Log24

Geometer H. S. M. Coxeter died on this date in 2003.

This evening’s daily number from the Keystone state:   822.

The following excerpts from Coxeter's Projective Geometry
sketch his attitude toward geometry in characteristic two.
"… we develop a self-contained account… made
more 'modern' by allowing the field to be general
(though not of characteristic 2) instead of real or complex."

The "modern" in quotation marks may have been an oblique
reference to Segre's Lectures on Modern Geometry  (1948, 1961).
(See Coxeter's reference 15 below.)

Click to enlarge.

"It is interesting to see what happens…."

Another thing that happens if 1 + 1 = 0 —

It is no longer true that every finite reflection group
is a Coxeter group (provided we use Chevalley's
fixed-hyperplane definition of "reflection").

The discovery of "square ice" is discussed in
Nature 519, 443–445 (26 March 2015).

Remarks related, if only by squareness —
this  journal on that same date, 26 March 2015 —

The above figure is part of a Log24 discussion of the fact that 
adjacency in the set of 16 vertices of a hypercube is isomorphic to
adjacency in the set of 16 subsquares of a square 4×4 array,
provided that opposite sides of the array are identified.  When
this fact was first  observed, I do not know. It is implicit, although
not stated explicitly, in the 1950 paper by H.S.M. Coxeter from
which the above figure is adapted (blue dots added).

Elijah Wood in "The Last Witch Hunter" —

See Graham Priest in this journal.

See also Coxeter + Discursive.

Symplectic.

Related material:

From the website of the American Mathematical Society today,
a column by John Baez that was falsely backdated to Sept. 1, 2015 —

Compare and contrast this Baez column 
with the posts in the above
Log24 search for "Symplectic."

Updates after 9 PM ET Sept. 17, 2015 —

Related wrinkles in time: 

Baez's preceding Visual Insight  post, titled 
"Tutte-Coxeter Graph," was dated Aug. 15, 2015.
This seems to contradict the AMS home page headline
of Sept. 5, 2015, that linked to Baez's still earlier post
"Heawood Graph," dated Aug. 1. Also, note the 
reference in "Tutte-Coxeter Graph" to Baez's related 
essay — dated August 17, 2015 — 

• "A Wrinkle in the Mathematical Universe" 
   at The n-Category Café .

(Continued)

A post of July 7, Haiku for DeLillo, had a link to posts tagged "Holy Field GF(3)."

As the smallest Galois field based on an odd prime, this structure 
clearly is of fundamental importance.  

It is, however, perhaps too  small  to be visually impressive.

A larger, closely related, field, GF(9), may be pictured as a 3×3 array…

… hence as the traditional Chinese  Holy Field.

Marketing the Holy Field

The above illustration of China's  Holy Field occurred in the context of
Log24 posts on Child Buyers.   For more on child buyers, see an excellent
condemnation today by Diane Ravitch of the U. S. Secretary of Education.

( A Chinese designation for the 3×3 square )

From an obituary for a Kennedy advisor
who reportedly died at 94 on February 23, 2014*—

“He favored withdrawing rural civilians
into what he called ‘strategic hamlets’
and spraying defoliants to cut off
the enemy’s food supply.”

Other rhetoric:  Hamlet and Infinite Space  in this journal,
as well as King of Infinite Space , Part I and Part II.

These “King” links, to remarks on Coxeter  and  Saniga ,
are about two human beings to whom Hamlet’s
phrase “king of infinite space” has been applied.

The phrase would, of course, be more accurately
applied to God.

* The date of the ‘God’s Architecture’ sermon
at Princeton discussed in this afternoon’s post.

Spidey Goes to Church

More realistically…

1. "Nick Bostrom … is a Swedish philosopher at 
   St. Cross College, University of Oxford…."
2. "The early location of St Cross was on a site in
   St Cross Road, immediately south of St Cross Church."
3. "The church building is located on St Cross Road
   just south of Holywell Manor."
4. "Balliol College has had a presence in the area since
   the purchase by Benjamin Jowett, the Master, in the 1870s
   of the open area which is the Balliol sports ground
   'The Master's Field.' "
5. Leaving Wikipedia, we find a Balliol field at Log24:
   
6. A different view of the same field, from 1950—
   .
7. A view from 1974, thanks to J. J. Seidel — 
   
8. Yesterday's Analogies.

Today's previous post, "For Odin's Day," discussed 
a mathematical object, the tesseract, from a strictly
narrative point of view.

In honor of George Balanchine, Odin might yield the
floor this evening to Apollo.

From a piece in today's online New York Times  titled
"How a God Finds Art (the Abridged Version)"—

"… the newness at the heart of this story,
in which art is happening for the first time…."

Some related art—

and, more recently—

This more recent figure is from Ian Stewart's 1996 revision 
of a 1941 classic, What Is Mathematics? , by Richard Courant
and Herbert Robbins.

Apollo might discuss with Socrates how the confused slave boy
of Plato's Meno  would react to Stewart's remark that

"The number of copies required to double an
 object's size depends on its dimension."

Apollo might also note an application of Socrates' Meno  diagram
to the tesseract of this afternoon's Odin post—

.

The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title, 
that of a novel by the author of The Exorcist .

The Ninth Configuration —

The ninth* in a list of configurations—

"There is a (2^d-1)_d  configuration
  known as the Cox configuration."

— MathWorld article on "Configuration"

For further details on the Cox 32₆ configuration's Levi graph,
a model of the 64 vertices of the six-dimensional hypercube γ₆  ,
see Coxeter, "Self-Dual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc.  Vol. 56, pages 413-455, 1950.
This contains a discussion of Kummer's 16₆ as it 
relates to  γ₆  , another form of the 4×4×4 Galois cube.

See also Solomon's Cube.

* Or tenth, if the fleeting reference to 11₃ configurations is counted as the seventh—
  and then the ninth  would be a 15₃ and some related material would be Inscapes.

The hypercube has 192 rotational symmetries.
Its full symmetry group, including reflections,
is of order 384.

See (for instance) Coxeter—

Related material—

The rotational symmetry groups of the Platonic solids
(from April 25, 2011)—

— and the figure in yesterday evening's post on the hypercube—

(Animation source: MIQEL.com)

Clearly hypercube rotations of this sort carry any
of the eight 3D subcubes to the central subcube
of a central projection of the hypercube—

The 24 rotational symmeties of that subcube induce
24 rigid rotations of the entire hypercube. Hence,
as in the logic of the Platonic symmetry groups
illustrated above, the hypercube has 8 × 24 = 192
rotational symmetries.

Suggested by an Oct. 18 piece in the Book Bench section
of the online New Yorker  magazine—

Related material suggested by the "Shouts and Murmurs" piece
in The New Yorker , issue dated Oct. 24, 2011—

"a series of e-mails from a preschool teacher planning to celebrate
the Day of the Dead instead of Halloween…"

A search for Coxeter + Graveyard in this journal yields…

Here the tombstone says "GEOMETRY… 600 BC — 1900 AD… R.I.P."

A related search for Plato + Tombstone yields an image from July 6, 2007…

Here Plato's poems to Aster suggested
the "Star and Diamond" tombstone.

The eight-rayed star is an ancient symbol of Venus
and the diamond is from Plato's Meno .

The star and diamond are combined in a figure from
12 AM on September 6th, 2011—

"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 8₃
are joined by four concurrent lines."
— H. S. M. Coxeter (see below)

Continued from Tuesday, Sept. 6—

For the title, see Palm Sunday.

"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987

From this date (April 22) last year—

Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"– "… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions: (1) Diamonds– informative, certain truths about the world– exist. (2) The theorems of Euclidean geometry are diamonds. Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry." Trudeau's book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory." Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds. * "Non-Euclidean" here means merely "other than  Euclidean." No violation of Euclid's parallel postulate is implied.

Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."

Geometry, a part of pure mathematics, is not  about the world. See G. H. Hardy, A Mathematician's Apology .

From last night's note on finite geometry—

"The (8₃, 8₃) Möbius-Kantor configuration here described by Coxeter is of course part of the larger (9₄, 12₃) Hesse configuration. Simply add the center point of the 3×3 Galois affine plane and the four lines (1 horizontal, 1 vertical, 2 diagonal) through the center point." An illustration—

This suggests a search for "diamond+star."

In memory of Jane Russell —

H.S.M. Coxeter's classic
Introduction to Geometry  (2nd ed.):

Note the resemblance of the central part to
a magical counterpart— the Ojo de Dios
of Mexico's Sierra Madre.

Related material— page 55 of Polly and the Aunt ,
by Mary E. Blatchford.

   For aficionados of mathematics and narrative —

Illustration from
"The Galois Quaternion— A Story"

This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—

The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean  plane, but rather with unit squares
representing points in a finite Galois  affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.

See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.

Google Logo July 11, 2010—

"Oog" is Dutch (and Afrikaans) for "eye."

Strong Emergence Illustrated
(May 23, 2007 — Figures from Coxeter)—

The 2007 "strong emergence" post compares the
center figure to an "Ojo de Dios."

Jamie James in The Music of the Spheres—

"The Pythagorean philosophy, like Zoroastrianism, Taoism, and every early system of higher thought, is based upon the concept of dualism. Pythagoras constructed a table of opposites from which he was able to derive every concept needed for a philosophy of the phenomenal world. As reconstructed by Aristotle in his Metaphysics, the table contains ten dualities (ten being a particularly important number in the Pythagorean system, as we shall see):

Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited (man, finite time, and so forth) and the unlimited (the cosmos, eternity, etc.) is not only the aim of Pythagoras's system but the central aim of all Western philosophy."

"Simplify, simplify." — Henry David Thoreau

"Because of their truly fundamental role in mathematics, even the simplest diagrams concerning finite reflection groups (or finite mirror systems, or root systems– the languages are equivalent) have interpretations of cosmological proportions."

— Alexandre Borovik, 2010 (See previous entry.)

Exercise: Discuss Borovik's remark
that "the languages are equivalent"
in light of the web page

A Simple Reflection Group
of Order 168.

Background:

Theorems 15.1 and 15.2 of Borovik's book (1st ed. Nov. 10, 2009)
Mirrors and Reflections: The Geometry of Finite Reflection Groups—

15.1 (p. 114): Every finite reflection group is a Coxeter group.

15.2 (p. 114): Every finite Coxeter group is isomorphic to a finite reflection group.

Consider in this context the above simple reflection group of order 168.

(Recall that "…there is only one simple Coxeter group (up to isomorphism); it has order 2…" —A.M. Cohen.)

Wikipedia:

“In the Roman Catholic tradition, the term ‘Body of Christ’ refers not only to the body of Christ in the spiritual realm, but also to two distinct though related things: the Church and the reality of the transubstantiated bread of the Eucharist….

According to the Catechism of the Catholic Church, ‘the comparison of the Church with the body casts light on the intimate bond between Christ and his Church. Not only is she gathered around him; she is united in him, in his body….’

….To distinguish the Body of Christ in this sense from his physical body, the term ‘Mystical Body of Christ’ is often used. This term was used as the first words, and so as the title, of the encyclical Mystici Corporis Christi of Pope Pius XII.”

Pope Pius XII:

“83. The Sacrament of the Eucharist is itself a striking and wonderful figure of the unity of the Church, if we consider how in the bread to be consecrated many grains go to form one whole, and that in it the very Author of supernatural grace is given to us, so that through Him we may receive the spirit of charity in which we are bidden to live now no longer our own life but the life of Christ, and to love the Redeemer Himself in all the members of His social Body.”