See as well Peplowski in this journal.
Also on January 16, 2016 —
Source of animated gif: https://cage.ugent.be/~hs/polyhedra/dodeca.html.
Related reading —
Unfortunately, the volume of the dodecahedron formed by
unfolding a cube as in the University of Ghent animation above
is not double that of the cube, since refolding the cube leaves
an empty space inside … not shown in the Ghent animation.
But taking six congruent square pyramids that form a cube and using
them to cover the faces of a second, congruent, cube yields a
rhombic dodecahedron. This does offer a sort of solution to the
Delian problem… provided the new rhombic-dodecahedral altar is
found to be fit for ceremonial use. See . . .
The Hopkins mentioned above is not the noted Jesuit
Gerard Manley Hopkins, but rather Jasper Hopkins .
See that Hopkins in this journal on Twelfth Night, 2015.
See as well . . .

"The Demolished Man was a novel that had fascinated De Palma
since the late 1950s and appealed to his background in mathematics
and avant-garde storytelling. Its unconventional unfolding of plot
(exemplified in its mathematical layout of dialogue) and its stress on
perception have analogs in De Palma's filmmaking." — Wikipedia
This, together with the Cuernavaca balcony in Deschooling MIT, is
perhaps enough of a clue for mystified theologians on St. Peter's Day.
From Martin Heidegger's
Contributions to Philosophy (From Enowning) ,
Translated by Parvis Emad and Kenneth Maly,
Indiana University Press, 1999 (first published in German
in 1989 as Beiträge zur Philosophie (Vom Ereignis) but
written in 1936-1938 —
"The 'between' [das Zwischen ] is the simple 'bursting open'
that enowns be-ing to a being, which up until then is held back
from what is ownmost to it and is not yet to be named a being.
This 'bursting open' is the clearing for the sheltered. But the
'bursting open' does not disperse. and the clearing is not a mere
emptiness.
The 'between' [das Zwischen ] which bursts open gathers
what it removes into the open of its strifing and refusing
belongingness, moves unto the ab-ground , out of which everything
(god, man, world, earth) recoils in swaying into itself and thus leaves
to be-ing the unique decidedness of en-ownment."
— 270, "The Essential Sway of Be-ing" (p. 341)
"Enownment and enstrifing, historical grounding and decision,
uniqueness and the onefold, what has the character of
the between [Zwischenhafte ] and the cleavage [Geklüft ] —
they never name the essential sway of be-ing as properties
but rather in each case the whole essential swaying* of its essential
sway."
— 270, "The Essential Sway of Be-ing" (p. 342)
* For "swaying" as "unfolding," see (for instance)
the Stanford Encyclopedia of Philosophy and
also George Kovacs, Thinking and Be-ing in Heidegger's
Beiträge zur Philosophie (Vom Ereignis) ,
Zeta Books, 2015.
Ereignis in the Stanford Encyclopedia of Philosophy —
|
Further aspects of the essential unfolding of Being are revealed by what is perhaps the key move in the Contributions—a rethinking of Being in terms of the notion of Ereignis, a term translated variously as ‘event’ (most closely reflecting its ordinary German usage), ‘appropriation’, ‘appropriating event’, ‘event of appropriation’ or ‘enowning’. (For an analysis which tracks Heidegger's use of the term Ereignis at various stages of his thought, see Vallega-Neu 2010). The history of Being is now conceived as a series of appropriating events in which the different dimensions of human sense-making—the religious, political, philosophical (and so on) dimensions that define the culturally conditioned epochs of human history—are transformed. Each such transformation is a revolution in human patterns of intelligibility, so what is appropriated in the event is Dasein and thus the human capacity for taking-as (see e.g., Contributions 271: 343). Once appropriated in this way, Dasein operates according to a specific set of established sense-making practices and structures. In a Kuhnian register, one might think of this as the normal sense-making that follows a paradigm-shift. — Michael Wheeler, 2011 |
See as well "reordering" in Sunday evening's post Tetrads for McLuhan
and in a Log24 search for Reordering + Steiner.
See also "Press Agent" in this journal and a post from Maundy Thursday,
the date of Sabinson's reported death . . .
From a New York Times book review of a new novel about
Timothy Leary that was in the Times online on April 10 —
"Most of the novel resides in the perspective
of Fitzhugh Loney, one of Leary’s graduate students."
"A version of this article appears in print on ,
on Page 10 of the Sunday Book Review with the headline:
Strange Days."
For material about one of Leary's non -fictional grad students,
Ralph Metzner, see posts now tagged Metzner's Pi Day.
Related material —
The reported publication date of Searching for the Philosophers' Stone
was January 1, 2019.
A related search published here on that date:
* Title suggested by two of Ralph Metzner's titles,
The Expansion of Consciousness and The Unfolding Self .
See also "The Unfolding" in this journal.
"You said something about the significance of spaces between
elements being repeated. Not only the element itself being repeated,
but the space between. I'm very interested in the space between.
That is where we come together." — Peter Eisenman, 1982
|
https://www.parrhesiajournal.org/ Parrhesia No. 3 • 2007 • 22–32
(Up) Against the (In) Between: Interstitial Spatiality by Clare Blackburne Blackburne — www.parrhesiajournal.org 24 — "The excessive notion of espacement as the resurgent spatiality of that which is supposedly ‘without space’ (most notably, writing), alerts us to the highly dynamic nature of the interstice – a movement whose discontinuous and ‘aberrant’ nature requires further analysis." Blackburne — www.parrhesiajournal.org 25 — "Espacement also evokes the ambiguous figure of the interstice, and is related to the equally complex derridean notions of chora , différance , the trace and the supplement. Derrida’s reading of the Platonic chora in Chora L Works (a series of discussions with the architect Peter Eisenman) as something which defies the logics of non-contradiction and binarity, implies the internal heterogeneity and instability of all structures, neither ‘sensible’ nor ‘intelligible’ but a third genus which escapes conceptual capture.25 Crucially, chora , spacing, dissemination and différance are highly dynamic concepts, involving hybridity, an ongoing ‘corruption’ of categories, and a ‘bastard reasoning.’26 Derrida identification of différance in Margins of Philosophy , as an ‘unappropriable excess’ that operates through spacing as ‘the becoming-space of time or the becoming-time of space,’27 chimes with his description of chora as an ‘unidentifiable excess’ that is ‘the spacing which is the condition for everything to take place,’ opening up the interval as the plurivocity of writing in defiance of ‘origin’ and ‘essence.’28 In this unfolding of différance , spacing ‘insinuates into presence an interval,’29 again alerting us to the crucial role of the interstice in deconstruction, and, as Derrida observes in Positions , its impact as ‘a movement, a displacement that indicates an irreducible alterity’: ‘Spacing is the impossibility for an identity to be closed on itself, on the inside of its proper interiority, or on its coincidence with itself. The irreducibility of spacing is the irreducibility of the other.’30"
25. Quoted in Jeffrey Kipnis and Thomas Leeser, eds., 26. Ibid, 25.
27. Derrida, Margins of Philosophy. 28. Derrida, Chora L Works , 19 and 10. 29. Ibid, 203. 30. Derrida, Positions , 94. |
"The stars and galaxies seem static, eternal, or moving slowly
in deterministic patterns, becoming the background stage
on which we move. But if we could speed up the sequence,
we would see how dramatic and unpredictable this background
really is — an actor, director, script and stage all at once.
Moreover, it is a unified universe, a single unfolding event
of which we are an embedded part, a narrative of highly
dangerous and fine-tuned events, something more like
a detective thriller with many crimes and last-minute escapes
than the impersonal account of astronomy textbooks.
We are only just beginning to decipher the plot and figure out
the Cosmic Code, as Heinz Pagels puts it."
— Charles Jencks, The Architecture of the Jumping Universe :
A Polemic (How Complexity Science is Changing Architecture
and Culture), Academy Editions, 1995, rev. ed. 1997
"A Grand Unified Theory (GUT) is a model in particle physics…."
— Wikipedia
"Under the GUT symmetry operation these field components
transform into one another. The reason quantum particles
appear to have different properties in nature is that the unifying
symmetry is broken. The various gluons, quarks and leptons
are analogous to the facets of a cut diamond, which appear
differently according to the way the diamond is held but in
fact are all manifestations of the same underlying object."
— Heinz Pagels, Perfect Symmetry , Bantam paperback, 1986, p. 284
See also the recent post Multifaceted Narrative.
From the online New York Times this afternoon:
Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.
From this journal this morning:
"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to
make a fancy amusement-park ride.
There had to be more—"
— Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168
"The sensation of folding space . . . ."
Or unfolding:
Click the above unfolded space for some background.
For the former, see the previous post.
For the latter, see yesterday's The Unfolding.
Related material —
(Screenshot at 9:08 PM ET)
Heimel on Dating —
|
She adapted “Sex Tips” and “But Enough About You,” a 1986 collection, into a play, “A Girl’s Guide to Chaos,” which opened later that year off Broadway at the American Place Theater. The play is largely a conversation among four friends, one of whom, Cynthia (played by Debra Jo Rupp in the original production) realizes to her horror that she will have to start dating again. “Please, God, no, don’t make me do it!” she says. “I’ll be good from now on, I promise! I’ll stop feeding the dog hashish! I’ll be kind, thoughtful, sober, industrious, anything. But please, God, not the ultimate torture of dating!”
— Richard Sandomir tonight in The New York Times |
<meta property="article:published"
itemprop="datePublished"
content="2018-02-27T19:37:54-05:00" />
From the website of Richard P. Gabriel —
" As part of my studies, I came up with a 'theory of poetry'
based loosely on Christopher Alexander’s 'Nature of Order.' "
[The Alexander link is mine, not Gabriel’s.]
A phrase from this journal a year ago today — "poetic order" —
links to the theory of Gabriel —
From Gabriel's "The Nature of Poetic Order" —
Positive Space
• Positive space is the characteristic of a center
that moves outward from itself, seemingly oozing life
rather than collapsing on itself
• An image that resonates is showing positive space
• A word that has many connotations that fit with the
other centers in the poem is showing positive space
• It is an expansion outward rather than a contraction
inward, and it shows that the poem is unfolding
in front of us and not dying
Related material —
From a post of April 26, 2017 —
A sketch, adapted tonight from Girl Scouts of Palo Alto —

From the April 14 noon post High Concept —

From the April 14 3 AM post Hudson and Finite Geometry —
From the April 24 evening post The Trials of Device —
Note that Hudson’s 1905 “unfolding” of even and odd puts even on top of
the square array, but my own 2013 unfolding above puts even at its left.
On a new HBO series that opens at 9 PM ET tonight —
|
Watching Westworld , you can sense a grand mythology unfolding before your eyes. The show’s biggest strength is its world-building, an aspect of screenwriting that many television series have botched before. Often shows will rush viewers into plot, forgetting to instill a sense of place and of history, that you’re watching something that doesn’t just exist in a vacuum but rather is part of some larger ecosystem. Not since Lost can I remember a TV show so committed to immersing its audience into the physical space it inhabits. (Indeed, Westworld can also be viewed as a meta commentary on the art of screenwriting itself: brainstorming narratives, building characters, all for the amusement of other people.) Westworld is especially impressive because it builds two worlds at once: the Western theme park and the futuristic workplace. The Western half of Westworld might be the more purely entertaining of the two, with its shootouts and heists and chases through sublime desert vistas. Behind the scenes, the theme park’s workers show how the robot sausage is made. And as a dystopian office drama, the show does something truly original. — Adam Epstein at QUARTZ, October 1, 2016 |
"… committed to immersing its audience
into the physical space it inhabits…."
See also, in this journal, the Mimsy Cube —
|
"Mimsy Were the Borogoves," "… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play." |
"I could a tale unfold…" — Hamlet's father's ghost
Or not.
The following Log24 excerpts are from a noted mathematician's
recent date of death, and the preceding date.
The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander's book
The Timeless Way of Building (Oxford University Press, 1979).
A quote from the publisher:
"Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself."
Three paragraphs from the book (pp. xiii-xiv):
19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.
20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.
21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.
Compare to, and contrast with, these illustrations of "Boolean space":
(See also similar illustrations from Berkeley and Purdue.)
Detail of the above image —
Note the "unfolding," as Christopher Alexander would have it.
These "Boolean" spaces of 1, 2, 4, 8, and 16 points
are also Galois spaces. See the diamond theorem —
A print copy of next Sunday’s New York Times Book Review
arrived in today’s mail. From the front-page review:
Marcel Theroux on The Book of Strange New Things ,
a novel by Michel Faber —
“… taking a standard science fiction premise and
unfolding it with the patience and focus of a
tai chi master, until it reveals unexpected
connections, ironies and emotions.”
What is a tai chi master, and what is it that he unfolds?
Perhaps the taijitu symbol and related material will help.
The Origin of Change
“Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined
On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come.”
— Wallace Stevens,
“Notes Toward a Supreme Fiction,”
Canto IV of “It Must Change”
(Continued from Seize the Dia, April 6)
Two chess games by Fischer, against two brothers—
1956: "In this game, Fischer (playing Black) demonstrates
noteworthy innovation and improvisation." — Wikipedia
1963: "Fischer [playing Black] had engineered a brilliantly
disguised trap for him and … he had fallen into it." — NY Times
See also this evening's Times obituaries and The Unfolding.
Some context: The Crosswicks Curse.
A chess set previously mentioned in this journal—

These chessmen appeared in the weblog Minimalissimo
on Sept. 20, 2010. In Log24 on that date, the issue was
not so much the chessmen as the underlying board.
See "The Unfolding." See also the following from
the Occupy Space gallery in Limerick today—
| C A V E S – Anthony Murphy Solo Exhibition Opening 7 pm Thursday 1st Dec Exhibition 2nd – 22nd Dec 2011 Plato's allegory of the cave describes prisoners, inhabiting the cave since childhood, immobile, facing an interior wall. A large fire burns behind the prisoners, and as people pass this fire their shadows are cast upon the cave's wall, and these shadows of the activity being played out behind the prisoner become the only version of reality that the prisoner knows. C A V E S is an exhibition of three large scale works, each designed to immerse the viewer, and then to confront the audience with a question regarding how far they, as privileged viewers of the shadows and reflections being played out upon the walls, are willing to allow themselves to believe what they know to be a false reality. The works are based on explorations of simple 2D shapes; regular polygons are exploded to create fractured pattern, or layered upon one another until intricate forms emerge, upon which the projections can begin to draw out a third dimension. |
Two pictures suggested by recent comments on
Peter J. Cameron's Sept. 17 post about T.S. Eliot—

For some further background, see Symmetry of Walsh Functions.
In memory of an historian of Mexico
who died on Tuesday, July 6, 2010—

Related material—
and
In the latter, click on
the link Eleven.
or: Catullus vs. Ovid
(Today's previous post, "Coxeter vs. Fano,"
might also have been titled "Toronto vs. Rome.")
ut te postremo donarem munere mortis
Explicatio
Image by Christopher Thomas at Wikipedia —
Unfolding of a hypercube and of a cube —


The metaphor for metamorphosis no keys unlock.
— Steven H. Cullinane, "Endgame"
The current New Yorker has a translation of
the above line of Catullus by poet Anne Carson.
According to poets.org, Carson "attended St. Michael's College
at the University of Toronto and, despite leaving twice,
received her B.A. in 1974, her M.A. in 1975 and her Ph.D. in 1981."
Carson's translation is given in a review of her new book Nox.
The title, "The Unfolding," of the current review echoes an earlier
New Yorker piece on another poet, Madeleine L'Engle—
Cynthia Zarin in The New Yorker, issue dated April 12, 2004–
“Time, for L’Engle, is accordion-pleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded– or it’s a story without a book.’”
(See also the "harrow up" + Hamlet link in yesterday's 6:29 AM post.)
The Unfolding
A post for Florencio Campomanes,
former president of the World Chess Federation.
Campomanes died at 83 in the Philippines
at 1:30 PM local time (1:30 AM Manhattan time)
on Monday, May 3, 2010.
From this journal on the date of his death —
"There is such a thing as a tesseract."
– Madeleine L'Engle
Image by Christopher Thomas at Wikipedia —
Unfolding of a hypercube and of a cube —
Related material from a story of the Philippines —
“Music and mathematics are among the pre-eminent wonders of the race. Levi-Strauss sees in the invention of melody ‘a key to the supreme mystery’ of man– a clue, could we but follow it, to the singular structure and genius of the species. The power of mathematics to devise actions for reasons as subtle, witty, manifold as any offered by sensory experience and to move forward in an endless unfolding of self-creating life is one of the strange, deep marks man leaves on the world. Chess, on the other hand, is a game in which thirty-two bits of ivory, horn, wood, metal, or (in stalags) sawdust stuck together with shoe polish, are pushed around on sixty-four alternately coloured squares. To the addict, such a description is blasphemy. The origins of chess are shrouded in mists of controversy, but unquestionably this very ancient, trivial pastime has seemed to many exceptionally intelligent human beings of many races and centuries to constitute a reality, a focus for the emotions, as substantial as, often more substantial than, reality itself. Cards can come to mean the same absolute. But their magnetism is impure. A mania for whist or poker hooks into the obvious, universal magic of money. The financial element in chess, where it exists at all, has always been small or accidental.
To a true chess player, the pushing about of thirty-two counters on 8×8 squares is an end in itself, a whole world next to which that of a mere biological or political or social life seems messy, stale, and contingent. Even the patzer, the wretched amateur who charges out with his knight pawn when the opponent’s bishop decamps to R4, feels this daemonic spell. There are siren moments when quite normal creatures otherwise engaged, men such as Lenin and myself, feel like giving up everything– marriage, mortgages, careers, the Russian Revolution– in order to spend their days and nights moving little carved objects up and down a quadrate board. At the sight of a set, even the tawdriest of plastic pocket sets, one’s fingers arch and a coldness as in a light sleep steals over one’s spine. Not for gain, not for knowledge or reknown, but in some autistic enchantment, pure as one of Bach’s inverted canons or Euler’s formula for polyhedra.”
— George Steiner in “A Death of Kings,” The New Yorker, issue dated September 7, 1968, page 133
“Examples are the stained-glass windows of knowledge.” —Nabokov

Click above images for some context.

Hypercube and Cube
For every kind of vampire,
there is a kind of cross.
— Gravity’s Rainbow
Kate Beckinsale, poster for
Underworld: Evolution
(DVD release date 6/6/6)
evolve:
1641, “to unfold, open out, expand,”
from L. evolvere “unroll,” from ex- “out”
+ volvere “to roll” (see vulva).
— Online Eymology Dictionary
Related material:
Introduction to Multispeech,
All Hallows’ Eve, 2005
From Jung's Map of the Soul, by Murray Stein:
"… Jung thinks of the self as undergoing continual transformation during the course of a lifetime…. At the end of his late work Aion, Jung presents a diagram to illustrate the dynamic movements of the self…."
"The formula presents a symbol of the self, for the self is not just a stable quantity or constant form, but is also a dynamic process. In the same way, the ancients saw the imago Dei in man not as a mere imprint, as a sort of lifeless, stereotyped impression, but as an active force…. The four transformations represent a process of restoration or rejuvenation taking place, as it were, inside the self…."
"The formula reproduces exactly the essential features of the symbolic process of transformation. It shows the rotation of the mandala, the antithetical play of complementary (or compensatory) processes, then the apocatastasis, i.e., the restoration of an original state of wholeness, which the alchemists expressed through the symbol of the uroboros, and finally the formula repeats the ancient alchemical tetrameria, which is implicit in the fourfold structure of unity.
What the formula can only hint at, however, is the higher plane that is reached through the process of transformation and integration. The 'sublimation' or progress or qualitative change consists in an unfolding of totality into four parts four times, which means nothing less than its becoming conscious. When psychic contents are split up into four aspects, it means that they have been subjected to discrimination by the four orienting functions of consciousness. Only the production of these four aspects makes a total description possible. The process depicted by our formula changes the originally unconscious totality into a conscious one."
— Jung, Collected Works,
Vol. 9, Part 2, Aion: Researches into the Phenomenology of the Self (1951)
Related material:
"Although 'wholeness' seems at first sight to be nothing but an abstract idea (like anima and animus), it is nevertheless empirical in so far as it is anticipated by the psyche in the form of spontaneous or autonomous symbols. These are the quaternity or mandala symbols, which occur not only in the dreams of modern people who have never heard of them, but are widely disseminated in the historical recods of many peoples and many epochs. Their significance as symbols of unity and totality is amply confirmed by history as well as by empirical psychology. What at first looks like an abstract idea stands in reality for something that exists and can be experienced, that demonstrates its a priori presence spontaneously. Wholeness is thus an objective factor that confronts the subject independently of him… Unity and totality stand at the highest point on the scale of objective values because their symbols can no longer be distinguished from the imago Dei. Hence all statements about the God-image apply also to the empirical symbols of totality."
— Jung, Aion, as quoted in
Carl Jung and Thomas Merton
Wikipedia on the tesseract:
Robert A. Heinlein in Glory Road:
Johnny Cash: “And behold, a white horse.”
On The Last Battle, a book in the Narnia series by C. S. Lewis:
Lewis said in “The Weight of Glory”—
On enchantments that need to be broken:
See the description of the Eater of Souls in Glory Road and of Scientism in
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