(Continued from Nov. 28, 2010)
Thursday, August 8, 2019
Saturday, August 3, 2019
Wednesday, July 31, 2019
Icelandic Fantasy
"In the fantasy, Owen is still working on his Rubik’s Cube.
Finally, he finishes — he’s put together all 6 sides."
— "Maniac" Season 1, Episode 9 recap: ‘Utangatta’
by Cynthia Vinney at showsnob.com, Oct. 9, 2018
Related material —
See also Exploded in this journal.
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“A Certain Character of Permanence” — G. H. Hardy
See the above title in this journal. See also . . .
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Monday, July 22, 2019
Space, Time, Form
Click the image for some remarks on a related novel.
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Tuesday, July 16, 2019
Morf Vandewalt, Social Prism
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Tuesday, July 9, 2019
Schoolgirl Space: 1984 Revisited
Cube Bricks 1984 —
From "Tomorrowland" (2015) —
From John Baez (2018) —
See also this morning's post Perception of Space
and yesterday's Exploring Schoolgirl Space.
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Perception of Space
The three previous posts have now been tagged . . .
Tetrahedron vs. Square and Triangle vs. Cube.
Related material —
Tetrahedron vs. Square:
Labeling the Tetrahedral Model (Click to enlarge) —
Triangle vs. Cube:
… and, from the date of the above John Baez remark —
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Saturday, July 6, 2019
Mythos and Logos
Mythos
Logos
The six square patterns which, applied as above to the faces of a cube,
form "diamond" and "whirl" patterns, appear also in the logo of a coal-
mining company —
.
Related material —
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Tuesday, July 2, 2019
Depth Psychology Meets Inscape Geometry
An illustration from the previous post may be interpreted
as an attempt to unbokeh an inscape —
The 15 lines above are Euclidean lines based on pairs within a six-set.
For examples of Galois lines so based, see Six-Set Geometry:
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Monday, July 1, 2019
Desperately Seeking Clarity
"Let us consider the crux of Hopkins' sensibility…"
Seeking claritas :
From a "cube tales" post of June 21 —
The number "six" in the second tale above counts faces of the cube,
as shown in a post of June 23 —
.
". . . Then the universe exploded into existence . . . ."
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Sunday, June 30, 2019
Creepy Peepers
In memory of Susan Bernard , who reportedly died on June 21, 2019* —
Image from the 2016 post A Paris Review . . .
* See as well this journal on June 21,
"Cube Tales for Solstice Day."
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Tuesday, June 25, 2019
Sunday, June 23, 2019
Thursday, June 20, 2019
Friday, May 31, 2019
Sunday, May 26, 2019
Thursday, May 23, 2019
May the Fourth for Spaceheads
An image posted here two years earlier, on May the Fourth, 2017 —
From Betty and Veronica Double Digest #229 —
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Sunday, May 19, 2019
The Building Blocks of Geometry
From "On the life and scientific work of Gino Fano"
by Alberto Collino, Alberto Conte, and Alessandro Verra,
ICCM Notices , July 2014, Vol. 2 No. 1, pp. 43-57 —
|
" Indeed, about the Italian debate on foundations of Geometry, it is not rare to read comments in the same spirit of the following one, due to Jeremy Gray13. He is essentially reporting Hans Freudenthal’s point of view: ' When the distinguished mathematician and historian of mathematics Hans Freudenthal analysed Hilbert’s Grundlagen he argued that the link between reality and geometry appears to be severed for the first time in Hilbert’s work. However, he discovered that Hilbert had been preceded by the Italian mathematician Gino Fano in 1892. . . .' " 13 J. Gray, "The Foundations of Projective Geometry in Italy," Chapter 24 (pp. 269–279) in his book Worlds Out of Nothing , Springer (2010). |
Restoring the severed link —
See also Espacement and The Thing and I.
Related material —
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Monday, May 6, 2019
One Stuff
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Saturday, May 4, 2019
The Chinese Jars of Shing-Tung Yau
The title refers to Calabi-Yau spaces.
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A less "cosmic" but still noteworthy code — The Golay code.
This resides in a 12-dimensional space over GF(2).
Related material from Plato and R. T. Curtis —
A related Calabi-Yau "Chinese jar" first described in detail in 1905 —
A figure that may or may not be related to the 4x4x4 cube that
holds the classical Chinese "cosmic code" — the I Ching —
ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf
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Wednesday, May 1, 2019
For the First of May
"The purpose of mathematics cannot be derived from an activity
inferior to it but from a higher sphere of human activity, namely,
religion."
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also Ultron Cube.
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Thursday, April 18, 2019
Expanding the Unfolding*
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 .
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Wednesday, April 17, 2019
The Royal Society Diamond
The phrase "pattern recognition" in a news story about the
April 13 death of Princeton neuroscientist Charles Gross,
and yesterday's post about a fanciful "purloined diamond,"
suggest a review of a less fanciful diamond.
See also earlier posts tagged Fitch
and my own, much earlier and very
different, approach to such patterns —
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Wednesday, April 3, 2019
Nocturnal Object of Beauty
What is going on in this picture?
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Tuesday, April 2, 2019
Plan
"If this weren't a public situation, I'd be tempted to get into this on a
psychiatric level." — Christopher Alexander to Peter Eisenman, 1982
Scene from the sequel to Unbreakable and Split —
Not to mention elevation .
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Monday, March 25, 2019
Espacement
(Continued from the previous post.)
|
In-Between "Spacing" and the "Chôra " (Ch. 2 in Henk Oosterling & Ewa Plonowska Ziarek (Eds.), Intermedialities: Philosophy, Arts, Politics , Lexington Books, October 14, 2010) "The term 'spacing' ('espacement ') is absolutely central to Derrida's entire corpus, where it is indissociable from those of différance (characterized, in the text from 1968 bearing this name, as '[at once] spacing [and] temporizing' 1), writing (of which 'spacing' is said to be 'the fundamental property' 2) and deconstruction (with one of Derrida's last major texts, Le Toucher: Jean-Luc Nancy , specifying 'spacing ' to be 'the first word of any deconstruction' 3)." 1 Jacques Derrida, “La Différance,” in Marges – de la philosophie (Paris: Minuit, 1972), p. 14. Henceforth cited as D . 2 Jacques Derrida, “Freud and the Scene of Writing,” trans. A. Bass, in Writing and Difference (Chicago: University of Chicago Press, 1978), p. 217. Henceforth cited as FSW . 3 Jacques Derrida, Le Toucher, Jean-Luc Nancy (Paris: Galilée, 2000), p. 207. . . . . "… a particularly interesting point is made in this respect by the French philosopher, Michel Haar. After remarking that the force Derrida attributes to différance consists simply of the series of its effects, and is, for this reason, 'an indefinite process of substitutions or permutations,' Haar specifies that, for this process to be something other than a simple 'actualisation' lacking any real power of effectivity, it would need “a soubassement porteur ' – let’s say a 'conducting underlay' or 'conducting medium' which would not, however, be an absolute base, nor an 'origin' or 'cause.' If then, as Haar concludes, différance and spacing show themselves to belong to 'a pure Apollonism' 'haunted by the groundless ground,' which they lack and deprive themselves of,16 we can better understand both the threat posed by the 'figures' of space and the mother in the Timaeus and, as a result, Derrida’s insistent attempts to disqualify them. So great, it would seem, is the menace to différance that Derrida must, in a 'properly' apotropaic gesture, ward off these 'figures' of an archaic, chthonic, spatial matrix in any and all ways possible…." 16 Michel Haar, “Le jeu de Nietzsche dans Derrida,” Revue philosophique de la France et de l’Etranger 2 (1990): 207-227. . . . . … "The conclusion to be drawn from Democritus' conception of rhuthmos , as well as from Plato's conception of the chôra , is not, therefore, as Derrida would have it, that a differential field understood as an originary site of inscription would 'produce' the spatiality of space but, on the contrary, that 'differentiation in general' depends upon a certain 'spatial milieu' – what Haar would name a 'groundless ground' – revealed as such to be an 'in-between' more 'originary' than the play of differences it in-forms. As such, this conclusion obviously extends beyond Derrida's conception of 'spacing,' encompassing contemporary philosophy's continual privileging of temporization in its elaboration of a pre-ontological 'opening' – or, shall we say, 'in-between.' |
For permutations and a possible "groundless ground," see
the eightfold cube and group actions both on a set of eight
building blocks arranged in a cube (a "conducting base") and
on the set of seven natural interstices (espacements ) between
the blocks. Such group actions provide an elementary picture of
the isomorphism between the groups PSL(2,7) (acting on the
eight blocks) and GL(3,2) (acting on the seven interstices).
Espacements
For the Church of Synchronology —
See also, from the reported publication date of the above book
Intermedialities , the Log24 post Synchronicity.
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Thursday, March 21, 2019
Geometry of Interstices
Finite Galois geometry with the underlying field the simplest one possible —
namely, the two-element field GF(2) — is a geometry of interstices :
For some less precise remarks, see the tags Interstice and Interality.
The rationalist motto "sincerity, order, logic and clarity" was quoted
by Charles Jencks in the previous post.
This post was suggested by some remarks from Queensland that
seem to exemplify these qualities —
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Saturday, March 16, 2019
Monday, March 11, 2019
Ant-Man Meets Doctor Strange
The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .
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Thursday, February 28, 2019
Wikipedia Scholarship
Besides omitting the name Cullinane, the anonymous Wikipedia author
also omitted the step of representing the hypercube by a 4×4 array —
an array called in this journal a Galois tesseract.
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Tuesday, February 26, 2019
Citation
Some related material in this journal — See a search for k6.gif.
Some related material from Harvard —
Elkies's "15 simple transpositions" clearly correspond to the 15 edges of
the complete graph K6 and to the 15 2-subsets of a 6-set.
For the connection to PG(3,2), see Finite Geometry of the Square and Cube.
The following "manifestation" of the 2-subsets of a 6-set might serve as
the desired Wikipedia citation —
See also the above 1986 construction of PG(3,2) from a 6-set
in the work of other authors in 1994 and 2002 . . .
-
Gonzalez-Dorrego, Maria R. (Maria del Rosario),
(16,6) Configurations and Geometry of Kummer Surfaces in P3.
American Mathematical Society, Providence, RI, 1994. -
Dolgachev, Igor, and Keum, JongHae,
"Birational Automorphisms of Quartic Hessian Surfaces."
Trans. Amer. Math. Soc. 354 (2002), 3031-3057.
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Thursday, February 21, 2019
A Tale of Eight Building Blocks*
* For another such tale, see Eightfold Cube in this journal.
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Sunday, January 13, 2019
Sunday the Thirteenth (Revisited)
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Sunday, January 6, 2019
For Broom Bridge*
GL(2,3) is not unrelated to GL(3,2).
See Quaternion Automorphisms
and Spinning in Infinity.
* See Wikipedia.
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Tuesday, January 1, 2019
Child’s Play Continues — La Despedida
This post was suggested by the phrase "Froebel Decade" from
the search results below.
This journal a decade ago had a post on the late Donald Westlake,
an author who reportedly died of a heart attack in Mexico on Dec. 31,
2008, while on his way to a New Year's Eve dinner.
One of Westlake's books —
Related material —
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Sunday, December 30, 2018
Also Sprach Aitchison
The New Yorker reviewing "Bumblebee" —
"There is one reliable source for superhero sublimity,
and it’s all the more surprising that it’s a franchise with
no sacred inspiration whatsoever but, rather, of purely
and unabashedly mercantile origins: the 'Transformers'
series, based on a set of toys, in which Michael Bay’s
exhilarating filmmaking offers phantasmagorical textures
of an uncanny unconscious resonance."
— Richard Brody on December 29, 2018
"Before time began, there was the Cube."
— Optimus Prime
Some backstory — A Riddle for Davos, Jan. 22, 2014.
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Sunday, December 9, 2018
Quaternions in a Small Space
The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.
Click to enlarge —
Three links from the above finitegeometry.org webpage on the
quaternion group —
-
Visualizing GL(2,p) — A 1985 note illustrating group actions
on the 3×3 (ninefold) square. -
Another 1985 note showing group actions on the 3×3 square
transferred to the 2x2x2 (eightfold) cube. - Quaternions in an Affine Galois Plane — A webpage from 2010.
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
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Wednesday, December 5, 2018
The Hassenfeld Legacy
The Finkelstein Talisman —
"Before time began, there was the Cube."
— Optimus Prime in "Transformers" (Paramount, 2007)
Wikipedia on Hasbro —
Three American Jewish brothers,[6] Herman, Hillel, and Henry Hassenfeld[7]
founded Hassenfeld Brothers in Providence, Rhode Island in 1923 . . . .
The Hassenfeld Auction —
Also on September 16, 2015 —
The Hindman Image —
The Hood Warenkorb —
Under the Hood —
Megan Fox in "Transformers" (2007) —
This Way to the Egress —
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Monday, December 3, 2018
The Relativity Problem at Hiroshima
“This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
See also Relativity Problem and Diamonds and Whirls.
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Review
|
Tuesday, March 31, 2015
Würfel-Märchen
|
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Sunday, December 2, 2018
Iacta Est.
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Wednesday, November 28, 2018
Geometry and Experience
Einstein, "Geometry and Experience," lecture before the
Prussian Academy of Sciences, January 27, 1921–
|
… This view of axioms, advocated by modern axiomatics, purges mathematics of all extraneous elements, and thus dispels the mystic obscurity, which formerly surrounded the basis of mathematics. But such an expurgated exposition of mathematics makes it also evident that mathematics as such cannot predicate anything about objects of our intuition or real objects. In axiomatic geometry the words "point," "straight line," etc., stand only for empty conceptual schemata. That which gives them content is not relevant to mathematics. Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need which was felt of learning something about the behavior of real objects. The very word geometry, which, of course, means earth-measuring, proves this. For earth-measuring has to do with the possibilities of the disposition of certain natural objects with respect to one another, namely, with parts of the earth, measuring-lines, measuring-wands, etc. It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of real objects of this kind, which we will call practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We will call this completed geometry "practical geometry," and shall distinguish it in what follows from "purely axiomatic geometry." The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience. …. |
Later in the same lecture, Einstein discusses "the theory of a finite
universe." Of course he is not using "finite" in the sense of the field
of mathematics known as "finite geometry " — geometry with only finitely
many points.
Nevertheless, his remarks seem relevant to the Fano plane , an
axiomatically defined entity from finite geometry, and the eightfold cube ,
a physical object embodying the properties of the Fano plane.
|
I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental picture to which, with some practice, we shall soon grow accustomed. First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To "visualize" a theory therefore means to bring to mind that abundance of sensible experiences for which the theory supplies the schematic arrangement. In the present case we have to ask ourselves how we can represent that behavior of solid bodies with respect to their mutual disposition (contact) that corresponds to the theory of a finite universe. |
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Thursday, November 8, 2018
Reality vs. Axiomatic Thinking
From https://blogs.scientificamerican.com/…
|
A Few of My Favorite Spaces:
The intuition-challenging Fano plane may be By Evelyn Lamb on October 24, 2015
"…finite projective planes seem like |
For Fano's axiomatic approach, see the Nov. 3 Log24 post
"Foundations of Geometry."
For the Fano plane's basis in reality , see the eightfold cube
at finitegeometry.org/sc/ and in this journal.
See as well "Two Views of Finite Space" (in this journal on the date
of Lamb's remarks — Oct. 24, 2015).
Some context: Gödel's Platonic realism vs. Hilbert's axiomatics
in remarks by Manuel Alfonseca on June 7, 2018. (See too remarks
in this journal on that date, in posts tagged "Road to Hell.")
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Saturday, November 3, 2018
Foundations of Geometry
"costruire (o, dirò meglio immaginare) un ente" — Fano, 1892
"o, dirò meglio, costruire" — Cullinane, 2018
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Wednesday, October 24, 2018
Shadowlands
The previous post suggests a review.
Following the above reference to March 30, 2016 —
Following the above reference to Lovasz —
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Friday, October 19, 2018
Kaleidoscope and Old Lace
See posts now tagged "Kaleidoscope Society" and, more generally,
a search in this journal for "Kaleidoscope."
Related material —
Photo caption in a news story today:
"Father Gary Thomas attends the premiere of Warner Brothers’
'The Rite' at Grauman’s Chinese Theatre, in Los Angeles,
on January 26, 2011. Thomas is holding a special Mass
on Thursday and Saturday [Oct. 18 and 20] to counter
a planned hex on Supreme Court Justice Brett Kavanaugh."
See as well posts tagged "Rubik Exorcism."
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Monday, October 15, 2018
History at Bellevue
The previous post, "Tesserae for a Tesseract," contains the following
passage from a 1987 review of a book about Finnegans Wake —
"Basically, Mr. Bishop sees the text from above
and as a whole — less as a sequential story than
as a box of pied type or tesserae for a mosaic,
materials for a pattern to be made."
A set of 16 of the Wechsler cubes below are tesserae that
may be used to make patterns in the Galois tesseract.
Another Bellevue story —
“History, Stephen said, is a nightmare
from which I am trying to awake.”
— James Joyce, Ulysses
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Saturday, September 15, 2018
Eidetic Reduction in Geometry
|
"Husserl is not the greatest philosopher of all times. — Kurt Gödel as quoted by Gian-Carlo Rota Some results from a Google search — Eidetic reduction | philosophy | Britannica.com Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart … Phenomenology Online » Eidetic Reduction
The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research Eidetic reduction – New World Encyclopedia Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience. |
For example —
The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between* the subsquares or subcubes.
See the diamond theorem and the eightfold cube.
* Cf. posts tagged Interality and Interstice.
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Friday, August 31, 2018
Perception of Number
Review of yesterday's post Perception of Space —
From Harry Potter and the Philosopher's Stone (1997),
republished as "… and the Sorcerer's Stone ," Kindle edition:
In a print edition from Bloomsbury (2004), and perhaps in the
earliest editions, the above word "movements" is the first word
on page 168:
Click the above ellipse for some Log24 posts on the eightfold cube,
the source of the 168 automorphisms ("movements") of the Fano plane.
"Refined interpretation requires that you know that
someone once said the offspring of reality and illusion
is only a staggering confusion."
— Poem, "The Game of Roles," by Mary Jo Bang
Related material on reality and illusion —
an ad on the back cover of the current New Yorker —
"Hey, the stars might lie, but the numbers never do." — Song lyric
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Thursday, August 30, 2018
Perception* of Space
* A footnote in memory of a dancer who reportedly died
yesterday, August 29 — See posts tagged Paradigm Shift.
"Birthday, death-day — what day is not both?" — John Updike
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Saturday, August 25, 2018
“Waugh, Orwell. Orwell, Waugh.”
Suggested by a review of Curl on Modernism —
Related material —
Waugh + Orwell in this journal and …
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Wednesday, August 15, 2018
An Illusion of Brilliance
” . . . the 3 by 3, the six-sided, three-layer configuration of
the original Rubik’s Cube, which bestows an illusion of brilliance
on those who can solve it.”
— John Branch in the online New York Times today,
“Children of the Cube”:
https://www.nytimes.com/2018/08/15/sports/
cubing-usa-nationals-max-park.html
Cube-solving, like other sports, allows for displays of
impressive and admirable skill, if not “brilliance.”
The mathematics — group theory — that is sometimes associated
with Rubik’s Cube is, however, not a sport. See Rubik + Group
in this journal.
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Monday, August 6, 2018
The Girl with Kaleidoscope Eyes
|
“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.
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Thursday, August 2, 2018
Sunday, July 29, 2018
The Materialization
McCarthy's "materialization of plot and character" does not,
for me, constitute a proof that "there is being, after all,
beyond the arbitrary flux of existence."
Neither does the above materialization of 281 as the page
number of her philosophical remark.
See also the materialization of 281 as a page number in
the book Witchcraft by Charles Williams —
The materialization of 168 as a page number in a
Stephen King novel is somewhat more convincing,
but less convincing than the materialization of Klein's
simple group of of 168 elements in the eightfold cube.
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Sunday, July 22, 2018
Saturday, July 21, 2018
Building-Block Theory
(A sequel to yesterday’s Geometry for Jews)
From this journal on the above UCI posting date — April 6, 2018 —
From this journal on the above lecture date — April 26, 2018 —
illustrations in a post titled Defining Form —
For the relevance of the above material to building blocks,
see Eightfold Cube in this journal.
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Wednesday, July 18, 2018
Mira and the Intruders* Present …
Epiphany 2012: An Exercise
in Bulk Apperception
This post was suggested by a Marrific photo posted today,
by a New Orleans song as played by Bix Beiderbecke,
by a trailer for a new Zemeckis film that appeared at YouTube
on the way to the New Orleans song, and by
the longing for Bix by Mira Sorvino in "Intruders."
The YouTube Bix date, Jan. 6, 2012, suggested a trip back
to that date via a Zemeckis Cube (see "Ready Player One.")
From Log24 on Epiphany 2012 —
A version of the Zemeckis Cube —
From the Zemeckis trailer —
* See a Log24 search for Mira + Intruders.
Comments Off on Mira and the Intruders* Present …
Sunday, July 15, 2018
Jewish Oases
"… Lincoln Plaza Cinemas, the Juilliard String Quartet,
and the Strand Book Store remained oases
for cultural and intellectual stimulation."
— John S. Friedman in The Forward , Jan. 21, 2018
Read more:
https://forward.com/culture/392483/
how-fred-bass-dan-talbot-robert-mann
-shaped-new-york-culture/
From the Oasis in Steven Spielberg's "Ready Player One" (2018) —
I prefer, from a Log24 search for Flux Capacitor …
From "Raiders of the Lost Images" —
"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."
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Saturday, July 14, 2018
Expanding the Spiel
The walkerart.org passage above is from Feb. 17, 2011.
See also this journal on Feb. 17, 2011 —
"… Only by the form, the pattern,
Can words or music reach
The stillness…."
— T. S. Eliot,
Four Quartets
For further details, see Time Fold.
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Friday, July 13, 2018
From the P. T. Barnum School of Science Writing
Related drama —
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Thursday, July 12, 2018
The Bell
Three hidden keys open three secret gates
Wherein the errant will be tested for worthy traits
And those with the skill to survive these straits
Will reach The End where the prize awaits
— Ready Player One , by Ernest Cline
“Look, my favorite expression is,
‘When you go up to the bell, ring it,
or don’t go up to the bell.’ …
We’ve gone too far. We have to ring the bell.”
— Mel Brooks on “The Producers”
in The New York Times today.
A 2016 Scribner edition of Stephen King’s IT —
Related material —
Mystery box merchandise from the 2011 J. J. Abrams film Super 8
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Wednesday, July 11, 2018
Clarity and Precision
"The whole meaning of the word is
looking into something with clarity and precision,
seeing each component as distinct,
and piercing all the way through
so as to perceive the most fundamental reality
of that thing."
For the word itself, try a Web search on
noteworthy phrases above.
“. . . the utterly real thing in writing is
the only thing that counts . . . ."
— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935
"168"
— Page number in a 2016 Scribner edition
of Stephen King's IT
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Sunday, July 8, 2018
Sixers*
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Monday, July 2, 2018
In Memoriam
This post is in memory of dancer-choreographer Gillian Lynne,
who reportedly died at 92 on Sunday, July 1, 2018.
For a scene from her younger days, click on Errol Flynn above.
The cube contemplated by Flynn is from Log24 on Sunday.
"This is how we enter heaven, enter dancing."
— Paraphrase of Lorrie Moore (See Oct. 18, 2003.)
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A Mykonos* Narrative …
For Cady Heron
"Why you gotta be so mean?" — Taylor Swift
* See references to that Greek island in this journal.
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Sunday, July 1, 2018
Deutsche Ordnung
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”
Related structures —
Greg Egan’s animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
| Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets or Mozart’s late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. |
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
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Friday, June 29, 2018
For St. Stanley
The phrase "Blue Dream" in the previous post
suggests a Web search for Traumnovelle .
That search yields an interesting weblog post
from 2014 commemorating the 1999 dies natalis
(birth into heaven) of St. Stanley Kubrick.
Related material from March 7, 2014,
in this journal —
That 2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.
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Thursday, June 28, 2018
Wednesday, June 27, 2018
Taken In
A passage that may or may not have influenced Madeleine L'Engle's
writings about the tesseract :
|
From Mere Christianity , by C. S. Lewis (1952) —
"Book IV – Beyond Personality: I warned you that Theology is practical. The whole purpose for which we exist is to be thus taken into the life of God. Wrong ideas about what that life is, will make it harder. And now, for a few minutes, I must ask you to follow rather carefully. You know that in space you can move in three ways—to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two, you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body, say, a cube—a thing like a dice or a lump of sugar. And a cube is made up of six squares. Do you see the point? A world of one dimension would be a straight line. In a two-dimensional world, you still get straight lines, but many lines make one figure. In a three-dimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways—in ways you could not imagine if you knew only the simpler levels. Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings—just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine. In God's dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something super-personal—something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already. You may ask, "If we cannot imagine a three-personal Being, what is the good of talking about Him?" Well, there isn't any good talking about Him. The thing that matters is being actually drawn into that three-personal life, and that may begin any time —tonight, if you like. . . . . |
But beware of being drawn into the personal life of the Happy Family .
https://www.jstor.org/stable/24966339 —
"The colorful story of this undertaking begins with a bang."
And ends with …
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
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Monday, June 11, 2018
Arty Fact
The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.
The Eye of ARTI —
See also a post of May 19, "Uh-Oh" —
— and a post of June 6, "Geometry for Goyim" —
Mystery box merchandise from the 2011 J. J. Abrams film Super 8
An arty fact I prefer, suggested by the triangular computer-eye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
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Sunday, June 10, 2018
Number Concept
The previous post was suggested by some April 17, 2016, remarks
by James Propp on the eightfold cube.
Propp's remarks included the following:
"Here’s a caveat about my glib earlier remark that
'There are only finitely many numbers ' in a finite field.
It’s a bit of a stretch to call the elements of finite fields
'numbers'. Elements of GF(q ) can be thought of as
the integers mod q when q is prime, and they can be
represented by 0, 1, 2, …, q–1; but when q is a prime
raised to the 2nd power or higher, describing the
elements of GF(q ) is more complicated, and the word
'number' isn’t apt."
Related material —
See also this journal on the date of Propp's remarks — April 17, 2016.
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Thursday, June 7, 2018
Wednesday, June 6, 2018
Geometry for Goyim
Mystery box merchandise from the 2011 J. J. Abrams film Super 8 —
A mystery box that I prefer —
Click image for some background.
See also Nicht Spielerei .
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Monday, June 4, 2018
The Trinity Stone Defined
“Unsheathe your dagger definitions.” — James Joyce, Ulysses
The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .
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Thursday, May 31, 2018
Eightfold Suffering:
A New, Improved Version of Quantum Suffering !
Background for group actions on the eightfold cube —
See also other posts now tagged Quantum Suffering
as well as — related to the image above of the Great Wall —
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Quaternion Group Models
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Sunday, May 20, 2018
Some Style
Dialogue from the 1984 fourth draft of the script, as found on the Web,
for "Back to the Future" (1985) (apparently some changes were made
in the filming) —
A sort of "flux capacitor" (see previous post) —
… plus "e" for Einstein …
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Saturday, May 19, 2018
Flux Capacitor
For Tom Hanks and Dan Brown —
From "Raiders of the Lost Images" —
"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."
See as well a Google search for flux philosophy —
https://www.google.com/search?q=flux+philosophy.
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Thursday, May 17, 2018
Speak, Memory
On the film "Anna" in the previous post —
See also the above world premiere date in the posts of October 2013 —
esp. the post Conundrum.
Related material — An early scene in "Mindscape" . . .
. . . and "The Abacus Conundrum" in this journal.
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Friday, May 4, 2018
Art & Design
A star figure and the Galois quaternion.
The square root of the former is the latter.
See also a passage quoted here a year ago today
(May the Fourth, "Star Wars Day") —
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Sunday, April 29, 2018
Sunday School
From a search in this journal for Desmic —
"As the chaos grew . . . ."
"We have, in fact, the corners of a cube . . . ."
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Monday, April 23, 2018
Blockbuster Exhibition
Mike Hale in The New York Times online today —
Review: ‘Genius’ Paints Picasso by the Numbers
"… the production’s tinselly soul.
For instance, it’s on the record that Picasso’s lovers
Dora Maar and Marie-Thérèse Walter had
a wrestling match in his studio while he was
painting 'Guernica.' 'Genius' includes that
scene, naturally, but adds its own detail:
The altercation helps Picasso overcome a creative block
and gleefully set to work on the gigantic painting.
It may be news to scholars that one of art’s
greatest testaments to the horror of war was
inspired, in part, by the excitement of being
fought over by a pair of jealous women."
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Sunday, April 8, 2018
Design
From a Log24 post of Feb. 5, 2009 —
An online logo today —
See also Harry Potter and the Lightning Bolt.
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Saturday, April 7, 2018
Sides
The FBI holding cube in "The Blacklist" —
" 'The Front' is not the whole story . . . ."
— Vincent Canby, New York Times film review, 1976,
as quoted in Wikipedia.
See also Solomon's Cube in this journal.
Some may view the above web page as illustrating the
Glasperlenspiel passage quoted here in Summa Mythologica —
“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”
A less poetic meditation on the above 4x4x4 design cube —
"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."
See also a related remark by Lévi-Strauss in 1955:
"…three different readings become possible:
left to right, top to bottom, front to back."
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Monday, April 2, 2018
Review
See the posts of April 1 three years ago.
Some context from a personal Kindle library —
"Elementary Mathematics from an Ad" suggests . . .
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Sunday, April 1, 2018
Saturday, March 31, 2018
Illusion
“The greatest obstacle to discovery
is not ignorance —
it is the illusion of knowledge.”
— Daniel J. Boorstin,
Librarian of Congress,
quoted here in 2006.
Related material —
Remarks on Rubik’s Cube from June 13, 2014 and . . .
See as well a different Gresham, author of Nightmare Alley ,
and Log24 posts on that book and the film of the same name .
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Friday, March 30, 2018
Unorthodox Drama
The online New York Times this evening has an obituary
for "an unorthodox … drama scholar" who reportedly died
on Thursday, March 22, 2018.
Some drama in this journal from around that date — in posts
tagged "The Cubes" — includes the following excerpt from
a graphic novel:
"Program or be programmed."
— A saying by the author of the above graphic novel.
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Thursday, March 29, 2018
To Imagine (or, Better, to Construct)
The title reverses a phrase of Fano —
“costruire (o, dirò meglio immaginare).”
Illustrations of imagining (the Fano plane) and of constructing (the eightfold cube) —
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“Before Creation Itself . . .”
From the Diamond Theorem Facebook page —
A question three hours ago at that page —
“Is this Time Cube?”
Notes toward an answer —
And from Six-Set Geometry in this journal . . .
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Wednesday, March 28, 2018
On Unfairly Excluding Asymmetry
A comment on the the Diamond Theorem Facebook page —
Those who enjoy asymmetry may consult the "Expert's Cube" —
For further details see the previous post.
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Tuesday, March 27, 2018
Compare and Contrast
Related material on automorphism groups —
The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.
Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure above
than they are an eight-cube mechanical puzzle like the one below.
Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."
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Saturday, March 24, 2018
Sure, Whatever.
The search for Langlands in the previous post
yields the following Toronto Star illustration —
From a review of the recent film "Justice League" —
"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."
For other cubic adventures, see yesterday's post on A Piece of Justice
and the block patterns in posts tagged Design Cube.
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Friday, March 23, 2018
From the Personal to the Platonic
On the Oslo artist Josefine Lyche —
"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."
— Ann Cathrin Andersen,
http://bryggmagasin.no/2017/behind-the-glitter/
Personal —
The Rushkoff Logo
— From a 2016 graphic novel by Douglas Rushkoff.
See also Rushkoff and Talisman in this journal.
Platonic —
Compare and contrast the shifting hexagon logo in the Rushkoff novel above
with the hexagon-inside-a-cube in my "Diamonds and Whirls" note (1984).
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Wednesday, March 21, 2018
WISC RISC
WISC = Wechsler Intelligence Scale for Children
RISC = Reduced Instruction Set Computer or
Rust Inventory of Schizotypal Cognitions
See related material in earlier WISC RISC posts.
See also . . .
"Many parents ask us about the Block Design section
on the WISC and hope to purchase blocks and exercises
like those used on the WISC test. We explain that doing that
has the potential to invalidate their child's test results.
These Froebel Color Cubes will give you a tool to work with
your child on the skills tested for in the Block Design section
of the WISC in an ethical and appropriate way. These same
skills are applicable to any test of non-verbal reasoning like
the NNAT, Raven's or non-verbal sections of the CogAT or OLSAT. "
For a webpage that is perhaps un ethical and in appropriate,
see Block Designs in Art and Mathematics.
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Monday, March 12, 2018
Stein
Stein reportedly died at 100 last Friday (March 9).
Related material —
Textiles by Stein arranged on the six faces of a cube —
Ethel Stein, "Circus & Slapstick," 1996
See also a less amusing approach to
patterns on the faces of a cube.
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Wednesday, March 7, 2018
Unite the Seven.
Related material —
The seven points of the Fano plane within
|
|
"Before time began . . . ."
— Optimus Prime
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Tuesday, February 27, 2018
Raiders of the Lost Images
On the recent film "Justice League" —
From DC Extended Universe Wiki, "Mother Box" —
"However, during World War I, the British rediscovered
mankind's lost Mother Box. They conducted numerous studies
but were unable to date it due to its age. The Box was then
shelved in an archive, up until the night Superman died,
where it was then sent to Doctor Silas Stone, who
recognized it as a perpetual energy matrix. . . ." [Link added.]
The cube shape of the lost Mother Box, also known as the
Change Engine, is shared by the Stone in a novel by Charles Williams,
Many Dimensions . See the Solomon's Cube webpage.
See too the matrix of Claude Lévi-Strauss in posts tagged
Verwandlungslehre .
Some literary background:
Who speaks in primordial images speaks to us
as with a thousand trumpets, he grips and overpowers,
and at the same time he elevates that which he treats
out of the individual and transitory into the sphere of
the eternal. — C. G. JUNG
"In the conscious use of primordial images—
the archetypes of thought—
one modern novelist stands out as adept and
grand master: Charles Williams.
In The Place of the Lion he incarnates Plato’s
celestial archetypes with hair-raising plausibility.
In Many Dimensions he brings a flock of ordinary
mortals face to face with the stone bearing
the Tetragrammaton, the Divine Name, the sign of Four.
Whether we understand every line of a Williams novel
or not, we feel something deep inside us quicken
as Williams tells the tale.
Here, in The Greater Trumps , he has turned to
one of the prime mysteries of earth . . . ."
— William Lindsay Gresham, Preface (1950) to
Charles Williams's The Greater Trumps (1932)
For fans of what the recent series Westworld called "bulk apperception" —
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Saturday, February 17, 2018
The Binary Revolution
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —
- “The digital revolution based on the 2 symbols (0,1)”
- “The algebra of George Boole”
- “Binary codes”
- “Dirac’s spinors, with their up/down dichotomy”
These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .
I prefer other, purely geometric, reasons for the importance of GF(2) —
- The 2×2 square
- The 2x2x2 cube
- The 4×4 square
- The 4x4x4 cube
See Finite Geometry of the Square and Cube.
See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
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Tuesday, February 13, 2018
Attention Must Be Paid
For the late Anne M. Treisman, who reportedly died Friday, Feb. 9:
From "A Feature-Integration Theory of Attention" —
"The controversy between analytic and synthetic theories
of perception goes back many years: the Associationists
asserted that the experience of complex wholes is built
by combining more elementary sensations, while the
Gestalt psychologists claimed that the whole precedes
its parts, that we initially register unitary objects and
relationships, and only later, if necessary, analyze these
objects into their component parts or properties. This view
is still active now . . . ."
— Anne M. Treisman, University of British Columbia,
and Garry Gelade, Oxford University, in
Cognitive Psychology 12, 97-136 (1980)
"Before time began, there was the Cube." — Optimus Prime
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Tuesday, February 6, 2018
For Times Square Church*
The metaphor for metamorphosis no keys unlock.
— Steven H. Cullinane, "Endgame"
* See Times Square Church in this journal and
the posts of July 2010. Related material:
A Monday night death —
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Wednesday, January 17, 2018
At Heaven’s Gate
(Continued from September 12, 2005)
The previous post contrasted the number-triple 11-7-8 below
with number triples 12-9-5 and 12-5-9.
A perhaps more logical counterpart of the triple 11-7-8, based
on opposite locations of star-points or cube-edges, is
the triple 9-12-5. For a theological interpretation, see 9/12/05.
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Tuesday, January 16, 2018
An Orison for Ha-Why
Lines from characters played in the film by Tom Hanks and Halle Berry —
— Cloud Atlas , by David Mitchell (2004).
An orison of sorts from a post on Martin Scorsese's
birthday, Sunday, Nov. 18, 2007 —
Displayed on the BlackBerry are parts
of Log24 posts from October 25, 2007,
and October 24, 2007.
Related pattern geometry
From a Log24 search for Angleton + Brotherhood:
A photo of Angleton in a post from 12/9/5 —
From a post of 11/7/8 —
A cryptic note for Dan Brown:
The above dates 11/7/8 and 12/9/5 correspond to the corner-labels
(read clockwise and counter-clockwise) of the two large triangles
in the Finkelstein Talisman —
Above: More symbology for Tom Hanks from
this morning's post The Pentagram Papers.
The above symbology is perhaps better suited to Hanks in his
role as Forrest Gump than in his current role as Ben Bradlee.
For Hanks as Dan Brown's Harvard symbologist
Robert Langdon, see the interpretation 12/5/9, rather
than 12/9/5, of the above triangle/cube-corner label.
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The Pentagram Papers
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Saturday, January 6, 2018
Report from Red Mountain
Tom Wolfe in The Painted Word (1975):
“It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde understood, they
were not only theories but … hot news,
straight from the studios, from the scene.”
Harold Rosenberg in The New Yorker (click to enlarge)—
See also Interality and the Eightfold Cube .
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Friday, January 5, 2018
Seven Types of Interality*
* See the term interality in this journal.
For many synonyms, see
“The Human Seriousness of Interality,”
by Peter Zhang, Grand Valley State University,
China Media Research 11(2), 2015, 93-103.
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Wednesday, December 27, 2017
For Day 27 of December 2017
See the 27-part structure of
the 3x3x3 Galois cube
as well as Autism Sunday 2015.
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Tuesday, December 26, 2017
Raiders of the Lost Stone
Two Students of Structure
A comment on Sean Kelly's Christmas Morning column on "aliveness"
in the New York Times philosophy series The Stone —
Diana Senechal's 1999 doctoral thesis at Yale was titled
"Diabolical Structures in the Poetics of Nikolai Gogol."
Her mother, Marjorie Senechal, has written extensively on symmetry
and served as editor-in-chief of The Mathematical Intelligencer .
From a 2013 memoir by Marjorie Senechal —
|
"While I was in Holland my enterprising student assistant at Smith had found, in Soviet Physics – Crystallography, an article by N. N. Sheftal' on tetrahedral penetration twins. She gave it to me on my return. It was just what I was looking for. The twins Sheftal' described had evidently begun as (111) contact twins, with the two crystallites rotated 60o with respect to one another. As they grew, he suggested, each crystal overgrew the edges of the other and proceeded to spread across the adjacent facet. When all was said and done, they looked like they'd grown through each other, but the reality was over-and-around. Brilliant! I thought. Could I apply this to cubes? No, evidently not. Cube facets are all (100) planes. But . . . these crystals might not have been cubes in their earliest stages, when twinning occurred! I wrote a paper on "The mechanism of certain growth twins of the penetration type" and sent it to Martin Buerger, editor of Neues Jarbuch für Mineralogie. This was before the Wrinch symposium; I had never met him. Buerger rejected it by return mail, mostly on the grounds that I hadn't quoted any of Buerger's many papers on twinning. And so I learned about turf wars in twin domains. In fact I hadn't read his papers but I quickly did. I added a reference to one of them, the paper was published, and we became friends.[5]
After reading Professor Sheftal's paper I wrote to him in Moscow; a warm and encouraging correspondence ensued, and we wrote a paper together long distance.[6] Then I heard about the scientific exchanges between the Academies of Science of the USSR and USA. I applied to spend a year at the Shubnikov Institute for Crystallography, where Sheftal' worked. I would, I proposed, study crystal growth with him, and color symmetry with Koptsik. To my delight, I was accepted for an 11-month stay. Of course the children, now 11 and 14, would come too and attend Russian schools and learn Russian; they'd managed in Holland, hadn't they? Diana, my older daughter, was as delighted as I was. We had gone to Holland on a Russian boat, and she had fallen in love with the language. (Today she holds a Ph.D. in Slavic Languages and Literature from Yale.) . . . . |
Philosophy professors and those whose only interest in mathematics
is as a path to the occult may consult the Log24 posts tagged Tsimtsum.
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Thursday, December 21, 2017
Friday, December 15, 2017
Matter
The previous post, "Mind," suggests a search for "n+1" in this journal.
From that search —
The above psychoanalytic remarks suggest . . .
See also "Transformers" (2007).
"Before time began, there was the Cube."
— Optimus Prime
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Wednesday, November 22, 2017
Goethe on All Souls’ Day
David E. Wellbery on Goethe
From an interview published on 2 November 2017 at
http://literaturwissenschaft-berlin.de/interview-with-david-wellbery/
as later republished in
The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .
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Sunday, November 19, 2017
Galois Space
This is a sequel to yesterday's post Cube Space Continued.
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Sunday, October 29, 2017
File System… Unlocked
Logo from the above webpage —
See also the similar structure of the eightfold cube, and …
Related dialogue from the new film "Unlocked" —
1057
01:31:59,926 –> 01:32:01,301
Nice to have you back, Alice.
1058
01:32:04,009 –> 01:32:05,467
Don't be a stranger.
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Tuesday, October 24, 2017
Visual Insight
The most recent post in the "Visual Insight" blog of the
American Mathematical Society was by John Baez on Jan. 1, 2017 —
A visually related concept — See Solomon's Cube in this journal.
Chronologically related — Posts now tagged New Year's Day 2017.
Solomon's cube is the 4x4x4 case of the diamond theorem —
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Thursday, October 19, 2017
Graphic Design: Fast Forward
Typographical: »
Eightfold Cube:
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Friday, October 13, 2017
Speak, Memra
The above was suggested by a Log24 review of October 13, 2002,
which in turn suggested a Log24 search for Carousel that yielded
(from Bloomsday Lottery) —
See as well Asimov's "prime radiant," and an illustration
of the number 13 as a radiant prime …
"The Prime Radiant can be adjusted to your mind,
and all corrections and additions can be made
through mental rapport. There will be nothing to
indicate that the correction or addition is yours.
In all the history of the Plan there has been no
personalization. It is rather a creation of all of us
together. Do you understand?"
"Yes, Speaker!"
— Isaac Asimov,
Second Foundation , Ch. 8: Seldon's Plan
"Before time began, there was the Cube."
— Optimus Prime
See also Transformers in this journal.
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