"….mirando il punto 
a cui tutti li tempi son presenti"
— Dante, Paradiso , XVII, 17-18
For instance…
Click image for higher quality.
"….mirando il punto 
a cui tutti li tempi son presenti"
— Dante, Paradiso , XVII, 17-18
For instance…
Click image for higher quality.
The title of yesterday's 11:22 PM post was "The Place of the Lion."
This is also the title of a novel by Charles Williams.
See, too, Midsummer Eve's Dream and Midsummer Night 2007.
For C. S. Lewis, who was born on this date in 1898,
and Natalie Wood, who died on this date in 1981
"He was accustomed to receiving manuscripts from strangers…."
— C. P. Snow on mathematician G. H. Hardy
"Whoever you are— I have always depended on
the kindness of strangers." — A Streetcar Named Desire
From this journal on September 24, 2012—
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"A single self-transcendence" — Aldous Huxley From an anonymous author at the website Kill Devil Hill— "This little story… has that climactic moment of Kill Devil Hills also appears in a 1983 film—
"Suppose it were possible to transfer — Trailer for "Brainstorm" (1983), |
C. P. Snow on G. H. Hardy, in Snow's foreword to A Mathematician's Apology—
"One morning early in 1913, he found, among the letters on his breakfast table, a large untidy envelope decorated with Indian stamps. When he opened it, he found sheets of paper by no means fresh, on which, in a non-English holograph, were line after line of symbols. Hardy glanced at them without enthusiasm. He was by this time, at the age of thirty-six, a world famous mathematician: and world famous mathematicians, he had already discovered, are unusually exposed to cranks. He was accustomed to receiving manuscripts from strangers, proving the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion, or the cryptograms that Bacon has inserted in the plays of the so-called Shakespeare."
Some related material (click to enlarge)—
The author links to, but does not name, the source of the above
"line after line of symbols." It is "Visualizing GL(2,p)." See that webpage
for some less esoteric background.
See also the two Wikipedia articles Finite geometry and Hesse configuration
and an image they share—

Quotes from the Bremen site
http://dada.compart-bremen.de/ —

" 'compArt | center of excellence digital art' is a project
at the University of Bremen, Germany. It is dedicated
to research and development in computing, design,
and teaching. It is supported by Rudolf Augstein Stiftung,
the University of Bremen, and Karin und Uwe Hollweg Stiftung."
See also Stiftung in this journal.
From a New York Times weblog last night—
The Reconstruction of Rome
By ROBERT BEASER
The New York Times , Nov. 27, 2012, 9:00 PM
Logic Pro software enables us to layer complex
technolike tracks and simulate meta, sampled orchestras—
fake orchestras that have on more than one occasion
fooled a jury of the most discriminating composers
into thinking it was the real thing. …
Several decades of cultural relativism has helped to
hasten the decline of the dominance of Western canon….
This next generation is becoming adept at taking small
bits of information, unformed, and assembling it
into larger asynchronous maps, of nonlinear order.
IT from BITS*
These failures of number agreement—
orchestras… it, decades… has, bits… it —
suggest a look at synesis.
Synesis is a traditional grammatical/rhetorical term
derived from Greek σύνεσις (originally meaning "unification,
meeting, sense, conscience, insight, realization, mind, reason").
A constructio kata synesin (or constructio ad sensum in Latin)
means a grammatical construction in which a word takes
the gender or number not of the word with which it should
regularly agree, but of some other word implied in that word.
It is effectively an agreement of words with the sense,
instead of the morphosyntactic form. Example:
"If the band are popular, they will play next month." —Wikipedia
The conclusion of Wikipedia's synesis article is of particular interest:
See also…. Elohim , a Hebrew word whose number varies.
* A nod to the late John Archibald Wheeler.
A pdf of a 1977 three-page article with this title
has been added at finitegeometry.org/sc.
The non-Coxeter simple reflection group of order 168
is a counterexample to the statement that
"Every finite reflection group is a Coxeter group."
The counterexample is based on a definition of "reflection group"
that includes reflections defined over finite fields.
Today I came across a 1911 paper that discusses the counterexample.
Of course, Coxeter groups were undefined in 1911, but the paper, by
Howard H. Mitchell, discusses the simple order-168 group as a reflection group .
(Naturally, Mitchell's definition of "reflection" and his statement that
"The discussion of the binary groups
applies also to the case p = 2."
should be approached with care.)
A review of this topic might be appropriate for Jessica Fintzen's 2012 fall tutorial at Harvard
on reflection groups and Coxeter groups. The syllabus for the tutorial states that
"finite Coxeter groups correspond precisely to finite reflection groups." This statement
is based on Fintzen's definition of "reflection group"—
"Reflection groups are— as their name indicates—
groups generated by reflections across
hyperplanes of Rn which contain the origin."
For some background, see William Kantor's 1981 paper "Generation of Linear Groups"
(quoted at the finitegeometry.org page on the simple order-168 counterexample).
Kantor discusses Mitchell's work in some detail, but does not mention the
simple order-168 group explicitly.
… Meets "The Master"—

Today's midday NY Lottery: 333 and 5885.
"Continue a search for thirty-three and three." — The Eight (1988)
"Make me young." — Kilgore Trout in
Breakfast of Champions . Trout was modeled after
author Theodore Sturgeon… who died on 5/8/85.
(An example of Sturgeon's work: The Dreaming Jewels (1950).)
Related illustrations from the eighth day of 2012—
See also "I'm sorry to be catechizing you like this."
The New York Lottery:
CATCH OUR LIVE DRAWINGS:
NUMBERS and WIN 4
Draw Day: Twice Daily
Draw Time: Midday: 12:20 p.m. – 12:30 p.m.
Evening: 7:30 p.m. – 7:40 p.m.
NY Lottery this evening: 674 and 1252 —
.
Contrapuntal themes:
Related Log24 posts today—
at 11:30 AM ET (see post 674 ) and
at 7 PM ET (see post 1252).
The Devil at Midday:
Interpreting the midday numbers,
172 and 7817, is more difficult. Perhaps 172
refers to a Zen page number in a post from
the Feast of Saint Louis in 2003, and perhaps,
in a less saintly manner, 7817 refers to
two posts in which these four digits appear
in product numbers within links— namely,
the Garden Party "Background" link and
the Seven Bridges "wild" link.
Then again, perhaps not.
"It's a grim joke." — Amy Adams in "The Master"
When Irish Eyes Are Smiling…
Click diagram for some background from 3/17.
See, too, some background on Amy Adams and on Leap Day.
For related Harvard humor, see Venn Diagram.
… For the Boston Church of the Advent coffee hour:
Suggested topic:
The Klee picture on the above cup— a graphic rendition of the poem
Once Emerged from the Gray of Night — as presented on the cover
and discussed in the text of Martha B. Helfer's The Word Unheard .
Talk amongst yourselves.
Today's sermon is for Martha B. Helfer, author of
the treatise on Darstellung in today's previous post
and of the following—
(Click for clearer image.)
Helfer's The Word Unheard was published by Northwestern University Press
on St. Andrew's Day, 2011. Log24 posts on that day—
Lines, Grids, and Fatuity for St. Andrew's Day.
The last of these warned of an upcoming Jewish Book Week event
on February 22, 2012.
That date turned out to be Ash Wednesday. See a Log24 post on that topic
that quotes a poet, T.S. Eliot, with anti-Semitic proclivities—
"And the light shone in darkness and
Against the Word the unstilled world still whirled
About the centre of the silent Word."
— T. S. Eliot, "Ash Wednesday"
This is perhaps not entirely irrelevant to Helfer's title, The Word Unheard .
* A concept of Schopenhauer and Hitler, and the first name of
a fictional Boston mathematician.
The title of yesterday's post Will and Representation is of course
a reference to Schopenhauer's philosophical work of that name.
As the post itself indicates, the title is also a punning reference to
mathematical representation theory .
To avoid confusion, it should be noted that Schopenhauer's
representation , in the original German, was Vorstellung .
The German for mathematical representation theory is,
on the other hand, Darstellungstheorie . (The mathematical
use of Vorstellung is non-technical, referring to concepts
of pedagogy. (A group presentation is a Präsentation .))
For a discussion of the Vorstellung-Darstellung distinction
in philosophy, not mathematics, see…
The Retreat of Representation: The Concept of Darstellung
in German Critical Discourse , by Martha B. Helfer,
State University of New York Press, 1996, esp. pp. 24-26.
Robert A. Wilson, in an inaugural lecture in April 2008—
Representation theory
A group always arises in nature as the symmetry group of some object, and group
theory in large part consists of studying in detail the symmetry group of some
object, in order to throw light on the structure of the object itself (which in some
sense is the “real” object of study).
But if you look carefully at how groups are used in other areas such as physics
and chemistry, you will see that the real power of the method comes from turning
the whole procedure round: instead of starting from an object and abstracting
its group of symmetries, we start from a group and ask for all possible objects
that it can be the symmetry group of .
This is essentially what we call Representation theory . We think of it as taking a
group, and representing it concretely in terms of a symmetrical object.
Now imagine what you can do if you combine the two processes: we start with a
symmetrical object, and find its group of symmetries. We now look this group up
in a work of reference, such as our big red book (The ATLAS of Finite Groups),
and find out about all (well, perhaps not all) other objects that have the same
group as their group of symmetries.
We now have lots of objects all looking completely different, but all with the same
symmetry group. By translating from the first object to the group, and then to
the second object, we can use everything we know about the first object to tell
us things about the second, and vice versa.
As Poincaré said,
Mathematicians do not study objects, but relations between objects.
Thus they are free to replace some objects by others, so long as the
relations remain unchanged.
Fano plane transformed to eightfold cube,
and partitions of the latter as points of the former:
* For the "Will" part, see the PyrE link at Talk Amongst Yourselves.
For the title, see the phrase "reappearing number" in this journal.
Some related mathematics—
the Greek labyrinth of Borges, as well as…
Note that "0" here stands for "23," while ∞ corresponds to today's date.
New today…
John Markoff in this morning's New York Times—
"…what is new in recent months is the growing speed and accuracy
of deep-learning programs, often called artificial neural networks
or just 'neural nets' for their resemblance to the neural connections
in the brain."
Not so new…
From the early days of Mike Lynch's Autonomy Inc.—
"Autonomy 's intelligent agents use neural networking
to search for patterns of information, rather than
specific words or phrases, thereby distinguishing relevant
from irrelevant information." (Business Wire , October 8, 1996)


For the answer, click here.
Connoisseurs of synchronicity may note two posts
from the date the above photo was taken:
First Draft of History and The Swedish Solution.
(The late Larry Hagman was, according to NBC News, "a longtime member
of the Peace and Freedom Party, a minor leftist organization in California.")
(Continued from 1986)
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S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: A 4×4 array. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
A representative coordinatization:
0000 0001 0010 0011
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.
Jennifer Scott at IT Pro , Feb. 16, 2012, on Autonomy—
Mike Lynch, founder of Autonomy and vice president
of information management at HP, took to the stage
at his new parent company’s global partner conference
to impart his philosophy to the 3,000 partners gathered.
‘It is no longer about the data but about the meaning
of that data,’ he said. ‘There is a fundamental revolution
going on in information and the industry is now about
the “I” not the “T” in IT.'”
Click on the logo below for the source.
See also today’s previous post and…
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“After A Wrinkle in Time was finally published, |
"Any sufficiently advanced technology is indistinguishable from magic."
"The HP/Autonomy Debacle," by John C. Dvorak at pcmag.com on Tuesday, Nov. 20, 2012—
"The whole Autonomy thing was weird since the company seemed to be performing magic. On co-founder Michael Richard Lynch's Wikipedia page, the company is described as 'a leader in the area of computer understanding of unstructured information, an area which is becoming known as meaning-based computing .'
I do not know how gullible HP's board of directors is, but when I see the sudden emergence of something called 'meaning-based computing,' the alarms sound and the bullcrap meter begins to tag the red line."
A story by Terence K. Huwe in Online magazine, Sept.-Oct. 2011, defines meaning-based computing (MBC), discusses Autonomy , and llnks to…
John Markoff in The New York Times , March 4, 2011—
"Engineers and linguists at Cataphora, an information-sifting company based in Silicon Valley, have their software mine documents for the activities and interactions of people— who did what when, and who talks to whom. The software seeks to visualize chains of events. It identifies discussions that might have taken place across e-mail, instant messages and telephone calls.
Then the computer pounces, so to speak, capturing 'digital anomalies' that white-collar criminals often create in trying to hide their activities.
For example, it finds 'call me' moments— those incidents when an employee decides to hide a particular action by having a private conversation. This usually involves switching media, perhaps from an e-mail conversation to instant messaging, telephone or even a face-to-face encounter."
For example…

Jonathan Weil at Bloomberg.com yesterday
on the HP-Autonomy story—
"The goodwill figure is especially telling."
From remarks by Wallace Stevens featured in recent Log24 posts:
"The poet finds that as between these two sources: the imagination and reality, the imagination is false, whatever else may be said of it, and reality is true; and being concerned that poetry should be a thing of vital and virile importance, he commits himself to reality, which then becomes his inescapable and ever-present difficulty and innamorata. In any event, he has lost nothing; for the imagination, while it might have led him to purities beyond definition, never yet progressed except by particulars. Having gained the world, the imaginative remains available to him in respect to all the particulars of the world. Instead of having lost anything, he has gained a sense of direction and a certainty of understanding. He has strengthened himself to resist the bogus."
— Bard College speech of 1951
Related material:
With Autonomy , H-P Bought An Old-Fashioned
Accounting Scandal. Here's How It Worked.
(Forbes , Nov. 20, 2012)
Hewlett-Packard is losing a star in Mike Lynch
(The Guardian , May 25, 2012)
The Quest for Meaning: The world's smartest search engine
took 250 years to build. Autonomy is here.
(Wired , Feb. 2000)
"And forth the particulars of rapture come."
— Wallace Stevens, "Notes Toward a Supreme Fiction,"
Canto IV of "It Must Change," quoted here yesterday.
A death yesterday: Sir Philip Ledger.
Happy birthday to…

Today's sermon, by Marie-Louise von Franz—
For more on the modern physicist analyzed by von Franz,
see The Innermost Kernel , by Suzanne Gieser.
Another modern physicist, Niels Bohr, died
on this date in 1962…
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The circle above is marked with a version For the square, see the diamond theorem. "Two things of opposite natures seem to depend — Wallace Stevens, |
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