Log24

Breach's 1981 approach is not axiomatic,
but instead graphic. Another such approach —

Also on December 16  (Click to enlarge) —

See instances of the title in this journal.

Material related to yesterday evening's post
"Bright and Dark at Christmas" —

The Buddha of Rochester:

See also the Gelman (i.e., Gell-Mann) Prize
in the film "Dark Matter" and the word "Eightfold"
in this journal.

" A fanciful mark is a mark which is invented
for the sole purpose of functioning as a trademark,
e.g., 'Kodak.' "

"… don't take my Kodachrome away." — Paul Simon

From a Dec. 21 obituary posted by the
University of Tennessee at Knoxville —

"Wade was ordained as a pastor and served
at Oakwood Baptist Church in Knoxville."

Other information —

In a Log24 post, "Seeing the Finite Structure,"
of August 16, 2008, Wade appeared as a co-author
of the Walsh series book mentioned above —

Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp et al.,
Taylor & Francis, 1990

From the 2008 post —

The patterns on the faces of the cube on the cover
of Walsh Series above illustrate both the 
Walsh functions of order 3 and the same structure
in a different guise, subspaces of the affine 3-space 
over the binary field. For a note on the relationship
of Walsh functions to finite geometry, see 
Symmetry of Walsh Functions.

Consider Stevens’s verse from “The Rock” (1954):

“That in which space itself is contained.” 

Consider also Whitehead in 1906 —

"This is proved by the consideration
of a three dimensional geometry in which
there are only fifteen points."

— and Stevens on the sublime (1935):

"And the sublime comes down
To the spirit itself,

The spirit and space,
The empty spirit
In vacant space."

See also a Terry Gilliam film on "crunching entities."

For some commentary,
see Spider in this journal.

*The Hudson of the title is the author of Kummer's Quartic Surface  (1905).
The Rosenhain of the title is the author for whom Hudson's 4×4 diagrams
of "Rosenhain tetrads" are named. For the "complexity to simplicity" of
the title, see Roger Fry in the previous post.

Cézanne "showed how it was possible to pass
from the complexity of the appearance of things
to the geometrical simplicity which design demands."

— Roger Fry in the catalogue for the 1910 London 
exhibition "Manet and the Post-Impressionists,"
according to …

See also A Roger Fry Reader 
(edited by Christopher Reed,
University of Chicago Press, 1996).

Commentary —

   Backstory:  Aug. 21, 2016, and Quora.com.

"After finishing high school in Miami,
Ms. Reno attended Cornell University,
graduating in 1960 with a degree in chemistry."

— The online New York Times  today

* Folk quotation.

  An example that is blatantly not  "just the facts," from a Cornell author
  found via last midnight's link "Ghost Light"  —

 
         — David W. Henderson, Cornell University

Or:  Pentagram Meets Counting-Pattern, Continued

Arts & Letters Daily  today links to a Chronicle of Higher Education
piece on philosophy with an illustration by the late Paul Laffoley …

This suggests a review of Laffoley's work. In particular —

For a larger view of the above Laffoley pentagram, click here.
Contrast with Wittgenstein's "counting-pattern" above, which
is, in fact, a hyperspace.

The previous post, Nobel Perspective, suggests a review of
the following passage pictured here on August 27, 2013.

Click image for a better view of the original.

There are, of course, more sophisticated approaches
to the place of perspective in the history of art.

"Sultan" was a pseudonym of Peter Lindbergh, now a 
well-known fashion photographer. Click image for the source.

Related art — Diamond Theory Roullete, by Radames Ajna,
2013 (Processing  code at ReCode Project based on
"Diamond Theory" by Steven H. Cullinane, 1977).

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).

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.

From "The Cerebral Savage," by Clifford Geertz —

(Encounter, Vol. 28 No. 4 (April 1967), pp. 25-32.)

From http://www.diamondspace.net/about.html —

"May, / The months [sic ] of understanding" — Wallace Stevens

Good question.

See Bedtime Story (Sept. 1, 2016).

A star figure and the Galois quaternion.

The square root of the former is the latter.

See also a search in this journal for "Set a Structure."

Harry Potter, star of the new film
"Imperium," with lightning-bolt
scar on his forehead

Fugue No. 21
B-Flat Major
Well-Tempered Clavier Book II
Johann Sebastian Bach
by Timothy A. Smith

Theme and Variations
by Steven H. Cullinane

The beginning of each —

Some context —

(Continued from April 2013 and later)

This is what I called "the large Desargues configuration" 
in posts of April 2013 and later.

The novel Blood on Snow , set in Oslo, was published
by Knopf on April 7, 2015.  This journal on that date —

A related image —

From this journal —

See (for instance) Sacred Order, July 18, 2006 —

From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —

See also Log24 on the publication date of Dark Matter .

"Benedict Cumberbatch Says a Journey
From Fact to Faith Is at the Heart of Doctor Strange"

— io9 yesterday

" 'This man comes from a binary universe where it’s all about logic,' 
the actor told us at San Diego Comic-Con . . . .

'And there’s a lot of humor in the collision between Easter [sic ]
mysticism and Western scientific, sort of logical binary.' "

Related material — Strange Awards, April 14, 2016.

I prefer a different sort of journey. See Boole vs. Galois.

"… which grounds the self" . . .

Popular Mechanics  online today —

"Verizon exec Marni Walden seemed to
indicate Mayer's future may still be up in the air."

See also 5×5 in this  journal —

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Those who want a serious  approach to the mathematics
of Clifford algebras — via finite geometry, the natural setting
of the four-group  of the previous post — should consult

"Finite Geometry, Dirac Groups and the Table of
Real Clifford Algebras," by Ron Shaw (1995).

Click to enlarge.

See also the word  "contrapuntal" in this journal. 

Somewhere Over the Rainbow …

"What I'm aiming for are moments of strong sensation —
moments of total physical experience of the landscape,
when weather just reaches out and sucks you in."
— The late Jane Wilson —

See also the previous post and, from the date of Wilson's death,

Geometry for Jews (Continued) —

The post Outer, Inner of July 16, 2016, contained the following
illustration of a quote from "An Ordinary Evening in New Haven" —

An image from yesterday morning pictured a link to the
Feb. 10, 2014, post Mystery Box III: Inside, Outside.

That post, shown below, offers a deeper interpretation of the
Stevens quote "an interior made exterior."

(Click image below to use the post's links.)

Further details from Edmund Hess in 1889* related to
last night's remarks on the Klein 60₁₅ configuration 
and the Kummer 16₆ configuration —

* Edmund Hess, "Beiträge zur Theorie der räumlichen Configurationen.
Ueber die Klein'sche Configuration Cf. (60₁₅, 30₆) und einige
bemerkenswerthe aus dieser ableitbare räumliche Configurationen."

Verhandlungen der Kaiserlichen Leopoldinisch-Carolinischen 
Deutschen Akademie der Naturforscher, Vol.55, No. 2, pp. 98-167