See a search for Nocciolo in this journal.
An image from that search —
Recall also Hamlet's
"O God… bad dreams."
See a search for Nocciolo in this journal.
An image from that search —
Recall also Hamlet's
"O God… bad dreams."
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
— GianCarlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:
Related material:
Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.
See also Rota and Psychoshop.
For the new Jesuit pope (see previous post)
Now among Log24 posts tagged "Khora" is one
from July 15, 2010, dealing with a book called
Deconstruction in a Nutshell: A Conversation with
Jacques Derrida , edited and with a commentary by
John D. Caputo (Fordham University Press, 1997).
Related material:
"Khora is the felix culpa of a passion for the impossible,
the happy fault of a poetics of the possible, the heartless
heart of an ethical and religious eschatology.
Khora is the devil that justice demands we give his due."
— John D. Caputo, conclusion of "Abyssus Abyssum Invocat :
A Response to Kearney." Caputo's remarks followed
Richard Kearney's "Khora or God?," pp. 107122 in
A Passion for the Impossible: John D. Caputo in Focus ,
edited by Mark Dooley, State University of New York Press,
Albany, 2003. See "Abyssus " on pp. 123127.
See also other uses here of the phrase "In a Nutshell."
The Kernel of the Concept of the Object…
according to the New York Lottery yesterday—
From 4/27
From 11/24
A page numbered 176
A page numbered 187
"The yarns of seamen have a direct simplicity,
the whole meaning of which
lies within the shell of a cracked nut.
But Marlow was not typical
(if his propensity to spin yarns be excepted),
and to him the meaning of an episode
was not inside like a kernel but outside,
enveloping the tale which brought it out
only as a glow brings out a haze,
in the likeness of one of these misty halos
that sometimes are made visible by
the spectral illumination of moonshine."
— Joseph Conrad in Heart of Darkness
A revision of the above diagram showing
the Galoisadditiontable structure —
Related tables from August 10 —
See "Schoolgirl Space Revisited."
The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Note: There is no Galois (i.e., finite) field with six elements, but
the theory of finite fields underlies applications of sixset geometry.
<title datarh="true">Frank Heart, Who Linked Computers Before the Internet, Dies at 89 – The New York Times</title> 
See also yesterday's "For 6/24" and …
Text —
"A field is perhaps the simplest algebraic structure we can invent."
— Hermann Weyl, 1952
Context —
See also yesterday's Personalized Book Search.
Full text of Symmetry – Internet Archive — https://archive.org/details/Symmetry_482
A field is perhaps the simplest algebraic 143 structure 
From a Log24 search for Mathematics+Nutshell —
Published as the final chapter, Chapter 13, in
Episodes in the History of Modern Algebra (18001950) ,
edited by Jeremy J. Gray and Karen Hunger Parshall,
American Mathematical Society, July 18, 2007, pages 301326.
See also this journal on the above McLarty date —
May 24, 2003: Mental Health Month, Day 24.
This post's title is from the tags of the previous post —
The title's "shift" is in the combined concepts of …
Space and Number
From Finite Jest (May 27, 2012):
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —
"Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange"
— io9 , July 29, 2016
" 'This man comes from a binary universe
where it’s all about logic,' the actor told us
at San Diego ComicCon . . . .
'And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.' "
[Typo now corrected, except in a comment.]
For Scarlett
From a search for "Preparation" in this journal —
"In a nutshell, the book serves as an introduction to
Gauss' theory of quadratic forms and their composition laws
(the cornerstone of his Disquisitiones Arithmeticae ) from the
modern point of view (ideals in quadratic number fields)."
From a film in which Scarlett portrays a goddess —
Madness related to several recent posts —
Then, with an unheard splash which sent from the silver water to the shore a line of ripples echoed in fear by my heart, a swimming thing emerged beyond the breakers. The figure may have been that of a dog, a human being, or something more strange. It could not have known that I watched—perhaps it did not care—but like a distorted fish it swam across the mirrored stars and dived beneath the surface. After a moment it came up again, and this time, since it was closer, I saw that it was carrying something across its shoulder. I knew, then, that it could be no animal, and that it was a man or something like a man, which came toward the land from a dark ocean. But it swam with a horrible ease.
— From "The Night Ocean," by H. P. Lovecraft 
Related news —
"When hardliners seized power in Moscow in August 1991
and imprisoned Mr. Gorbachev in his vacation house on the
Black Sea, Mr. Chernyaev, a guest there and a powerful swimmer,
offered to smuggle out a note by swimming to a beach more than
three miles away. Uncertain where he could take the note, they
dropped the plan. The coup quickly failed in any case."
This is a followup to Tuesday's post on the Nov. 15 American
Mathematical Society (AMS) obituary of Joseph J. Rotman.
Detail of a page in "Notes on Finite Geometry, 19781986,"
"An outer automorphism of S_{6} related to M_{24}" —
Related work of Rotman —
"Outer Automorphisms of S_{6}," by
Gerald Janusz and Joseph Rotman,
The American Mathematical Monthly ,
Vol. 89, No. 6 (Jun. – Jul., 1982), pp. 407410
Some background —
"In a Nutshell: The Seed," Log24 post of Sept. 4, 2006:
Previous references in this journal to the "Church of Synchronology"
suggest a review of that phrase's source —
"The fine line between hokum and rational thinking
is precisely the point of The Lost Time Accidents ;
a brick of a book not just because of its length but
because of the density of both the prose and the
ideas it contains.
It is, in a nutshell, a sweeping historical novel that's
also a love story but is rooted in timetravel
science fiction and takes on as its subject
the meaning of time itself. This is no small endeavor."
— Janelle Brown in The Los Angeles Times
on February 4, 2016
See also …
In memory of physicist David Ritz Finkelstein,
who reportedly died yesterday —
"His sense of irony and precision was appreciated" ….
Precision
Irony
An illustration of the song "Stuck in the Middle with You"
(from the Tarantino film "Reservoir Dogs") was posted by
an academic at Christmas 2015 —
See also, in this journal,
The Jewel in the Lotus Meets the Kernel in the Nutshell
(December 16, 2015).
See pages 36 and 37 of Suzanne Gieser's The Innermost Kernel
as well as PyrE in The Stars My Destination and Old St. Patrick's*
in "Gangs of New York."
For some related aesthetic remarks, see a New Yorker essay
published onlne today and this journal's previous post.
* The older version of the "Old St. Patrick's"
of The Stars My Destination . (Update of 4/21/16.)
Meets the Kernel in the Nutshell.
This post was suggested by the title of Natalie Wolchover’s
article in Quanta Magazine today,
“A Fight for the Soul of Science.”
The post continues a meditation on the number 6
as the kernel in the nutshell of 15.
For an illustration of the 6 in the 15,
see nocciolo in this journal.
For an illustration of the jewel in the lotus,
see that phrase in this journal.
A recent nottoobright book from Princeton —
Some older, brighter books from Tony Zee —
Fearful Symmetry (1986) and
Quantum Field Theory in a Nutshell (2003).
* Continued.
"O God, I could be bounded in a nutshell
and count myself a king of infinite space,
were it not that I have bad dreams." — Hamlet
The New York Review of Books , in a review
of two books on video games today, quotes an author
who says that the Vikings believed the sky to be
“the blue skull of a giant.”
See as well posts tagged The Nutshell.
The American Mathematical Society yesterday:
Harvey Cohn (19232014)
Wednesday September 10th 2014
Cohn, an AMS Fellow and a Putnam Fellow (1942), died May 16 at the age of 90. He served in the Navy in World War II and following the war received his PhD from Harvard University in 1948 under the direction of Lars Ahlfors. He was a member of the faculty at Wayne State University, Stanford University, Washington University in St. Louis, the University of Arizona, and at City College of New York, where he was a distinguished professor. After retiring from teaching, he also worked for the NSA. Cohn was an AMS member since 1942.
Paid death notice from The New York Times , July 27, 2014:
COHN–Harvey. Fellow of the American Mathematical Society and member of the Society since 1942, died on May 16 at the age of 90. He was a brilliant Mathematician, an adoring husband, father and grandfather, and faithful friend and mentor to his colleagues and students. Born in New York City in 1923, Cohn received his B.S. degree (Mathematics and Physics) from CCNY in 1942. He received his M.S. degree from NYU (1943), and his Ph.D. from Harvard (1948) after service in the Navy (Electronic Technicians Mate, 194446). He was a member of Phi Beta Kappa (Sigma Chi), won the William Lowell Putnam Prize in 1942, and was awarded the Townsend Harris Medal in 1972. A pioneer in the intensive use of computers in an innovative way in a large number of classical mathematical problems, Harvey Cohn held faculty positions at Wayne State University, Stanford, Washington University Saint Louis (first Director of the Computing Center 195658), University of Arizona (Chairman 19581967), University of Copenhagen, and CCNY (Distinguished Professor of Mathematics). After his retirement from teaching, he worked in a variety of capacities for the National Security Agency and its research arm, IDA Center for Computing Sciences. He is survived by his wife of 63 years, Bernice, of Laguna Woods, California and Ft. Lauderdale, FL, his son Anthony, daughter Susan Cohn Boros, three grandchildren and one greatgranddaughter.
— Published in The New York Times on July 27, 2014
See also an autobiographical essay found on the web.
None of the above sources mention the following book, which is apparently by this same Harvey Cohn. (It is dedicated to "Tony and Susan.")
Advanced Number Theory, by Harvey Cohn
Courier Dover Publications, 1980 – 276 pages
(First published by Wiley in 1962 as A Second Course in Number Theory )
Publisher's description:
" 'A very stimulating book … in a class by itself.'— American Mathematical Monthly
Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.
The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups.
Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.
In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory."
User Review –
"In a nutshell, the book serves as an introduction to Gauss' theory of quadratic forms and their composition laws (the cornerstone of his Disquisitiones Arithmeticae) from the modern point of view (ideals in quadratic number fields). I strongly recommend it as a gentle introduction to algebraic number theory (with exclusive emphasis on quadratic number fields and binary quadratic forms). As a bonus, the book includes material on Dirichlet Lfunctions as well as proofs of Dirichlet's class number formula and Dirichlet's theorem in primes in arithmetic progressions (of course this material requires the reader to have the background of a onesemester course in real analysis; on the other hand, this material is largely independent of the subsequent algebraic developments).
Better titles for this book would be 'A Second Course in Number Theory' or 'Introduction to quadratic forms and quadratic fields'. It is not a very advanced book in the sense that required background is only a onesemester course in number theory. It does not assume prior familiarity with abstract algebra. While exercises are included, they are not particularly interesting or challenging (if probably adequate to keep the reader engaged).
While the exposition is *slightly* dated, it feels fresh enough and is particularly suitable for selfstudy (I'd be less likely to recommend the book as a formal textbook). Students with a background in abstract algebra might find the pace a bit slow, with a bit too much time spent on algebraic preliminaries (the entire Part I—about 90 pages); however, these preliminaries are essential to paving the road towards Parts II (ideal theory in quadratic fields) and III (applications of ideal theory).
It is almost inevitable to compare this book to BorevichShafarevich 'Number Theory'. The latter is a fantastic book which covers a large superset of the material in Cohn's book. BorevichShafarevich is, however, a much more demanding read and it is out of print. For gentle selfstudy (and perhaps as a preparation to later read BorevichShafarevich), Cohn's book is a fine read."
Or: The Nutshell
What about Pascal?
For some background on Pascal's mathematics,
not his wager, see…
Richmond, H. W.,
"On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics ,
Volume 31 (1900), pp. 125160,
dated by Richmond March 30,1899
Richmond, H. W.,
"The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen ,
Volume 53 (1900), Issue 12, pp 161176,
dated by Richmond February 1, 1899
See also Nocciolo in this journal.
Recall as well that six points in space may,
if constrained to lie on a circle, be given
a religious interpretation. Richmond's
six points are secular and more general.
See Coxeter + Aleph in this journal.
Epigraph to "The Aleph," a 1945 story by Borges:
"O God! I could be bounded in a nutshell,
and count myself a King of infinite space…"
– Hamlet, II, 2
The geometry posts of Sunday and Monday have been
placed in finitegeometry.org as
Classical Geometry in Light of Galois Geometry.
Some background:
See Baker, Principles of Geometry , Vol. II, Note I
(pp. 212218)—
On Certain Elementary Configurations, and
on the Complete Figure for Pappus's Theorem
and Vol. II, Note II (pp. 219236)—
On the Hexagrammum Mysticum of Pascal.
Monday's elucidation of Baker's Desarguestheorem figure
treats the figure as a 15_{4}20_{3 }configuration (15 points,
4 lines on each, and 20 lines, 3 points on each).
Such a treatment is by no means new. See Baker's notes
referred to above, and
"The Complete Pascal Figure Graphically Presented,"
a webpage by J. Chris Fisher and Norma Fuller.
What is new in the Monday Desargues post is the graphic
presentation of Baker's frontispiece figure using Galois geometry :
specifically, the diamond theorem square model of PG(3,2).
See also Cremona's kernel, or nocciolo :
Baker on Cremona's approach to Pascal—
"forming, in Cremona's phrase, the nocciolo of the whole."
A related nocciolo :
Click on the nocciolo for some
geometric background.
Randy Kennedy in tomorrow's print edition
of The New York Times—
Art collector Albert C. Barnes "viewed his foundation
less as a museum than as a school."
Roberta Smith in the New York Times
print edition of May 18, 2012, on
art arrangements by Albert C. Barnes—
"Barnes’s arrangements are as eyeopening,
intoxicating and, at times, maddening as ever, maybe more so.
They mix major and minor in relentlessly symmetrical patchworks
that argue at once for the idea of artistic genius and the
pervasiveness of talent. Nearly every room is an exhibition
unto itself— a kind of art wunderkammer, or cabinet of curiosities…."
This journal at noon on the same day, May 18, 2012—
Balakrishnan's Banners
See also Brightness at Noon from March 25.
Rachel Dodes in The Wall Street Journal
on All Souls' Day, 2012—
"In one of the first lines uttered by Daniel DayLewis, playing Abraham Lincoln in the new Steven Spielberg film opening Nov. 9, he says, 'I could be bounded in a nutshell, and count myself a king of infinite space— were it not that I have bad dreams.'
The line was ripped straight from 'Hamlet,' by Lincoln's favorite writer, William Shakespeare. Tony Kushner, the Pulitzer Prizewinning playwright ('Angels in America') who wrote the script for the film, says that Shakespeare, much like Lincoln, 'had extraordinary mastery over the darkest parts of the human spirit.'"
The above quotation omits Shakespeare's words prefacing the nutshell part— "O God."
These same words in a different tongue— "Hey Ram"— have often been quoted as the last words of Gandhi. (See yesterday's noon post.)
"… for the Highest Essence (brahman ),
which is the core of the world, is identical
with the Highest Self (ātman ), the kernel
of man's existence."
— Heinrich Zimmer, Myths and Symbols
in Indian Art and Civilization , Pantheon
Books, 1946, page 142
Related material: A post linked to here on Friday night
that itself links to a different Shakespeare speech.
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Commentary—
“Harriot has given no indication of how to resolve
such problems, but he has pasted in in English,
at the bottom of his page, these three enigmatic
lines:
‘Much ado about nothing.
Great warres and no blowes.
Who is the foole now?’
Harriot’s sardonic vein of humour, and the subtlety of
his logical reasoning still have to receive their full due.”
— “Minimum and Maximum, Finite and Infinite:
Bruno and the Northumberland Circle,” by Hilary Gatti,
Journal of the Warburg and Courtauld Institutes ,
Vol. 48 (1985), pp. 144163
Continued from Banderas (Aug. 18, 2011)—
Balakrishnan's Banners
See also The Colors of Halloween and Smiling Buddha.
In a nutshell —
Epigraph to "The Aleph," a 1945 story by Borges:
O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2
The story in book form, 1949
A 2006 biography of geometer H.S.M. Coxeter:
The Aleph (implicit in a 1950 article by Coxeter):
The details:
Related material: Group Actions, 19842009.
"What exactly was Point Omega?"
This is Robert Wright in Nonzero: The Logic of Human Destiny.
Wright is discussing not the novel Point Omega by Don DeLillo,
but rather a (related) concept of the Jesuit philosopher Pierre Teilhard de Chardin.
My own idiosyncratic version of a personal "point omega"—
The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."
For some other sorts of points, see the drawings
on the wall and Geometry Simplified—
The two points of the trivial affine space are represented by squares,
and the one point of the trivial projective space is represented by
a line segment separating the affinespace squares.
For related darkness at noon, see Derrida on différance
as a version of Plato's khôra—
The above excerpts are from a work on and by Derrida
published in 1997 by Fordham University,
a Jesuit institution— Deconstruction in a Nutshell—
For an alternative to the Villanova view of Derrida,
see Angels in the Architecture.
"The eye you see him with is the same
eye with which he sees you."
– Father Egan on page 333
of Robert Stone's A Flag for Sunrise
(Knopf hardcover, 1981)
Part I– Bounded in a Nutshell
Ian McKellen at a mental hospital's diamondshaped window in "Neverwas"
Part II– The Royal Castle
Ian McKellen at his royal castle's diamondshaped window in "Neverwas"
Part III– King of Infinite Space
H.S.M. Coxeter crowns himself "King of Infinite Space"
Related material:
See Coxeter in this journal.
“For every kind of vampire,
there is a kind of cross.”
— Thomas Pynchon in
Gravity’s Rainbow
“Since 1963, when Pynchon’s first novel, V., came out, the writer– widely considered America’s most important novelist since World War II– has become an almost mythical figure,
— Nancy Jo Sales in the November 11, 1996, issue of New York Magazine
(Click on images for their
source in past entries.)
In a Nutshell:
“Plato’s Ghost evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”
— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics (by Jeremy Gray, September 2008)

The Simplest Terms
“Broken down in the simplest terms, the story centres around two warring factions, the ‘Fathers’ and the ‘Friends.'”
Today’s birthdays:
Kirk Douglas,
Buck Henry,
John Malkovich.
In a nutshell:
The Soul’s Code and
today’s previous entry.
“The Ambition of the Short Story,” the essay by Steven Millhauser quoted here on Tuesday, September 30, is now online.
(See also Hamlet’s Transformation.)
The moral of this story,
it’s simple but it’s true:
Hey, the stars might lie,
but the numbers never do.
"I have another far more solid and central ground for submitting to it as a faith, instead of merely picking up hints from it as a scheme. And that is this: that the Christian Church in its practical relation to my soul is a living teacher, not a dead one. It not only certainly taught me yesterday, but will almost certainly teach me tomorrow. Once I saw suddenly the meaning of the shape of the cross; some day I may see suddenly the meaning of the shape of the mitre. One free morning I saw why windows were pointed; some fine morning I may see why priests were shaven. 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 tomorrow, 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 tomorrow at breakfast. He is always expecting to see some truth that he has never seen before."
— G. K. Chesterton, Orthodoxy, Ch. IX
From Plato, Pegasus, and the Evening Star (11/11/99):
"Nonbeing must in some sense be, otherwise what is it that there is not? This tangled doctrine might be nicknamed Plato's beard; historically it has proved tough, frequently dulling the edge of Occam's razor…. I have dwelt at length on the inconvenience of putting up with it. It is time to think about taking steps." "The Consul could feel his glance at Hugh becoming a cold look of hatred. Keeping his eyes fixed gimletlike upon him he saw him as he had appeared that morning, smiling, the razor edge keen in sunlight. But now he was advancing as if to decapitate him." 
"O God, I could be
bounded in a nutshell
and count myself
a king of infinite space,
were it not that
I have bad dreams."
— Hamlet
From today's newspaper:
Notes:
For an illustration of
the phrase "solid and central,"
see the previous entry.
For further context, see the
five Log24 entries ending
on September 6, 2006.
For background on the word
"hollow," see the etymology of
"hole in the wall" as well as
"The GodShaped Hole" and
"Is Nothing Sacred?"
For further ado, see
Macbeth, V.v
("signifying nothing")
and The New Yorker,
issue dated tomorrow.
"O God, I could be bounded in a nutshell
and count myself a king of infinite space,
were it not that I have bad dreams."
— Hamlet
Background:
"… Something have you heard
Of Hamlet's transformation; so call it,
Sith nor the exterior nor the inward man
Resembles that it was…."
The transformation:
Click on picture for details.
Related material:
Figures of Speech (June 7, 2006) and
Ursprache Revisited (June 9, 2006).
"The symmetric group S_{6} of permutations of 6 objects is the only symmetric group with an outer automorphism….
This outer automorphism can be regarded as the seed from which grow about half of the sporadic simple groups…."
This "seed" may be pictured as
within what Burkard Polster has called "the smallest perfect universe"– PG(3,2), the projective 3space over the 2element field.
Related material: yesterday's entry for Sylvester's birthday.
The following figure from a June 11, 1986, note illustrates Sylvester's "duads" and "synthemes" using the concept of an "inscape" (part B of the figure). As R. T. Curtis and Noam Elkies have explained, the duads and synthemes lead to constructions of many of the sporadic simple groups.
Inscape
My entry for New Year's Day links to a paper by Robert T. Curtis*
from The Arabian Journal for Science and Engineering
(King Fahd University, Dhahran, Saudi Arabia),
Volume 27, Number 1A, January 2002.
From that paper:
"Combinatorially, an outer automorphism [of S_{6}] can exist because the number of unordered pairs of 6 letters is equal to the number of ways in which 6 letters can be partitioned into three pairs. Which is to say that the two conjugacy classes of odd permutations of order 2 in S_{6} contain the same number of elements, namely 15. Sylvester… refers to the unordered pairs as duads and the partitions as synthemes. Certain collections of five synthemes… he refers to as synthematic totals or simply totals; each total is stabilized within S_{6} by a subgroup acting triply transitively on the 6 letters as PGL_{2}(5) acts on the projective line. If we draw a bipartite graph on (15+15) vertices by joining each syntheme to the three duads it contains, we obtain the famous 8cage (a graph of valence 3 with minimal cycles of length 8)…."
Here is a way of picturing the 8cage and a related configuration of points and lines:
Diamond Theory shows that this structure
can also be modeled by an "inscape"
made up of subsets of a
4×4 square array:
The illustration below shows how the
points and lines of the inscape may
be identified with those of the
CremonaRichmond configuration.
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.)
Reciprocity
From my entry of Sept. 1, 2003:
"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….
… 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, review of Himmelfarb, New York Times Book Review, October 30, 1994
Last year's entry on this date:
Today's birthday:
"Mathematics is the music of reason."
Sylvester, a nineteenthcentury mathematician, coined the phrase "synthematic totals" to describe some structures based on 6element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory. 
The picture above is of the complete graph
Diamond theory describes how the 15 twoelement subsets of a sixelement set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to grouptheoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M_{24}.
If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. "Reciprocity" in the sense of Lao Tzu. See
Reciprocity and Reversal in Lao Tzu.
For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in
Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the
Click on the design for details.
Those who prefer a Jewish approach to physics can find the star of David, in the form of
A Graphical Representation
of the Dirac Algebra.
The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.
Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See
Today's birthday: James Joseph Sylvester
"Mathematics is the music of reason." — J. J. Sylvester
Sylvester, a nineteenthcentury mathematician, coined the phrase "synthematic totals" to describe some structures based on 6element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory. See also the abstract of a December 7, 2000, talk, Mathematics and the Art of M. C. Escher, in which Curtis notes that graphic designs can "often convey a mathematical idea more eloquently than pages of symbolism."
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