Sunday, December 9, 2018
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 —
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
Friday, December 7, 2018
(Continued from this morning)
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also other Log24 posts tagged Kummerhenge.
Tuesday, December 4, 2018
March 8, 2018, was the date of death for Melbourne author Peter Temple.
Monday, December 3, 2018
“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.
Sunday, December 2, 2018
Continued . See as well a Log24 search for "Symmetric Generation."
Update of 2 PM ET —
A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —
http://www.math.sci.hiroshimau.ac.jp/
branched/files/2018/abstract/Aitchison.txt
Iain AITCHISON
Title:
Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II
Abstract:
Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness.
Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the HorrocksMumford bundle. Poincare's homology 3sphere, and Kummer's surface in real dimension 4 also play special roles.
In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other socalled `Arnol'd Trinities'.
Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss interrelationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set.
Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential interconnectedness of those exceptional objects considered.
Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective.
Some new results arising from this work will also be given, such as an alternative graphicillustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6 and 8component links, the latter related by Thurston to Klein's quartic curve.

See also yesterday morning's post, "Character."
Update: For a followup, see the next Log24 post.
Friday, November 30, 2018
Continued from March 13, 2011 —
"…as we saw, there are two different Latin squares of order 4…."
— Peter J. Cameron, "The Shrikhande Graph," August 26, 2010
Cameron counts Latin squares as the same if they are isotopic .
Some further context for Cameron's remark—
A new website illustrates a different approach to Latin squares of order 4 —
https://shc7596.wixsite.com/website .
Thursday, November 29, 2018
Clicking on Zong in the above post leads to a 2005 article
in the Bulletin of the American Mathematical Society .
See also the eightfold cube and interality .
Wednesday, November 28, 2018
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 earthmeasuring, proves this. For earthmeasuring has to do with the possibilities of the disposition of certain natural objects with respect to one another, namely, with parts of the earth, measuringlines, measuringwands, 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 practicallyrigid bodies. To be able to make such assertions, geometry must be stripped of its merely logicalformal 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 practicallyrigid 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 geometricalphysical 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.

Saturday, November 24, 2018
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Friday, November 23, 2018
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Thursday, November 22, 2018
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"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
Note also the four 4×4 arrays surrounding the central diamond
in the chi of the chirho page of the Book of Kells —
From a Log24 post
of March 17, 2012
"Interlocking, interlacing, interweaving"
— Condensed version of page 141 in Eddington's
1939 Philosophy of Physical Science
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Tuesday, November 20, 2018
(Continued)
Musical accompaniment from Sunday morning —
Update of Nov. 21 —
The reader may contrast the above Squarespace.com logo
(a rather serpentine version of the acronym SS) with a simpler logo
for a square space (the Galois window ):
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Sunday, November 18, 2018
Update of Nov. 19 —
"Design is how it works." — Steve Jobs
See also www.cullinane.design.
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The title is a useful search phrase:
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Saturday, November 17, 2018
See as well . . .
. . . and posts tagged Alperin.
On its current homepage, the American Mathematical Society
links to a Nov. 15 blog post illustrating the Stan Lee approach
to mathematics:
Stories: "Math needs more stories. All kinds of stories…" —
See too Mathematics and Narrative in this journal.
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Friday, November 16, 2018
I.e. (click to enlarge) —
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Thursday, November 15, 2018
Mathematics —
See (for instance) a research group at Ghent University.
For those who prefer narrative . . .
See also . . .
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Wednesday, November 14, 2018
"Open the pod bay doors, Bernard."
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Tuesday, November 13, 2018
From the 1955 film "Blackboard Jungle" —
From a trailer for the recent film version of A Wrinkle in Time —
Detail of the phrase "quantum tesseract theorem":
From the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
Related mathematics from Koen Thas that some might call a
"quantum tesseract theorem" —
Some background —
See also posts tagged Dirac and Geometry. For more
background on finite geometry, see a web page
at Thas's institution, Ghent University.
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Thursday, November 8, 2018
From https://blogs.scientificamerican.com/…
A Few of My Favorite Spaces:
The Fano Plane
The intuitionchallenging Fano plane may be
the smallest interesting configuration
of points and lines.
By Evelyn Lamb on October 24, 2015
"…finite projective planes seem like
a triumph of purely axiomatic thinking
over any hint of reality. . . ."

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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From "The Trials of Device" (April 24, 2017) —
See also Wittgenstein in a search for "Ein Kampf " in this journal.
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Tuesday, November 6, 2018
A phrase from the previous post —
"a sizeeight dame in a sizesix dress" —
suggests a review . . .
See as well the diamondtheorem correlation and . . .
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Saturday, November 3, 2018
"costruire (o, dirò meglio immaginare) un ente" — Fano, 1892
"o, dirò meglio, costruire" — Cullinane, 2018
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Thursday, November 1, 2018
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Tuesday, October 23, 2018
Click the image for some context.
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Saturday, October 20, 2018
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Related Log24 posts — See Blade + Chalice.
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Wednesday, October 17, 2018
"… what once seemed pure abstractions have turned out to
underlie real physical processes."
— https://breakthroughprize.org/Prize/3
Related material from the current New Yorker —
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From "The Phenomenology of Mathematical Beauty,"
by GianCarlo Rota —
The Lightbulb Mistake
. . . . Despite the fact that most proofs are long, and despite our need for extensive background, we think back to instances of appreciating mathematical beauty as if they had been perceived in a moment of bliss, in a sudden flash like a lightbulb suddenly being lit. The effort put into understanding the proof, the background material, the difficulties encountered in unraveling an intricate sequence of inferences fade and magically disappear the moment we become aware of the beauty of a theorem. The painful process of learning fades from memory, and only the flash of insight remains.
We would like mathematical beauty to consist of this flash; mathematical beauty should be appreciated with the instantaneousness of a lightbulb being lit. However, it would be an error to pretend that the appreciation of mathematical beauty is what we vaingloriously feel it should be, namely, an instantaneous flash. Yet this very denial of the truth occurs much too frequently.
The lightbulb mistake is often taken as a paradigm in teaching mathematics. Forgetful of our learning pains, we demand that our students display a flash of understanding with every argument we present. Worse yet, we mislead our students by trying to convince them that such flashes of understanding are the core of mathematical appreciation.
Attempts have been made to string together beautiful mathematical results and to present them in books bearing such attractive titles as The One Hundred Most Beautiful Theorems of Mathematics . Such anthologies are seldom found on a mathematician’s bookshelf. The beauty of a theorem is best observed when the theorem is presented as the crown jewel within the context of a theory. But when mathematical theorems from disparate areas are strung together and presented as “pearls,” they are likely to be appreciated only by those who are already familiar with them.
The Concept of Mathematical Beauty
The lightbulb mistake is our clue to understanding the hidden sense of mathematical beauty. The stark contrast between the effort required for the appreciation of mathematical beauty and the imaginary view mathematicians cherish of a flashlike perception of beauty is the Leitfaden that leads us to discover what mathematical beauty is.
Mathematicians are concerned with the truth. In mathematics, however, there is an ambiguity in the use of the word “truth.” This ambiguity can be observed whenever mathematicians claim that beauty is the raison d’être of mathematics, or that mathematical beauty is what gives mathematics a unique standing among the sciences. These claims are as old as mathematics and lead us to suspect that mathematical truth and mathematical beauty may be related.
Mathematical beauty and mathematical truth share one important property. Neither of them admits degrees. Mathematicians are annoyed by the graded truth they observe in other sciences.
Mathematicians ask “What is this good for?” when they are puzzled by some mathematical assertion, not because they are unable to follow the proof or the applications. Quite the contrary. Mathematicians have been able to verify its truth in the logical sense of the term, but something is still missing. The mathematician who is baffled and asks “What is this good for?” is missing the sense of the statement that has been verified to be true. Verification alone does not give us a clue as to the role of a statement within the theory; it does not explain the relevance of the statement. In short, the logical truth of a statement does not enlighten us as to the sense of the statement. Enlightenment , not truth, is what the mathematician seeks when asking, “What is this good for?” Enlightenment is a feature of mathematics about which very little has been written.
The property of being enlightening is objectively attributed to certain mathematical statements and denied to others. Whether a mathematical statement is enlightening or not may be the subject of discussion among mathematicians. Every teacher of mathematics knows that students will not learn by merely grasping the formal truth of a statement. Students must be given some enlightenment as to the sense of the statement or they will quit. Enlightenment is a quality of mathematical statements that one sometimes gets and sometimes misses, like truth. A mathematical theorem may be enlightening or not, just as it may be true or false.
If the statements of mathematics were formally true but in no way enlightening, mathematics would be a curious game played by weird people. Enlightenment is what keeps the mathematical enterprise alive and what gives mathematics a high standing among scientific disciplines.
Mathematics seldom explicitly acknowledges the phenomenon of enlightenment for at least two reasons. First, unlike truth, enlightenment is not easily formalized. Second, enlightenment admits degrees: some statements are more enlightening than others. Mathematicians dislike concepts admitting degrees and will go to any length to deny the logical role of any such concept. Mathematical beauty is the expression mathematicians have invented in order to admit obliquely the phenomenon of enlightenment while avoiding acknowledgment of the fuzziness of this phenomenon. They say that a theorem is beautiful when they mean to say that the theorem is enlightening. We acknowledge a theorem’s beauty when we see how the theorem “fits” in its place, how it sheds light around itself, like Lichtung — a clearing in the woods. We say that a proof is beautiful when it gives away the secret of the theorem, when it leads us to perceive the inevitability of the statement being proved. The term “mathematical beauty,” together with the lightbulb mistake, is a trick mathematicians have devised to avoid facing up to the messy phenomenon of enlightenment. The comfortable oneshot idea of mathematical beauty saves us from having to deal with a concept that comes in degrees. Talk of mathematical beauty is a copout to avoid confronting enlightenment, a copout intended to keep our description of mathematics as close as possible to the description of a mechanism. This copout is one step in a cherished activity of mathematicians, that of building a perfect world immune to the messiness of the ordinary world, a world where what we think should be true turns out to be true, a world that is free from the disappointments, ambiguities, and failures of that other world in which we live.

How many mathematicians does it take to screw in a lightbulb?
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Monday, October 15, 2018
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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The structure at top right is that of the
ROMAORAMMAROAMOR square
in the previous post.
* "Zingari shoolerim" is from
Finnegans Wake .
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Friday, October 5, 2018
The above figure illustrating 24 permutations is dated "2007."
An earlier permutations figure, archived on June 17, 2006 —
An illustration of the above title "I Want To Be a Mathematician" —
Wanting does not make it so.
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Wednesday, October 3, 2018
A Catholic philosopher —
Related art —
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Sunday, September 30, 2018
Found today in an Internet image search, from the website of
an anonymous amateur mathematics enthusiast —
Forming Gray codes in the eightfold cube with the eight
I Ching trigrams (bagua ) —
This journal on Nov. 7, 2016 —
A different sort of cube, from the makers of the recent
Netflix miniseries "Maniac" —
See also Rubik in this journal.
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Saturday, September 29, 2018
The title is from Warburg. The Zwischenraum lines and shaded "cuts"
below are to be added together in characteristic two, i.e., via the
settheoretic symmetric difference operator.
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Sunday, September 23, 2018
The New York Times 's Sunday School today —
I prefer the three bricks of the Miracle Octad Generator —
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Saturday, September 22, 2018
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Venturi reportedly died on Tuesday, September 18.*
See also this journal on that date.
* Fact check:
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Norwegian artist Josefine Lyche —
Lyche's shirt honors the late Kurt Cobain.
"Here we are now, entertain us."
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Friday, September 21, 2018
Monday, September 17, 2018
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Or: Zero Dark Zero
" Lying at the axis of everything, zero is both real and imaginary. Lovelace was fascinated by zero; as was Gottfried Leibniz, for whom, like mathematics itself, it had a spiritual dimension. It was this that let him to imagine the binary numbers that now lie at the heart of computers: 'the creation of all things out of nothing through God's omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing.' He also wrote, 'The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and nonbeing.' "
— A footnote from page 229 of Sydney Padua's
April 21, 2015, book on Lovelace and Babbage

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Saturday, September 15, 2018
Update of 10:18 AM the same day —
See also Logicomix in this journal and, at Harvard,
http://www.math.harvard.edu/~mazur/ —

September 6, 2018: Eric Maskin, Amartya Sen and I
are giving a course this semester: 'Axiomatic Reasoning'
(PHIL 273B). Introduction to Axiomatic Reasoning gives a
general sense of what we intend to cover.
Update of 10:48 AM the same day —
See Log24 on the date of Tieszen's death.
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"Husserl is not the greatest philosopher of all times.
He is the greatest philosopher since Leibniz."
— Kurt Gödel as quoted by GianCarlo 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.
Terminology: Eidos

For example —
The reduction of twocolorings and fourcolorings 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, September 14, 2018
Underlying the I Ching structure is the finite affine space
of six dimensions over the Galois field with two elements.
In this field, "1 + 1 = 0," as noted here Wednesday.
See also other posts now tagged Interstice.
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Thursday, September 13, 2018
The title is from the 2013 paper by Latsis in the previous post.
The symmetries of the interstices at right underlie
the symmetries of the images at left.
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The previous post, "One Plus One," suggests some further
arthistorical remarks on interality —
From Third Text , 2013, Vol. 27, No. 6, pp. 774–785 —
"Genealogy of the Image in Histoire(s) du Cinéma : Godard, Warburg and the Iconology of the Interstice"
By Dimitrios S. Latsis
* * * * P. 775 —
My discussion will focus on the signiﬁcance of the concept of the ‘space inbetween,’ its importance for Godard’s work and its role in a relational historiography of images more broadly. I hope to corroborate how Godard functions as a twentyﬁrst century archaeologist of the moving image, constructing a metacinematic collage that, while consisting of an indexing of (almost exclusively) preexisting ﬁlmic samples, ends up becoming a hybrid work of art in its own right. Godard, in the ﬁnal analysis, expands the Warburgian programme of iconology into that of a cinematographic iconology of the interstice.
* * * * P. 777 —
Godard conceives of the image only in the plural, in the intermediate space between two images, be it a prolonged one (in Histoire(s) there are frequent instances of black screens) or a nonexistent one (superimposition, copresence of two images on screen). He comments: ‘[For me] it’s always two, begin by showing two images rather than one, that’s what I call image, the one made up of two’ [18] and elsewhere, ‘I perceived . . . cinema is that which is between things, not things [themselves] but between one and another.’ [19]
18. JeanLuc Godard and Youssef Ishaghpour, "Archéologie du cinéma et mémoire du siècle," Farrago ,Tours, 2000, p. 27. The title of this work is reﬂective of the Godardian agenda that permeates Histoire(s) .
19. JeanLuc Godard, "Introduction à une véritable histoire du cinéma," Albatros , Paris,1980, p. 145
* * * * P. 783 —
If it is in ‘the inbetween’ that thought is born, then for Godard cinematography as ‘a form that thinks . . . was born with the advent of modern painting.’ [62]
62. Godard and Ishaghpour, op. cit., pp 45–46.
* * * * P. 785 —
Warburg commented on the signiﬁcation of the black spaces that he placed between images in his analysis of the network of intervals in Mnemosyne , by quoting Johann Wolfgang Goethe’s dictum ‘the truth inhabits the middle space.’ [68] This citation induces a feeling of déjàvu for the viewer of Histoire(s). The link was not missed by Warburg himself, as one of his diary entries testiﬁes: ‘We can compare this phenomenon [the iconology of the interval] to that of the cinematic montage, the domain of the interpretation is an intervallic one.’ [69]
68. Warburg, Mnemosyne , pp 135–146.
69. Warburg is quoted in DidiHuberman, L’image survivante, p. 503. (Georges DidiHuberman, L’image survivante. Histoire de l’art et temps des fantômes selon Aby Warburg , Minuit, Paris, 2002)

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Sunday, September 9, 2018
"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.
Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."
— From p. 192 of "The Phenomenology of Mathematical Proof,"
by GianCarlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183196. Published by: Springer.
Stable URL: https://www.jstor.org/stable/20117627.
Related figures —
Note the 3×3 subsquare containing the triangles ABC, etc.
"That in which space itself is contained" — Wallace Stevens
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Saturday, September 8, 2018
For example —
See also Interality in this journal.
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Wednesday, September 5, 2018
See also, in this journal, 23cycle.
Update of Sept. 6, 2018, 9:05 AM ET: "The Cubist Method" —
Multifaceted narrative by James Joyce —
Multifaceted structures in pure mathematics, from Plato and R. T. Curtis —
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Saturday, September 1, 2018
The date of Ron Shaw's 2016 death appears to be June 21:
All other Internet sources I have seen omit the June 21 date.
This journal on that date —
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Friday, August 31, 2018
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 yellow 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
* A footnote in memory of a dancer who reportedly died
yesterday, August 29 — See posts tagged Paradigm Shift.
"Birthday, deathday — what day is not both?" — John Updike
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Monday, August 27, 2018
From …
Thinking in Four Dimensions
By Dusa McDuff
"I’ve got the rather foolhardy idea of trying to explain
to you the kind of mathematics I do, and the kind of
ideas that seem simple to me. For me, the search
for simplicity is almost synonymous with the search
for structure.
I’m a geometer and topologist, which means that
I study the structure of space …
. . . .
In each dimension there is a simplest space
called Euclidean space … "
— In Roman Kossak, ed.,
Simplicity: Ideals of Practice in Mathematics and the Arts
(Kindle Locations 705710, 735). Kindle Edition.
For some much simpler spaces of various
dimensions, see Galois Space in this journal.
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Saturday, August 25, 2018
Suggested by a review of Curl on Modernism —
Related material —
Waugh + Orwell in this journal and …
Cube Bricks 1984 —
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In literature —
In film —
In mathematics —
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Sunday, August 19, 2018
John Calder, an independent British publisher who built a prestigious list
of authors like Samuel Beckett and Heinrich Böll and spiritedly defended
writers like Henry Miller against censorship, died on Aug. 13 in Edinburgh.
He was 91.
— Richard Sandomir in the online New York Times this evening
On Beckett —
Also on August 13th —
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Thursday, August 16, 2018
(Continued)
Mathematics:
Narrative:
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Wednesday, August 15, 2018
" . . . the 3 by 3, the sixsided, threelayer 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/
cubingusanationalsmaxpark.html
Cubesolving, 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.
Comments Off on An Illusion of Brilliance
From previous posts on this topic —
Comments Off on An Exceptional Isomorphism
Tuesday, August 14, 2018
Monday, August 13, 2018
"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think,
Oxford University Press, 1999, page 268

". . . a perfect triptych of horsitude"
— James Parker on the 2007 film "Michael Clayton"
Related material —
Horsitude in the 4×2 grid, and
Comments Off on Trojan Horsitude
Friday, August 10, 2018
Comments Off on For Noah Cross in Chinatown
Thursday, August 9, 2018
From a search in this journal for "True Grid,"
a fanciful description of the 3×3 grid —
"This is the garden of Apollo,
the field of Reason…."
– John Outram, architect
A fanciful instance of the 4×2 grid in
a scene from the film "The Master" —
A fanciful novel referring to the number 8,
and a not sofanciful reference:
Illustrated above are Katherine Neville's novel The Eight and the
"knight" coordinatization of the 4×2 grid from a page on the exceptional
isomorphism between PSL(3,2) (alias GL(3,2)) and PSL(2,7) — groups
of, respectively, degree 7 and degree 8.
Literature related to the above remarks on grids:
Ross Douthat's New York Times column yesterday purported, following
a 1946 poem by Auden, to contrast students of the humanities with
technocrats by saying that the former follow Hermes, the latter Apollo.
I doubt that Apollo would agree.
Comments Off on True Grids
Wednesday, August 8, 2018
From mathoverflow.net on Dec. 7, 2016 —
The exceptional isomorphism between
PGL(3,2) and PSL(2,7): geometric origin?
Essentially the same question was asked earlier at
math.stackexchange.com on Aug. 2, 2010.
See also this journal in November 2017 —
"Read something that means something."
— New Yorker ad
^{.}
Background — Relativity Problem in Log24.
Comments Off on 8/8
Monday, August 6, 2018
“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. 247248).
Penguin Publishing Group. Kindle Edition.

Related material from Log24 on All Hallows' Eve 2013 —
Comments Off on The Girl with Kaleidoscope Eyes
Sunday, August 5, 2018
Comments Off on Board Structure
Saturday, August 4, 2018
A novel by Harry Albers featuring his fictional Pacific Science Institute:
See the real Pacific Science Institute (PSI) in the previous post.
Synchronology check —
Related literary remarks —
— Cloud Atlas , by David Mitchell (2004).
Comments Off on The Secret Life of Harry Albers
An alleged manifestation in physics, from Scientific American —
Manifestations in pure mathematics, from Plato and R. T. Curtis —
For some entertaining literary manifestations, see Wrinkle.
Comments Off on Manifestations of Exquisite Geometry
Tuesday, July 31, 2018
From the Wikipedia article on the 1994 film "North" —
"North is forced to ship himself home in a FedEx box.
He reaches his house … but as he runs toward his parents,
an assassin takes aim. As he squeezes the trigger,
North awakens in the mall, now empty. The Easter Bunny
takes him home . . . ."
The film's author —
Zweibel's FedEx box suggests a review of
the post Geometry for Goyim (June 6, 2018).
Comments Off on Easter Storytelling
"A blank underlies the trials of device." — Wallace Stevens
.
Comments Off on Lexicon
Sunday, July 29, 2018
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.
Comments Off on The Materialization
Saturday, July 28, 2018
Floyd: "You're trying to figure out this length.
That's the hypotenuse. So you have to
know this angle."
Comments Off on Angle
Wednesday, July 25, 2018
Update of 10:30 PM ET the same day —
For some philosophical background, including an I Ching diagram, see . . .
See as well my own 8×8 diagram (1989) related to the I Ching .
Comments Off on Bucharest Physics
Tuesday, July 24, 2018
Comments Off on Artistic Style
Monday, July 23, 2018
Click to enlarge:
Above are the 7 frames of an animated gif from a Wikipedia article.
* For the Furey of the title, see a July 20 Quanta Magazine piece —
See also the eightfold cube in this journal.
"Before time began . . . ." — Optimus Prime
Comments Off on Eightfold Cube for Furey*
From the April 1st publication date of "Interality Shows Through,"
by Geling Shang —
See too yesterday's post Space.
Comments Off on Space 101
Sunday, July 22, 2018
Saturday, July 21, 2018
"For many of us, the geometry course sounded the death knell
for our progress — and interest — in mathematics."
— "Shape and Space in Geometry"
© 19972003 Annenberg/CPB. All rights reserved.
Legal Policy

Comments Off on ComicCon 2018
(A sequel to yesterday's Geometry for Jews)
From Dr/ Yau's own website —
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.
Comments Off on BuildingBlock Theory
Friday, July 20, 2018
(Continued)
Click image to enlarge —
A portrait from the home page of David Eppstein,
a professor at the University of California, Irvine
"… how can an image with 8 points and 8 lines
possibly represent a space with 7 points and 7 lines???"
— David Eppstein, 21 December 2015
See " Projective spaces as 'collapsed vector spaces,' "
page 203 in Geometry and Symmetry by Paul B. Yale,
published by HoldenDay in 1968.
Comments Off on Geometry for Jews
Wednesday, July 18, 2018
From "The Educated Imagination: A Website Dedicated
to Northrop Frye" —
"In one of the notebooks for his first Bible book Frye writes,
'For at least 25 years I’ve been preoccupied by
the notion of a key to all mythologies.' . . . .
Frye made a valiant effort to provide a key to all mythology,
trying to fit everything into what he called the Great Doodle. . . ."
From a different page at the same website —
Here Frye provides a diagram of four sextets.
I prefer the Miracle Octad Generator of R. T. Curtis —
.
Comments Off on Doodle
Monday, July 16, 2018
"The novel has a parallel narrative that eventually
converges with the main story."
— Wikipedia on a book by Foer's novelist brother
Public Squares
An image from the online New York Times
on the date, July 6,
of the above Atlantic article —
An image from "Blackboard Jungle," 1955 —
"Through the unknown, remembered gate . . . ."
— T. S. Eliot, Four Quartets
Comments Off on Greatly Exaggerated Report
Sunday, July 15, 2018
"… 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/
howfredbassdantalbotrobertmann
shapednewyorkculture/
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."
Comments Off on Jewish Oases
Saturday, July 14, 2018
(Continued)
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.
Comments Off on Expanding the Spiel
Friday, July 13, 2018
From a Log24 post of March 13, 2003 —
"For many of us, the geometry course sounded the death knell
for our progress — and interest — in mathematics."
— "Shape and Space in Geometry"
© 19972003 Annenberg/CPB. All rights reserved.
Legal Policy

Comments Off on Segue for Harlan Ellison
Thursday, July 12, 2018
“… the utterly real thing in writing is the only thing that counts…."
— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935
"Omega is as real as we need it to be."
— Burt Lancaster in "The Osterman Weekend"
Comments Off on Kummerhenge Illustrated
Wednesday, July 11, 2018
"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
Comments Off on Clarity and Precision
Sunday, July 8, 2018
Friday, July 6, 2018
"… Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."
— T. S. Eliot, "Burnt Norton," 1936
"Read something that means something."
— Advertising slogan for The New Yorker
The previous post quoted some mystic meditations of Octavio Paz
from 1974. I prefer some less mystic remarks of Eddington from
1938 (the Tanner Lectures) published by Cambridge U. Press in 1939 —
"… we have sixteen elements with which to form a groupstructure" —
See as well posts tagged Dirac and Geometry.
Comments Off on Something
Thursday, July 5, 2018
Some context for what Heidegger called
das SpiegelSpiel des Gevierts
From Helen Lane's translation of El Mono Gramático ,
a book by Nobel winner Octavio Paz first published
in Barcelona by Seix Barral in 1974 —
Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oftrepeated and everdifferent metaphor of identity.
— Paz, Octavio. The Monkey Grammarian
(Kindle Locations 11851191).
Arcade Publishing. Kindle Edition.

A related 1960 meditation from Claude LéviStrauss taken from a
Log24 post of St. Andrew's Day 2017, "The Matrix for Quantum Mystics":
The Matrix of LéviStrauss —
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude LéviStrauss
I prefer the earlier, betterknown, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets
(from The Matrix Meets the Grid) —
.
Comments Off on Paz:
Tuesday, July 3, 2018
The phrase "quantum space" in today's 10:45 AM post
was used earlier in a book title —
Amazon.com gives the Quantum Space publication date
for its Kindle edition as April 10, 2017.
I prefer my own remarks of April 10, 2017 —
Comments Off on Propriation
Combining concepts from the two previous posts, we have the above title.
A more concise alternative title …
Lost in the Matrix
For some related non fiction, see posts tagged Dirac and Geometry.
Comments Off on Lost in Quantum Space
Monday, July 2, 2018
This post is in memory of dancerchoreographer 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.)
Comments Off on In Memoriam
Sunday, July 1, 2018
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 tetrahedral key to the arrangement of the 56 triangles within the above
structure, 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
Jan Zwicky, University of Victoria
09:0009:40 Friday, April 5, 2013
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 —
Comments Off on Deutsche Ordnung
Friday, June 29, 2018
From a post of July 25, 2008, "56 Triangles," on the Klein quartic
and the eightfold cube —
"Baez's discussion says that the Klein quartic's 56 triangles
can be partitioned into 7 eighttriangle Egan 'cubes' that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eighttriangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane."
Related material from 1975 —
More recently …
Comments Off on Triangles in the Eightfold Cube
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.
Comments Off on For St. Stanley
Wednesday, June 27, 2018
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:
or First Steps in the Doctrine of the Trinity"
. . . .
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 twodimensional world, you still get straight lines, but many lines make one figure. In a threedimensional 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 superpersonal—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 threepersonal 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 threepersonal 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 …
Martin Gardner on Galois—
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
Comments Off on Taken In
Monday, June 25, 2018
<title datarh="true">Frank Heart, Who Linked Computers Before the Internet, Dies at 89 – The New York Times</title>
. . . .
<meta datarh="true" name="description" itemprop="description" content="Mr. Heart’s team built the gateway device for the Arpanet, the precursor to the internet. Data networking was so new then, they made it up as they went."/>
. . . .
<meta datarh="true" property="article:published" itemprop="datePublished" content="20180625T19:16:17.000Z"/>

See also yesterday's "For 6/24" and …
Comments Off on The Gateway Device
Sunday, June 24, 2018
A clue to the relationship between the Kummer (16, 6)
configuration and the large Mathieu group M_{24} —
Related material —
See too the diamondtheorem correlation.
Comments Off on For 6/24
Saturday, June 23, 2018
Backstory for fiction fans, from Log24 on June 11 —
Related non fiction —
See as well the structure discussed in today's previous post.
Comments Off on Meanwhile …
From Nanavira Thera, "Early Letters," in Seeking the Path —
"nine possibilities arising quite naturally" —
Compare and contrast with Hudson's parametrization of the
4×4 square by means of 0 and the 15 2subsets of a 6set —
Comments Off on Plan 9 from Inner Space
Thursday, June 21, 2018
A Buddhist view —
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
A Christian view —
A formal view —
From a Log24 search for High Concept:
See also Galois Tesseract.
Comments Off on Models of Being
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
"… a treatise on Kummer's quartic surface."
The "supermathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.
See Hudson + Kummer in this journal.
See also posts tagged Dirac and Geometry.
Comments Off on Dirac and Geometry (continued)
See also the Omega Matrix in this journal.
Comments Off on Kummerhenge
Wednesday, June 20, 2018
"… what we’re witnessing is not a glitch. It’s a feature…."
— A Boston Globe columnist on June 19.
An image from this journal at the beginning of Bloomsday 2018 —
An encountered feature , from the midnight beginning of June 16 —
Literary Symbolism
"… what we’re witnessing is not a glitch. It’s a feature…."
The glitch encountered on Bloomsday by Agent Smith (who represents
the academic world) is the author of the above page, John P. Anderson.
The feature is the book that Anderson quotes, James Joyce
by Richard Ellmann (first published in 1959, revised in 1982).
Comments Off on Feature
Tuesday, June 19, 2018
Comments Off on Ici vient M. Jordan
Saturday, June 16, 2018
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See too "The Ruler of Reality" in this journal.
Related material —
A more esoteric artifact: The Kummer 16_{6} Configuration . . .
An array of Göpel tetrads appears in the background below.
"As you can see, we've had our eye on you
for some time now, Mr. Anderson."
Comments Off on Kummer’s (16, 6) (on 6/16)
"But perhaps the desire for story
is what gets us into trouble to begin with."
— Sarah Marshall on June 5, 2018
"Beckett wrote that Joyce believed fervently in
the significance of chance events and of
random connections. ‘To Joyce reality was a paradigm,
an illustration of a possibly unstateable rule…
According to this rule, reality, no matter how much
we try to manipulate it, can only shift about
in continual movement, yet movement
limited in its possibilities…’ giving rise to
‘the notion of the world where unexpected simultaneities
are the rule.’ In other words, a coincidence … is actually
just part of a continually moving pattern, like a kaleidoscope.
Or Joyce likes to put it, a ‘collideorscape’."
— Gabrielle Carey, "Breaking Up with James Joyce,"
Sydney Review of Books , 15 June 2018
Carey's carelessness with quotations suggests a look at another
author's quoting of Ellmann on Joyce —
Comments Off on For June 16
Monday, June 11, 2018
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, "UhOh" —
— 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 computereye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
Comments Off on Arty Fact
The "more complicated" material mentioned by James Propp
in the previous post was notably described by A. A. Albert in 1956:
Comments Off on Finite Fields in 1956
Sunday, June 10, 2018
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.
Comments Off on Number Concept
Saturday, June 9, 2018
Now out from behind a paywall . . .
The diamond theorem at SASTRA —
Comments Off on SASTRA paper
Thursday, June 7, 2018
See also Eightfold Trinity in this journal.
Comments Off on For Dan Brown
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