Log24

Yesterday evening's post Some Old Philosophy from Rome
(a reference, of course, to a Wallace Stevens poem)
had a link to posts now tagged Wittgenstein's Pentagram.

For a sequel to those posts, see posts with the term Inscape ,
a mathematical concept related to a pentagram-like shape.

The inscape concept is also, as shown by R. W. H. T. Hudson
in 1904, related to the square array of points I use to picture
PG(3,2), the projective 3-space over the 2-element field.

"… what we’re witnessing is not a glitch. It’s a feature…."

— A Boston Globe  columnist on June 19.

An image from this  journal at the beginning of Bloomsday 2018 —

An encountered feature , from the midnight beginning of June 16 —

Literary Symbolism

"… what we’re witnessing is not a glitch. It’s a feature…."

The glitch  encountered on Bloomsday by Agent Smith (who represents 
the academic world) is the author  of the above page, John P. Anderson.
The feature  is the book  that Anderson quotes, James Joyce 
by Richard Ellmann (first published in 1959, revised in 1982).

See also Inscape in this journal and, for a related Chapel Hill thesis,
the post Kummer and Dirac.

From Monday morning's post Advanced Study —

"Mathematical research currently relies on
a complex system of mutual trust
based on reputations."

— The late Vladimir Voevodsky,
Institute for Advanced Study, Princeton,
The Institute Letter , Summer 2014, p. 8

Related news from today's online New York Times —

A heading from the above screenshot: "SHOW US YOUR WALL."

This suggests a review of a concept from Galois geometry —

(On the wall — a Galois-geometry inscape .)

John Horgan and James (Jim) McClellan, according to Horgan
in Scientific American  on June 1, 2017 —

See also posts tagged Dirac and Geometry and Glitch.

The Ball-Weiner date above, 5 September 2011,
suggests a review of this journal on that date —

"Think of a DO NOT ENTER pictogram,
a circle with a diagonal slash, a type of ideogram.
It tells you what to do or not do, but not why.
The why is part of a larger context, a bigger picture."

— Customer review at Amazon.com

This passage was quoted here on August 10, 2009.

Also from that date:

The Sept. 5, 2011, Ball-Weiner paper illustrates the
"doily" view of the mathematical structure W(3,2), also
known as GQ(2,2), the Sp(4,2) generalized quadrangle.
(See Fig. 3.1 on page 33, exercise 13 on page 38, and
the answer to that exercise on page 55, illustrated by 
Fig. 5.1 on page 56.)

For "another view, hidden yet true," of GQ(2,2),
see Inscape and Symplectic Polarity in this journal.

Baker, Principles of Geometry, Vol. IV  (1925), Title:

Baker, Principles of Geometry, Vol. IV  (1925), Frontispiece:

Baker's Vol. IV frontispiece shows "The Figure of fifteen lines 
and fifteen points, in space of four dimensions."

Another such figure in a vector space of four dimensions
over the two-element Galois field  GF(2):

(Some background grid parts were blanked by an image resizing process.)

Here the "lines" are actually planes  in the vector 4-space over GF(2),
but as planes through the origin  in that space, they are projective  lines .

For some background, see today's previous post and Inscapes.

Update of 9:15 PM March 31—

The following figure relates the above finite-geometry
inscape  incidences to those in Baker's frontispiece. Both the inscape
version and that of Baker depict a Cremona-Richmond configuration.

A related image search:

Note particularly the following image:

This is from Inscapes.

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.

For T.S. Eliot's Birthday

Last night's post "Transformation" was suggested in part
by the title of a Sunday New York Times  article on
George Harrison, "Within Him, Without Him," and by
the song title "Within You Without You" in the post
Death and the Apple Tree.

Related material— "Hamlet's Transformation"—

Hamlet, 2.2:

Something have you heard
Of Hamlet’s transformation; so call it,
Sith nor the exterior nor the inward man
Resembles that it was….”

A transformation:

Click on picture for details.

See also, from this year's Feast of the Transfiguration,
Correspondences and Happy Web Day.

For those who prefer the paganism of Yeats to
the Christianity of Eliot, there is the sequel to
"Death and the Apple Tree," "Dancers and the Dance."

Today is the birthday of mathematician Jean-Pierre Serre.

Some remarks related to today's day number within the month, "15"—

The Wikipedia article on finite geometry has the following link—

Carnahan, Scott (2007-10-27), "Small finite sets", Secret Blogging Seminar, http://sbseminar.wordpress.com/2007/10/27/small-finite-sets/, notes on a talk by Jean-Pierre Serre on canonical geometric properties of small finite sets.

From Carnahan's notes (October 27, 2007)—

Serre has been giving a series of lectures at Harvard for the last month, on finite groups in number theory. It started off with some ideas revolving around Chebotarev density, and recently moved into fusion (meaning conjugacy classes, not monoidal categories) and mod p representations. In between, he gave a neat self-contained talk about small finite groups, which really meant canonical structures on small finite sets.

He started by writing the numbers 2,3,4,5,6,7,8, indicating the sizes of the sets to be discussed, and then he tackled them in order.

Related material on finite geometry and the indicated small numbers may, with one apparent exception, be found at my own Notes on Finite Geometry.

The apparent exception is "5." See, however, the role played in finite geometry by this number (and by "15") as sketched by Robert Steinberg at Yale in 1967—

See also …

(Click to enlarge.)

"Instead of a million count half a dozen." —Walden

"Of all the symmetric groups, S₆ is perhaps the most remarkable."
— Notes 2 (Autumn 2008), apparently by Robert A. Wilson,
   for Group Theory, MTH714U

For a connection of MTH714U with Walden, see "Window, continued."

For a connection of "Window" with the remarkable S₆, see Inscapes.

For some deeper background, see Wilson's "Exceptional Simplicity."

Jung’s Birthday, 5 AM:

Skewed Mirrors

Readings on Aesthetics for the
Feast of the Triumph of the Cross

Part I —

Bill Moyers and Julie Taymor

Director Taymor on her own passion play (see previous entry), “Frida“: 

“We always write stories of tragedies because that’s how we reach our human depth. How we get to the other side of it. We look at the cruelty, the darkness and horrific events that happened in our life whether it be a miscarriage or a husband who is not faithful. Then you find this ability to transcend. And that is called the passion, like the passion of Christ. You could call this the passion of Frida Kahlo, in a way.”

— 10/25/02 interview with Bill Moyers

Part II —

 Inside and Outside: Transformation

(Research note, July 11, 1986)

Click on the above typewritten note to enlarge.

Summary of
Parts I and II:

See also
Geometry for Jews.

“We’re not here to stick a mirror on you. Anybody can do that, We’re here to give you a more cubist or skewed mirror, where you get to see yourself with fresh eyes. That’s what an artist does. When you paint the Crucifixion, you’re not painting an exact reproduction.”

— Julie Taymor on “Frida” (AP, 10/22/02)

“She made ‘real’ an oxymoron, 
         she made mirrors, she made smoke.
She had a curve ball
          that wouldn’t quit,
                              a girlfriend for a joke.”

— “Arizona Star,” Guy Clark / Rich Alves