Log24

Sunday, March 12, 2017

Raise High the Ridgepole, Architects*

Filed under: Uncategorized — m759 @ 2:40 PM

A post suggested by remarks of J. D. Salinger in 
The New Yorker  of November 19, 1955 —

Wikipedia:  Taiji (philosophy)

Etymology

The word 太極 comes from I Ching : "易有太極,是生兩儀,兩儀生四象,四象生八卦,八卦定吉凶,吉凶生大業。"

Taiji  (太極) is a compound of tai   "great; grand; supreme; extreme; very; too" (a superlative variant of da   "big; large; great; very") and ji   "pole; roof ridge; highest/utmost point; extreme; earth's pole; reach the end; attain; exhaust". In analogy with the figurative meanings of English pole, Chinese ji  極 "ridgepole" can mean "geographical pole; direction" (e.g., siji  四極 "four corners of the earth; world's end"), "magnetic pole" (Beiji  北極 "North Pole" or yinji  陰極 "negative pole; cathode"), or "celestial pole" (baji  八極 "farthest points of the universe; remotest place"). Combining the two words, 太極 means "the source, the beginning of the world".

Common English translations of the cosmological Taiji  are the "Supreme Ultimate" (Le Blanc 1985, Zhang and Ryden 2002) or "Great Ultimate" (Chen 1989, Robinet 2008); but other versions are the "Supreme Pole" (Needham and Ronan 1978), "Great Absolute", or "Supreme Polarity" (Adler 1999).

See also Polarity in this journal.

* A phrase adapted, via Salinger,
from a poem by Sappho

Ἴψοι δὴ τὸ μέλαθρον,
     Υ᾽μήναον
ἀέρρετε τέκτονεσ ἄνδρεσ,
     Υ᾽μήναον
γάμβροσ ἔρχεται ἶσοσ Ά᾽ρευϊ,
     [Υ᾽μήναον]
ανδροσ μεγάλο πόλυ μείζων
     [Υ᾽μήναον]

Friday, May 13, 2016

Geometry and Kinematics

Filed under: Uncategorized — Tags: — m759 @ 10:31 PM

"Just as both tragedy and comedy can be written
by using the same letters of the alphabet, the vast
variety of events in this world can be realized by
the same atoms through their different arrangements
and movements. Geometry and kinematics, which
were made possible by the void, proved to be still
more important in some way than pure being."

— Werner Heisenberg in Physics and Philosophy

For more about geometry and kinematics, see (for instance)

"An introduction to line geometry with applications,"
by Helmut Pottmann, Martin Peternell, and Bahram Ravani,
Computer-Aided Design  31 (1999), 3-16.

The concepts of line geometry (null point, null plane, null polarity,
linear complex, Klein quadric, etc.) are also of interest in finite  geometry.
Some small finite spaces have as their natural models arrays of cubes .

Saturday, November 21, 2015

The Zero System

Filed under: Uncategorized — Tags: — m759 @ 12:00 AM

For the title phrase, see Encyclopedia of Mathematics .
The zero system  illustrated in the previous post*
should not be confused with the cinematic Zero Theorem .

* More precisely, in the part showing the 15 lines fixed under
   a zero-system polarity in PG(3,2).  For the zero system 
   itself, see diamond-theorem correlation.

Friday, November 13, 2015

A Connection between the 16 Dirac Matrices and the Large Mathieu Group

Filed under: Uncategorized — Tags: — m759 @ 2:45 AM



Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation
 
). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.

References:

Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Related material:

The 6-set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —

Background reading:

Ron Shaw on finite geometry, Clifford algebras, and Dirac groups 
(undated compilation of publications from roughly 1994-1995)—

Saturday, July 4, 2015

Context

Filed under: Uncategorized — Tags: , — m759 @ 10:00 AM

Some context for yesterday's post on a symplectic polarity —

This 1986 note may or may not have inspired some remarks 
of Wolf Barth in his foreword to the 1990 reissue of Hudson's
1905 Kummer's Quartic Surface .

See also the diamond-theorem correlation.  

Friday, July 3, 2015

Crunching Entities*

Filed under: Uncategorized — m759 @ 9:19 PM

A figure I prefer to the "Golden Tablet" of Night at the Museum —

IMAGE- The natural symplectic polarity in PG(3,2), illustrating a symplectic structure

The source — The Log24 post "Zero System" of July 31, 2014.

* For the title, see The New Yorker  of Sept. 22, 2014.

Thursday, February 26, 2015

Brit Award

Filed under: Uncategorized — m759 @ 1:06 AM

"The Brit Awards are the British equivalent
of the American Grammy Awards." — Wikipedia 

Detail of an image from yesterday's 5:30 PM ET post:

Related material:

From a review: "Imagine 'Raiders of the Lost Ark'
set in 20th-century London, and then imagine it
written by a man steeped not in Hollywood movies
but in Dante and the things of the spirit, and you
might begin to get a picture of Charles Williams's
novel Many Dimensions ."

See also Solomon's Seal (July 26, 2012).

Wednesday, February 25, 2015

Words and Images

Filed under: Uncategorized — Tags: — m759 @ 5:30 PM

The words:  "symplectic polarity"—

The images:

The Natural Symplectic Polarity in PG(3,2)

Symmetry Invariance in a Diamond Ring

The Diamond-Theorem Correlation

Picturing the Smallest Projective 3-Space

Quilt Block Designs

Saturday, February 21, 2015

High and Low Concepts

Filed under: Uncategorized — Tags: — m759 @ 4:30 PM

Steven Pressfield on April 25, 2012:

What exactly is High Concept?

Let’s start with its opposite, low concept.
Low concept stories are personal,
idiosyncratic, ambiguous, often European. 
“Well, it’s a sensitive fable about a Swedish
sardine fisherman whose wife and daughter
find themselves conflicted over … ”

ZZZZZZZZ.

Fans of Oslo artist Josefine Lyche know she has
valiantly struggled to find a high-concept approach
to the diamond theorem. Any such approach must,
unfortunately, reckon with the following low
(i.e., not easily summarized) concept —

The Diamond Theorem Correlation:

From left to right

http://www.log24.com/log/pix14B/140824-Diamond-Theorem-Correlation-1202w.jpg

http://www.log24.com/log/pix14B/140731-Diamond-Theorem-Correlation-747w.jpg

http://www.log24.com/log/pix14B/140824-Picturing_the_Smallest-1986.gif

http://www.log24.com/log/pix14B/140806-ProjPoints.gif

For some backstory, see ProjPoints.gif and "Symplectic Polarity" in this journal.

Saturday, December 27, 2014

More To Be Done

Filed under: Uncategorized — m759 @ 1:44 AM

  Ball and Weiner, 'An Introduction to Finite Geometry,' version of Sept. 5, 2011

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.

Wednesday, November 19, 2014

The Eye/Mind Conflict

Filed under: Uncategorized — Tags: — m759 @ 10:25 AM

Harold Rosenberg, "Art and Words," 
The New Yorker , March 29, 1969. From page 110:

"An advanced painting of this century inevitably gives rise
in the spectator to a conflict between his eye and his mind; 
as Thomas Hess has pointed out, the fable of the emperor's 
new clothes is echoed at the birth of every modemist art 
movement. If work in a new mode is to be accepted, the 
eye/mind conflict must be resolved in favor of the mind; 
that is, of the language absorbed into the work. Of itself, 
the eye is incapable of breaking into the intellectual system 
that today distinguishes between objects that are art and 
those that are not. Given its primitive function of 
discriminating among things in shopping centers and on 
highways, the eye will recognize a Noland as a fabric
design, a Judd as a stack of metal bins— until the eye's 
outrageous philistinism has been subdued by the drone of 
formulas concerning breakthroughs in color, space, and 
even optical perception (this, too, unseen by the eye, of 
course). It is scarcely an exaggeration to say that paintings 
are today apprehended with the ears. Miss Barbara Rose, 
once a promoter of striped canvases and aluminum boxes, 
confesses that words are essential to the art she favored 
when she writes, 'Although the logic of minimal art gained 
critical respect, if not admiration, its reductiveness allowed
for a relatively limited art experience.' Recent art criticism 
has reversed earlier procedures: instead of deriving principles 
from what it sees, it teaches the eye to 'see' principles; the 
writings of one of America's influential critics often pivot on 
the drama of how he failed to respond to a painting or 
sculpture the first few times he saw it but, returning to the 
work, penetrated the concept that made it significant and
was then able to appreciate it. To qualify as a member of the 
art public, an individual must be tuned to the appropriate 
verbal reverberations of objects in art galleries, and his 
receptive mechanism must be constantly adjusted to oscillate 
to new vocabularies."

New vocabulary illustrated:

Graphic Design and a Symplectic Polarity —

Background: The diamond theorem
and a zero system .

Thursday, July 31, 2014

Zero System

Filed under: Uncategorized — Tags: , — m759 @ 6:11 PM

The title phrase (not to be confused with the film 'The Zero Theorem')
means, according to the Encyclopedia of Mathematics,
a null system , and

"A null system is also called null polarity,
a symplectic polarity or a symplectic correlation….
it is a polarity such that every point lies in its own
polar hyperplane."

See Reinhold Baer, "Null Systems in Projective Space,"
Bulletin of the American Mathematical Society, Vol. 51
(1945), pp. 903-906.

An example in PG(3,2), the projective 3-space over the
two-element Galois field GF(2):

IMAGE- The natural symplectic polarity in PG(3,2), illustrating a symplectic structure

See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.

Sunday, June 8, 2014

Vide

Filed under: Uncategorized — Tags: , , — m759 @ 10:00 AM

Some background on the large Desargues configuration

“The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof.”

— Gian-Carlo Rota discussing the theorem of Desargues

What space  tells us about the theorem :  

In the simplest case of a projective space  (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel  lines and 20 Rosenhain  lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.

Vide  Classical Geometry in Light of Galois Geometry.

* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995.  The “simplest case” link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity 
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:

A coordinate-free approach to symplectic structure

Saturday, April 4, 2009

Saturday April 4, 2009

Filed under: Uncategorized — Tags: — m759 @ 8:00 AM
Annual Tribute to
The Eight

Katherine Neville's 'The Eight,' edition with knight on cover, on her April 4 birthday

Other knight figures:

Knight figures in finite geometry (Singer 7-cycles in the 3-space over GF(2) by Cullinane, 1985, and Curtis, 1987)

The knight logo at the SpringerLink site

Click on the SpringerLink
knight for a free copy
(pdf, 1.2 mb) of
the following paper
dealing with the geometry
underlying the R.T. Curtis
knight figures above:

Springer description of 1970 paper on Mathieu-group geometry by Wilbur Jonsson of McGill U.

Context:

Literature and Chess and
Sporadic Group References

Details:

 

Adapted (for HTML) from the opening paragraphs of the above paper, W. Jonsson's 1970 "On the Mathieu Groups M22, M23, M24…"–

"[A]… uniqueness proof is offered here based upon a detailed knowledge of the geometric aspects of the elementary abelian group of order 16 together with a knowledge of the geometries associated with certain subgroups of its automorphism group. This construction was motivated by a question posed by D.R. Hughes and by the discussion Edge [5] (see also Conwell [4]) gives of certain isomorphisms between classical groups, namely

PGL(4,2)~PSL(4,2)~SL(4,2)~A8,
PSp(4,2)~Sp(4,2)~S6,

where A8 is the alternating group on eight symbols, S6 the symmetric group on six symbols, Sp(4,2) and PSp(4,2) the symplectic and projective symplectic groups in four variables over the field GF(2) of two elements, [and] PGL, PSL and SL are the projective linear, projective special linear and special linear groups (see for example [7], Kapitel II).

The symplectic group PSp(4,2) is the group of collineations of the three dimensional projective space PG(3,2) over GF(2) which commute with a fixed null polarity tau…."

References

4. Conwell, George M.: The three space PG(3,2) and its group. Ann. of Math. (2) 11, 60-76 (1910).

5. Edge, W.L.: The geometry of the linear fractional group LF(4,2). Proc. London Math. Soc. (3) 4, 317-342 (1954).

7. Huppert, B.: Endliche Gruppen I. Berlin-Heidelberg-New York: Springer 1967.

Sunday, October 14, 2007

Sunday October 14, 2007

Filed under: Uncategorized — m759 @ 11:00 AM
The Dipolar God

Steven H. Cullinane, 'The Line'

“Logos and logic, crystal hypothesis,
Incipit and a form to speak the word
And every latent double in the word….”

— Wallace Stevens,
   “Notes Toward a Supreme Fiction

Yesterday’s meditation (“Simon’s Shema“) on the interpenetration of opposites continues:

Part I: The Jewel in the Lotus

“The fundamental conception of Tantric Buddhist metaphysics, namely, yuganaddha, signifies the coincidence of opposites.  It is symbolized by the conjugal embrace (maithuna or kama-kala) of a god and goddess or a Buddha and his consort (signifying karuna and sunyata or upaya and prajna, respectively), also commonly depicted in Tantric Buddhist iconography as the union of vajra (diamond sceptre) and padme (lotus flower).  Thus, yuganaddha essentially means the interpenetration of opposites or dipolar fusion, and is a fundamental restatement of Hua-yen theoretic structures.”

— p. 148 in “Part II: A Whiteheadian Process Critique of Hua-yen Buddhism,” in Process Metaphysics and Hua-Yen Buddhism: A Critical Study of Cumulative Penetration vs. Interpenetration (SUNY Series in Systematic Philosophy), by Steve Odin, State University of New York Press, 1982

Part II: The Dipolar God

And on p. 163 of Odin, op. cit., in “Part III: Theology of the Deep Unconscious: A Reconstruction of Process Theology,” in the section titled “Whitehead’s Dipolar God as the Collective Unconscious”–

“An effort is made to transpose Whitehead’s theory of the dipolar God into the terms of the collective unconscious, so that now the dipolar God is to be comprehended not as a transcendent deity, but the deepest dimension and highest potentiality of one’s own psyche.”

Part III: Piled High and Deep

Odin obtained his Ph.D. degree from the Department of Philosophy at the State University of New York (SUNY) at Stony Brook in 1980. (See curriculum vitae (pdf).)

For an academic review of Odin’s book, see David Applebaum, Philosophy East and West, Vol. 34 (1984), pp. 107-108.

It is perhaps worth noting, in light of the final footnote of Mark D. Brimblecombe’s Ph.D. thesis “Dipolarity and Godquoted yesterday, that “tantra” is said to mean “loom.” For some less-academic background on the Tantric iconography Odin describes, see the webpage “Love and Passion in Tantric Buddhist Art.” For a fiction combining love and passion with the word “loom” in a religious context, see Clive Barker’s Weaveworld.  This fiction– which is, if not “supreme” in the Wallace Stevens sense, at least entertaining– may correspond to some aspects of the deep Jungian psychological reality discussed by Odin.

Happy Birthday,
Hannah Arendt

(Oct. 14, 1906-
Dec. 4, 1975)

OPPOSITES:

Hannah (Arendt) and Martin (Heidegger) as portrayed in a play of that name

Actors portraying
Arendt and Heidegger

Click on image for details.

Saturday, October 13, 2007

Saturday October 13, 2007

Filed under: Uncategorized — m759 @ 9:22 AM
Simon’s Shema

“When times are mysterious
Serious numbers will always be heard
And after all is said and done
And the numbers all come home
The four rolls into three
The three turns into two
And the two becomes a
One”

Paul Simon, 1983


Related material:

Simon’s theology here, though radically reductive, is at least consistent with traditional Jewish thought. It may help counteract the thoughtless drift to the left of academic writing in recent decades. Another weapon against leftist nonsense appears, surprisingly, on the op-ed page of today’s New York Times:

“There is a Communist jargon recognizable after a single sentence. Few people in Europe have not joked in their time about ‘concrete steps,’ ‘contradictions,’ ‘the interpenetration of opposites,’ and the rest.”

— Doris Lessing, winner of this year’s Nobel Prize in Literature

The Times offers Lessing’s essay to counter Harold Bloom’s remark that this year’s award of a Nobel Prize to Lessing is “pure political correctness.” The following may serve as a further antidote to Bloom.

The Communist use of “interpenetration,” a term long used to describe the Holy Trinity, suggests– along with Simon’s hymn to the Unity, and the rhetorical advice of Norman Mailer quoted here yesterday—  a search for the full phrase “interpenetration of opposites” in the context* of theology.  Such a search yields a rhetorical gem from New Zealand:

“Dipolarity and God”
by Mark D. Brimblecombe,
Ph.D. thesis,
University of Auckland, 1999
.

* See the final footnote on the final page (249) of Brimblecombe’s thesis:

3 The Latin word contexo means to interweave, join, or braid together.

A check of the Online Eymology Dictionary supports this assertion:

context 1432, from L. contextus “a joining together,” orig. pp. of contexere “to weave together,” from com “together” + textere “to weave” (see texture).

See also Wittgenstein on “theology as grammar” and “context-sensitive” grammars as (unlike Simon’s reductive process) “noncontracting”– Log24, April 16, 2007: Happy Birthday, Benedict XVI.

Tuesday, May 24, 2005

Tuesday May 24, 2005

Filed under: Uncategorized — m759 @ 2:00 PM
Final Arrangements, continued:

Two Poles

From today’s New York Times:

The image “http://www.log24.com/log/pix05/050524-NYT.jpg” cannot be displayed, because it contains errors.

From erraticimpact.com on Paul Ricoeur:

“Ricoeur reserves his greatest admiration for
the narratologist Algirdas-Julien Greimas.
[See below.]
Ricoeur also explores the relationship
between the philosophical and religious
domains, attempting to reconcile
the two poles in his thought.”

From today’s NYT obituary of Sol Stetin:

“Mr. Stetin, who emigrated from Poland at the age of 10 and dropped out of high school in the ninth grade, was fond of saying he got his education in the labor movement.”

The image “http://www.log24.com/log/pix05/050524-JP2.jpg” cannot be displayed, because it contains errors.


“… it is not in isolation that the rhetorical power of such oppositions resides, but in their articulation in relation to other oppositions. In Aristotle’s Physics the four elements of earth, air, fire and water were said to be opposed in pairs. For more than two thousand years oppositional patterns based on these four elements were widely accepted as the fundamental structure underlying surface reality….


The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the ‘logical square’ of scholastic philosophy) as a means of analysing paired concepts more fully….”

Daniel Chandler, Semiotics for Beginners

Related material:

Poetry’s Bones and
Theme and Variations.

Other readings on polarity:

Log24, May 24, 2003, and
from July 26, 2003:

Bright Star and Dark Lady

“Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.”

Octavio Paz,
quoted by Homero Aridjis

Bright Star

Amen.

Dark Lady

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