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 太 "great; grand; supreme; extreme; very; too" (a superlative variant of da 大 "big; large; great; very") and ji 極 "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). |
Wednesday, October 25, 2017
The Source (Not by Michener)
Sunday, March 12, 2017
Raise High the Ridgepole, Architects*
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 太 "great; grand; supreme; extreme; very; too" (a superlative variant of da 大 "big; large; great; very") and ji 極 "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
"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
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
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
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*
A figure I prefer to the "Golden Tablet" of Night at the Museum —
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
"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
The words: "symplectic polarity"—
The images:
The Natural Symplectic Polarity in PG(3,2)
Symmetry Invariance in a Diamond Ring
The Diamond-Theorem Correlation
Saturday, February 21, 2015
High and Low Concepts
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
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
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
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):
See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.
Sunday, June 8, 2014
Vide
“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:
Saturday, April 4, 2009
Saturday April 4, 2009
The Eight
Other knight figures:
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:
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 M_{22}, M_{23}, M_{24}…"–
"[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
where A_{8} is the alternating group on eight symbols, S_{6} 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
“Logos and logic, crystal hypothesis, — Wallace Stevens, |
Yesterday’s meditation (“Simon’s Shema“) on the interpenetration of opposites continues:
“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
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.”
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 God” quoted 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.
Arendt and Heidegger
Click on image for details.
Saturday, October 13, 2007
Saturday October 13, 2007
“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” |
Related material:
— 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:
^{3} The Latin word contexo means to interweave, join, or braid together.
A check of the Online Eymology Dictionary supports this assertion:
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
Two Poles
From today’s New York Times:
From erraticimpact.com on Paul Ricoeur:
the narratologist Algirdas-Julien Greimas.
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.” |
continued:
“… 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
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, |
||
Bright Star |
Amen. |
Dark Lady |