urbane and dashing spymaster, Mr. Helms began his career with a reputation as a truthteller….”

Needless to say, that didn’t last.  I encountered this story this afternoon, after writing the entry below this morning.  The site I described there,


reads as though it were compiled by an intelligence officer, and may serve as a small memorial to Helms.  

Monday, October 21, 2002

Monday October 21, 2002

Filed under: General — Tags: — m759 @ 12:01 AM

Birthdays for a Small Planet

Today's birthdays:

The entry below, "Theology for a Small Planet," sketches an issue that society has failed to address since the fall of 1989, when it was first raised by the Harvard Divinity Bulletin.

In honor mainly of Ursula K. Le Guin, but also of her fellow authors above, I offer Le Guin's solution. It is not new. It has been ignored mainly because of the sort of hateful and contemptible arrogance shown by

  • executives in the tradition of Henry Ford and later Ford Foundation and Ford Motors employees McGeorge Bundy and Robert McNamara (see yesterday's entry below for Ford himself), by
  • theologians in the tradition of the Semitic religions — Judaism, Christianity, and Islam — and by
  • self-proclaimed "shamans of scientism" like Michael Shermer in the tradition of Scientific American magazine.

Here is an introduction to the theology that should replace the ridiculous and outdated Semitic religions.

According to Le Guin,

"Scholarly translators of the Tao Te Ching, as a manual for rulers, use a vocabulary that emphasizes the uniqueness of the Taoist 'sage,' his masculinity, his authority. This language is perpetuated, and degraded, in most popular versions. I wanted a Book of the Way accessible to a present-day, unwise, unpowerful, and perhaps unmale reader, not seeking esoteric secrets, but listening for a voice that speaks to the soul. I would like that reader to see why people have loved the book for 2500 years.

It is the most lovable of all the great religious texts, funny, keen, kind, modest, indestructibly outrageous and inexhaustibly refreshing. Of all the deep springs, this is the purest water. To me it is also the deepest spring."

Tao Te Ching: Chapter 6
translated by Ursula K. Le Guin

The valley spirit never dies
Call it the mystery, the woman.

The mystery,
the Door of the Woman,
is the root
of earth and heaven.

Forever this endures, forever.
And all its uses are easy.

Saturday, October 5, 2002

Saturday October 5, 2002

Filed under: General — m759 @ 12:00 PM

Zen holy day:

Bodhidharma Day

Epigraph to Chapter 23 of Contact, by Carl Sagan:

We have not followed cunningly devised fables….
— II Peter 1:16

Song lyric:

It’s still the same old story….
— Herman Hupfeld, 1931

From Chapter 23 of Contact, by Carl Sagan:

  “You mean you could decode a picture hiding in pi and it would be a mess of Hebrew letters?”
  “Sure.  Big black letters, carved in stone.”
  He looked at her quizzically.
  “Forgive me, Eleanor, but don’t you think you’re being a mite too… indirect?  You don’t belong to a silent order of Buddhist nuns.  Why don’t you just tell your story?”

Moonlight and love songs,
never out of date…. 

See also my journal note 
for Michaelmas, 2002,
Pi in the Sky.” 

Friday, October 4, 2002

Friday October 4, 2002

Filed under: General — m759 @ 4:17 AM

The Agony and the Ya-Ya

Today’s birthdays:

  • Charlton Heston
  • Anne Rice
  • Patti LaBelle

To honor the birth of these three noted spiritual leaders, I make the following suggestion: Use the mandorla as the New Orleans Mardi Gras symbol.  Rice lives in New Orleans and LaBelle’s classic “Lady Marmalade” deals with life in that colorful city.

What, you may well ask, is the mandorla? This striking visual symbol was most recently displayed prominently at a meeting of U.S. cardinals in the Pope’s private library on Shakespeare’s birthday.  The symbol appears in the upper half of a painting above the Pope.

From Church Anatomy:

The illustration below shows how Barbara G. Walker in her excellent book “The Woman’s Encyclopedia of Myths and Secrets” describes the mandorla.


The Agony
and the Ecstasy

Based on a novel by Irving Stone, this 1965 movie focuses on the relationship between Michelangelo (Charlton Heston) and Pope Julius II (Rex Harrison), who commissioned the artist to paint the Sistine Chapel ceiling.

Vesica piscis

Mandorla, “almond,” the pointed-oval sign of the yoni, is used in oriental art to signify the divine female genital; also called vesica piscis, the Vessel of the Fish. Almonds were holy symbols because of their female, yonic connotations.

Christian art similarly used the mandorla as a frame for figures of God, Jesus, and saints, because the artists forgot what it formerly meant. I. Frazer, G.B., 403

For further details on the mandorla (also known as the “ya-ya”) see my June 12, 2002, note The Ya-Ya Monologues.
A somewhat less lurid use of the mandorla in religious art — the emblem of the Episcopal Diocese of South Carolina, taken from the website of St. Michael’s Church in Charleston — is shown below.

Thursday, September 19, 2002

Thursday September 19, 2002

Filed under: General — m759 @ 1:11 AM

The winner of the self-promotion award
at the third annual Latin Grammys
last night was… 


Jennifer Love Hewitt in Rolling Stone An Awfully Big Adventure is the story of Stella, a headstrong, starry-eyed young teenager whose passion for the theatre leads her into a grownup world of sex and secrets, menace and manipulation.


Girl, you’re a hot-blooded woman-child

And it’s warm where you’re touchin’ me

But I can tell by your tremblin’ smile

You’re seein’ way too much in me

– Mac Davis,      1972

Sunday, September 8, 2002

Sunday September 8, 2002

Filed under: General — m759 @ 2:00 AM

In honor of the September 8 birthdays of

From a website on Donna Tartt‘s novel The Secret History… 

“It is like a storyteller looking up suddenly into the eyes of his audience across the embers of a once blazing fire…

…the reader feels privy to the secrets of human experience by their passage down through the ages; the telling and re-telling. A phrase from the ghost in Hamlet comes to mind:

‘I could a tale unfold whose lightest word /
Would harrow up thy soul…..’ “

This work of literature seems especially relevant at the start of a new school year, and in light of my remarks below about ancient Greek religion. One should, when praising Apollo, never forget that Dionysus is also a powerful god.

For those who prefer film to the written word, I recommend “Barton Fink” as especially appropriate viewing for the High Holy Days. Judy Davis (my favorite actress) plays a Faulkner-figure’s “secretary” who actually writes most of his scripts.

Tartt is herself from Faulkner country.  For her next book, see this page from Square Books, 160 Courthouse Square, Oxford, Misssissippi.

Let us pray that Tartt fares better in real life than Davis did in the movie.

As music for the High Holy Days, I recommend Don Henley’s “The Garden of Allah.” For some background on the actual Garden of Allah Hotel at 8080 Sunset Boulevard (where “Barton Fink” might have taken place), see


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