A remark on coordinatization linked to by John Baez today —
This suggests a more historical perspective:
See as well a search for Interpenetration in this journal.
A remark on coordinatization linked to by John Baez today —
This suggests a more historical perspective:
See as well a search for Interpenetration in this journal.
Or: A Candle for Sunrise
Commentary —
“Looking carefully at Golay’s code is like staring into the sun.”
Wikipedia— "The first Million Mask March occurred in 2013."
A check of the date of that march in this journal yields …
See as well, more generally, "Interpenetration" in this journal.
Shown below is an illustration from "The Puzzle Layout Problem" —
Exercise: Using the above numerals 1 through 24
(with 23 as 0 and 24 as ∞) to represent the points
∞, 0, 1, 2, 3 … 22 of the projective line over GF(23),
reposition the labels 1 through 24 in the above illustration
so that they appropriately* illustrate the cube-parts discussed
by Iain Aitchison in his March 2018 Hiroshima slides on
cube-part permutations by the Mathieu group M24.
A note for Northrop Frye —
Interpenetration in the eightfold cube — the three midplanes —
A deeper example of interpenetration:
Aitchison has shown that the Mathieu group M24 has a natural
action on the 24 center points of the subsquares on the eightfold
cube's six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.
* "Appropriately" — I.e. , so that the Aitchison cube octads correspond
exactly, via the projective-point labels, to the Curtis MOG octads.
"Let me say this about that." — Richard Nixon
Interpenetration in Weyl's epistemology —
Interpenetration in Mazzola's music theory —
Interpenetration in the eightfold cube — the three midplanes —
A deeper example of interpenetration:
Aitchison has shown that the Mathieu group M24 has a natural
action on the 24 center points of the subsquares on the eightfold
cube's six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.
Impenetrability vs. Interpenetration
The previous post discussed impenetrability .
To give the opposing concept of interpenetration
a fair hearing, see . . .
More generally, see a search for interpenetration in this journal.
“All right, Jessshica. It’s time to open the boxsssschhh.” “Gahh,” she said. She began to walk toward the box, but her heart failed her and she retreated back to the chair. “Fuck. Fuck.” Something mechanical purred. The seam she had found cracked open and the top of the box began to rise. She squeezed shut her eyes and groped her way into a corner, curling up against the concrete and plugging her ears with her fingers. That song she’d heard the busker playing on the train platform with Eliot, “Lucy in the Sky with Diamonds”; she used to sing that. Back in San Francisco, before she learned card tricks. It was how she’d met Benny: He played guitar. Lucy was the best earner, Benny said, so that was mainly what she sang. She must have sung it five times an hour, day after day. At first she liked it but then it was like an infection, and there was nothing she could do and nowhere she could go without it running across her brain or humming on her lips, and God knew she tried; she was smashing herself with sex and drugs but the song began to find its way even there. One day, Benny played the opening chord and she just couldn’t do it. She could not sing that fucking song. Not again. She broke down, because she was only fifteen, and Benny took her behind the mall and told her it would be okay. But she had to sing. It was the biggest earner. She kind of lost it and then so did Benny and that was the first time he hit her. She ran away for a while. But she came back to him, because she had nothing else, and it seemed okay. It seemed like they had a truce: She would not complain about her bruised face and he would not ask her to sing “Lucy.” She had been all right with this. She had thought that was a pretty good deal. Now there was something coming out of a box, and she reached for the most virulent meme she knew. “Lucy in the sky!” she sang. “With diamonds!” • • •
Barry, Max. Lexicon: A Novel (pp. 247-248). |
Related material from Log24 on All Hallows' Eve 2013 —
"Just another shake of the kaleidoscope" —
Related material:
Kaleidoscope Puzzle,
Design Cube 2x2x2, and
Through the Looking Glass: A Sort of Eternity.
* I.e., Hemingway's novel The Garden of Eden.
See also Northrop Frye and "interpenetration"
in this journal and a University of Montana master's
thesis from 1994 on the Hemingway novel,
"And a river went out of Eden," by Howard A. Schmid.
See as well remarks by Stanley Fish quoted here on May 7.
Or: Trinity Test Site
From the New York Times Book Review of
next Sunday, August 6, 2017 —
"In a more conventional narrative sequence,
even a sequence of poems,
this interpenetration would acquire
sequence and evolution." [Link added.]
The concept under review is that of the Holy Trinity.
See also, in this journal, Cube Trinity.
For a simpler Trinity model, see the three-point line …
“Perhaps the philosophically most relevant feature of modern science
is the emergence of abstract symbolic structures as the hard core
of objectivity behind— as Eddington puts it— the colorful tale of
the subjective storyteller mind.”
— Hermann Weyl, Philosophy of Mathematics and
Natural Science , Princeton, 1949, p. 237
Harvard University Press on the late Angus Fletcher, author of
The Topological Imagination and Colors of the Mind —
From the Harvard webpage for Colors of the Mind —
Angus Fletcher is one of our finest theorists of the arts,
the heir to I. A. Richards, Erich Auerbach, Northrop Frye.
This… book… aims to open another field of study:
how thought— the act, the experience of thinking—
is represented in literature.
. . . .
Fletcher’s resources are large, and his step is sure.
The reader samples his piercing vision of Milton’s
Satan, the original Thinker,
leaving the pain of thinking
as his legacy for mankind.
A 1992 review by Vinay Dharwadker of Colors of the Mind —
See also the above word "dianoia" in The Echo in Plato's Cave.
Some context …
This post was suggested by a memorial piece today in
the Los Angeles Review of Books —
A Florilegium for Angus Fletcher
By Kenneth Gross, Lindsay Waters, V. N. Alexander,
Paul Auster, Harold Bloom, Stanley Fish, K. J. Knoespel,
Mitchell Meltzer, Victoria Nelson, Joan Richardson,
Dorian Sagan, Susan Stewart, Eric Wilson, Michael Wood
Fletcher reportedly died on November 28, 2016.
"I learned from Fletcher how to apprehend
the daemonic element in poetic imagination."
— Harold Bloom in today's Los Angeles florilegium
For more on Bloom and the daemonic, see a Log24 post,
"Interpenetration," from the date of Fletcher's death.
Some backstory: Dharwadker in this journal.
Excerpts from James C. Nohrnberg, "The Master of the Myth of Literature: An Interpenetrative Ogdoad for Northrop Frye," Comparative Literature Vol. 53, No. 1 (Winter, 2001), pp. 58-82
From page 58 — * P. 22 of Rereading Frye: The Published and Unpublished Works , ed. David Boyd and Imre Salusinszky, Frye Studies [series] (Toronto: University of Toronto Press, 1998). [Abbreviated as RF .]
From page 62 —
From page 63 —
From page 69 —
From page 71 —
From page 77 — |
The orange and black Princeton colors in the previous post
suggest a review of Halloween 2013 —
From the Wikipedia article Bauhaus (band) —
"On 31 October 2013 (Halloween), David J and Jill Tracy released
'Bela Lugosi's Dead (Undead Is Forever),' a cinematic piano-led
rework of 'Bela Lugosi's Dead.'"
Halloween 2013 here (click to enlarge) —
* See "synchronolog…" in this journal.
See also Interpenet- in this journal.
"Interpenetration, that's what I say!"
— Adapted from Humpty Dumpty
Bead-Game Structuralism:
Excerpts from Comments by Robert de Marrais
on Interpenetration and The Raw and the Cooked
Click the image below for the webpage:
The title is from an essay by James C. Nohrnberg—
"Just another shake of the kaleidoscope" —
Related material:
Kaleidoscope Puzzle,
Design Cube 2x2x2, and
Through the Looking Glass: A Sort of Eternity.
"'Interpenetration'" — Stanley Fish in yesterday evening's online New York Times
"You want Frye's with that?" — A recent humanities graduate
(Continued from Epiphany and from yesterday.)
Detail from the current American Mathematical Society homepage—
Further detail, with a comparison to Dürer’s magic square—
The three interpenetrating planes in the foreground of Donmoyer‘s picture
provide a clue to the structure of the the magic square array behind them.
Group the 16 elements of Donmoyer’s array into four 4-sets corresponding to the
four rows of Dürer’s square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.
Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—
Thus the Donmoyer array also enjoys the structural symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.
Just as the decomposition theorem’s interpenetrating lines explain the structure
of a 4×4 square , the foreground’s interpenetrating planes explain the structure
of a 2x2x2 cube .
For an application to theology, recall that interpenetration is a technical term
in that field, and see the following post from last year—
Saturday, June 25, 2011
— m759 @ 12:00 PM “… the formula ‘Three Hypostases in one Ousia ‘ Ousia
|
Yesterday's All About Eve post featured Pope John Paul II
with his close friend and confidant Jerzy Kluger.
Their counterparts Xavier and Magneto in the recent film
"X-Men: First Class," together with Catholic doctrine on telepathy,
suggest the following meditations.
Douglas Hofstadter on interpenetration—
— as well as Trinity in this journal.
First the punchline—
Then the joke.
"It's still the same old story…"
See Glory in this journal.
'But "glory" doesn't mean "a nice knock-down argument",' Alice objected.
'When I use a word,' Humpty Dumpty said, in rather a scornful tone, 'it means just what I choose it to mean— neither more nor less.'
'The question is,' said Alice, 'whether you can make words mean so many different things.'
'The question is,' said Humpty Dumpty, 'which is to be master— that's all.'
Alice was too much puzzled to say anything; so after a minute Humpty Dumpty began again. 'They've a temper, some of them — particularly verbs: they're the proudest— adjectives you can do anything with, but not verbs— however, I can manage the whole lot of them! Impenetrability! That's what I say!'
'Would you tell me please,' said Alice, 'what that means?'
'Now you talk like a reasonable child,' said Humpty Dumpty, looking very much pleased. 'I meant by "impenetrability" that we've had enough of that subject, and it would be just as well if you'd mention what you mean to do next, as I suppose you don't mean to stop here all the rest of your life.'
'That's a great deal to make one word mean,' Alice said in a thoughtful tone.
'When I make a word do a lot of work like that,' said Humpty Dumpty, 'I always pay it extra.'
See also Interpenetration in this journal… and in Northrop Frye.
"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.
Click on image for details.
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” |
Related material:
“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:
* 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:
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.
The August 2007 issue of Notices of the American Mathematical Society contains a review of a new book by Douglas Hofstadter, I Am a Strange Loop. (2007, Basic Books, New York. $26.95, 412 pages.)
A better review, in the Los Angeles Times of March 18, 2007, notes an important phrase in the book, "interpenetration of souls," that the AMS Notices review ignores.
Here is an Amazon.com search on "interpenetration" in the Hofstadter book:
1. | on Page 217: |
"… described does not create a profound blurring of two people's identities. Tennis and driving do not give rise to deep interpenetrations of souls. …" | |
2. | on Page 237: |
"… What seems crucial here is the depth of interpenetration of souls the sense of shared goals, which leads to shared identity. Thus, for instance, Carol always had a deep, …" | |
3. | on Page 270: |
"… including the most private feelings and the most confidential confessions, then the interpenetration of our worlds becomes so great that our worldviews start to fuse. Just as I could jump to California when …" | |
4. | on Page 274: |
"… we choose to downplay or totally ignore the implications of the everyday manifestations of the interpenetration of souls. Consider how profoundly wrapped up you can become in a close friend's successes and failures, in their very …" | |
5. | on Page 276: |
"… Interpenetration of National Souls Earlier in this chapter, I briefly offered the image of a self as analogous to a country …" | |
6. | from Index: |
"… birthday party for, 350 "bachelor", elusiveness of concept, 178 bad-breath analogy, 150 bandwidth of communication as determinant of degree of interpenetration, 212 213, 220, …" | |
7. | from Index: |
"… phrases denying interpenetration of souls, 270 271; physical phenomena that lack consciousness, 281 282; physical structures lacking hereness, 283; potential personal attributes, 183; …" |
The American Mathematical Society editors and reviewer seem to share Hofstadter's ignorance of Christian doctrine; they might otherwise have remembered a rather famous remark: "This is not mathematics, it is theology."
For more on the theology of interpenetration, see Log24 on "Perichoresis, or Coinherence" (Jan. 22, 2004).
For a more mathematical approach to this topic, see Spirituality Today, Spring 1991:
"… the most helpful image is perhaps the ellipse often used to surround divine figures in ancient art, a geometrical figure resulting from the overlapping, greater or lesser, of two independent circles, an interpenetration or coinherence which will, in some sense, reunify divided humanity, thus restoring to some imperfect degree the original image of God."
See also the trinitarian doctrine implicit in related Log24 entries of July 1, 2007, which include the following illustration of the geometrical figure described, in a somewhat confused manner, above:
"Values are rooted
in narrative."
— Harvey Cox,
Hollis Professor
of Divinity
at Harvard,
Atlantic Monthly,
November 1995
Related material:
Steps Toward Salvation:
An Examination of
Co-Inherence and
Substitution in
the Seven Novels
of Charles Williams,
by Dennis L. Weeks
Taking Lucifer Seriously:
Michael Sprinker
versus
The Society of Jesus
As the previous entry indicates, I do not take Christian poetry too seriously. The Prince of Darkness is another matter. I encountered him this morning in a book on the Christian poet Hopkins by the late Michael Sprinker.
“You were never on the debating team when you were in high school, were you, ace? When you’re in a debate, you don’t try to convince the other side; they’re never going to agree with you. You try to convince the judges and the audience.”
— Michael Sprinker, quoted in The Minnesota Review, 2003
“For Hopkins, poetry was the act of producing the self, one version of that selving which he associated not only with Christ but with Lucifer.”
— Michael Sprinker, “A Counterpoint of Dissonance” — The Aesthetics and Poetry of Gerard Manley Hopkins, Johns Hopkins University Press, 1980, p. 95
A counterbalance to Sprinker on Hopkins and Lucifer is Hopkins, the Self, and God, by Walter J. Ong, S. J. (University of Toronto Press, 1986). From p. 119:
“The interior dynamism of the Three Persons in One God was not for Hopkins some sort of formula for theological juggling acts but was rather the centre of his personal devotional life and thus of his own ‘selving.’ …. He writes to Bridges 24 October 1883…
‘For if the Trinity… is to be explained by grammar and by tropes… where wd. be the mystery? the true mystery, the incomprehensible
one.’ “
For the dynamics of the Trinity, see the Jan. 22 entry, Perichoresis, or Coinherence. Another word for coinherence is “indwelling,” as expressed in what might be called the
Song of Lucifer:
Me into you,
You into me,
Me into you…
For a Christian version of this “indwelling,” see
Coinherence,
Interpenetration,
Mutual Indwelling
See also last year’s entries of 9/09.
Perichoresis, or Coinherence
Edward Gibbon, Decline and Fall of the Roman Empire, Chapter XXI —
Gibbon, discussing the theology of the Trinity, defines perichoresis as
“… the internal connection and spiritual penetration which indissolubly unites the divine persons59 ….
59 … The or ‘circumincessio,’ is perhaps the deepest and darkest corner of the whole theological abyss.”
“Whoever fights monsters should see to it that in the process he does not become a monster. And when you look long into an abyss, the abyss also looks into you.”
Perichoresis does NOT mean “dancing around” ….
From a mailing list message:
If [a correspondent] will but open a lexicon, she will see that perichoresis (with a long o, omega) has nothing to do with “the Greek word for dance,” which is spelt with a short o (omicron). As a technical term in trinitarian theology, perichoresis means “interpenetration.”
Interpenetration in Arthur Machen
Interpenetration in T. S. Eliot:
“Between two worlds
become much like each other….”
On the Novels of Charles Williams
Coinherence in Charles Williams
The Per Speculum link is to a discussion of coinherence and the four last films of Kieslowski —
La Double Vie de Veronique (1991),
Trois Couleurs: Bleu (1993),
Trois Couleurs: Blanc (1993), and
Trois Couleurs: Rouge (1994).
See, too, previous log24 entries related to Kieslowski’s work and to coinherence:
Moulin Bleu (12/16/03),
Quarter to Three (12/20/03), and
White, Geometric, and Eternal (12/20/03).
STAR WARS
opened on this date in 1977.
From the web page Amande:
Le Christ et la Vierge apparurent souvent entourés d’une auréole en forme d’amande: la mandorle.
Étymologiquement, le mot amande est une altération de amandala, qui dérive lui-même du latin classique amygdala….
L’amande a… une connotation symbolique, celle du sexe féminin. Elle figure souvent la vulve. Elle est alors en analogie avec la yoni du vocabulaire de l’hindouisme, la vulve ou la matrice, représentée par une amande ou une noix coupée en deux.
Screenshot of the online
New York Times, May 25, 2003:
Ariel the Hutt and Princess Amygdala
by Horia Cristescu and The Triangle (TRIKONA) The intersection of two geometric forms (lines, triangles, circles, etc.) represents forces that are even more intense than those generated by the simple forms. Such an interpenetration indicates a high level in the dynamic interaction of the correspondent energies. The empty spaces generated by such combinations are described as very efficient operational fields of the forces emanating from the central point of the YANTRA. That is why we can very often encounter representations of MANTRAS in such spaces. YANTRA and MANTRA are complementary aspects of SHIVA and their use together is much more efficient than the use of one alone.
|
AMEN.
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