Thursday, January 24, 2019

Name Space

Filed under: General,Geometry — Tags: — m759 @ 1:10 AM

A correction at Wikipedia  (Click to enlarge.) —

That this correction is needed indicates that the phrase 
"Cullinane space" might be useful. (Click to enlarge.)

On a 16-point space with some remarkable properties

Wednesday, January 23, 2019


Filed under: General,Geometry — Tags: — m759 @ 10:00 AM

For those who prefer more elaborate decorations —

1.  A Facebook image from last August … 

2.  The Facebook glider suggests a tune from "The Thomas Crown Affair"
     (1968) that appeared in a Dec. 16, 2018 post on Christianity and
     "interlocking names"—

'The Eddington Song'

The revised lyrics describe a square space.

3.  An even more  elaborate square space:
     the Dance of the Snowflakes from
     Balanchine's version of The Nutcracker —

Thursday, January 10, 2019

Archimedes at Hiroshima

Filed under: General,Geometry — Tags: , , — m759 @ 7:35 PM

Two examples from the Wikipedia article  "Archimedean solid" —

Iain Aitchison said in a talk last year at Hiroshima that
the Mathieu group M24  can be represented as permuting
naturally the 24 edges  of the cuboctahedron.

The 24 vertices  of the truncated  octahedron are labeled 
naturally by the 24 elements of S4  in a permutahedron

Can M24  be represented as permuting naturally
the 24 vertices  of the truncated octahedron?


Sunday, January 6, 2019

For Broom Bridge*

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

GL(2,3) is not unrelated to GL(3,2).

See Quaternion Automorphisms 
and Spinning in Infinity.

* See Wikipedia.

Wednesday, January 2, 2019

Wolf as Lamb

Filed under: General,Geometry — Tags: — m759 @ 10:30 PM

The above graphic design is by Noma Bar.

See as well the lamb-in-triangle of the Dec. 27 post
A Candle for Lily

Related material —

Remarks by Evelyn  Lamb on the Deathly Hallows symbol.

Saturday, December 22, 2018


Filed under: General,Geometry — Tags: — m759 @ 12:34 PM

The following are some notes on the history of Clifford algebras
and finite geometry suggested by the "Clifford Modules" link in a
Log24 post of March 12, 2005

A more recent appearance of the configuration —

Friday, December 14, 2018

Small Space Odyssey

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 11:00 AM

References in recent posts to physical space and 
to mathematical space suggest a comparison.

Physical space is well known, at least in the world
of mass entertainment.

Mathematical space, such as the 12-dimensional
finite space of the Golay code, is less well known.

A figure from each space —

The source of the Conway-Sloane brick —

Quote from a mathematics writer —

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

The former practice yields reflections like those of Conway and Sloane.
The latter practice is not recommended.

Wednesday, December 12, 2018

Kummerhenge Continues.

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 7:24 PM

Those pleased by what Ross Douthat today called
"The Return of Paganism" are free to devise rituals
involving what might be called "the sacred geometry
of the Kummer 166  configuration."

As noted previously in this journal, 

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

Geometric incarnation and the Kummer configuration

See also earlier posts also tagged "Kummerhenge" and 
another property of the remarkable Kummer 166 

The Kummer 16_6 Configuration and the Nordstrom-Robinson Code

For some related literary remarks, see "Transposed" in  this journal.

Some background from 2001 —

An Inscape for Douthat

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 9:41 AM

Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —

Detail of Feb. 20, 1986, note by Steven H. Cullinane on Weyl's 'relativity problem'

Kibler's 2008 'Variations on a theme' illustrated.

In finite geometry and combinatorics,
an inscape  is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:


Related material — the phrase
"Quantum Tesseract Theorem" and  

A.  An image from the recent
      film "A Wrinkle in Time" — 

B.  A quote from the 1962 book —

"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."

Sunday, December 9, 2018

Quaternions in a Small Space

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 2:00 PM

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 —

Iain Aitchison on the 'symmetric generation' of R. T. Curtis

See as well the two Log24 posts of December 1st, 2018 —

Character and In Memoriam.

Friday, December 7, 2018

The Angel Particle

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 7:15 PM

(Continued from this morning)

Majorana spinors and fermions at ncatlab

The Gibbons paper on the geometry of Majorana spinors and the Kummer configuration

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

Geometric incarnation and the Kummer configuration

See also other Log24 posts tagged Kummerhenge.

Tuesday, December 4, 2018

Melbourne Noir

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 11:30 AM

 March 8, 2018, was the date of death for Melbourne author Peter Temple.

Monday, December 3, 2018

The Relativity Problem at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 6:21 PM

“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

Iain Aitchison's 'dice-labelled' cuboctahedron at Hiroshima, March 2018

See also Relativity Problem and Diamonds and Whirls.

Sunday, December 2, 2018

Symmetric Generation …

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 12:00 PM

Continued .   See as well a Log24 search for "Symmetric Generation."

Iain Aitchison on symmetric generation of M24

Iain Aitchison on symmetric generation of M24

Update of 2 PM ET —

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 6:43 AM

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




Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II


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 2-fold 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 Horrocks-Mumford bundle. Poincare's homology 3-sphere, 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 so-called `Arnol'd Trinities'.

Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships 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 inter-connectedness 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 graphic-illustrated 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 8-component 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

Latin-Square Structure

Filed under: G-Notes,General,Geometry — m759 @ 2:56 AM

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

The White Cube

Filed under: G-Notes,General,Geometry — m759 @ 9:57 AM

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

Geometry and Experience

Filed under: G-Notes,General,Geometry — m759 @ 9:18 AM

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. 

Saturday, November 24, 2018


Filed under: G-Notes,General,Geometry — m759 @ 6:29 PM

A new portfolio site:

Portfolio on art and geometry of Steven H. Cullinane

Friday, November 23, 2018

Artfield Studio

Filed under: General,Geometry — m759 @ 12:10 PM

The title is that of an new Internet domain, artfield.studio,
that is used only to store branded links  such as . . .





artfield.studio/pinterest .

Some context —


Thursday, November 22, 2018

Rosenhain and Göpel Meet Kummer in Projective 3-Space

Filed under: General,Geometry — Tags: — m759 @ 2:07 PM

For further details, see finitegeometry.org/sc/35/hudson.html.

Geometric Incarnation

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

"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 chi-rho  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

Tuesday, November 20, 2018


Filed under: General,Geometry — m759 @ 12:21 PM


Musical accompaniment from Sunday morning

'The Eddington Song'

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 ):

Sunday, November 18, 2018

Space Music

Filed under: General,Geometry — Tags: , — m759 @ 9:27 AM

'The Eddington Song,' based on 'The Philosophy of Physical Science,' p. 141 (1939)

Update of Nov. 19 —

"Design is how it works." — Steve Jobs

See also www.cullinane.design.

Diamond Theorem Symmetry

Filed under: General,Geometry — m759 @ 1:00 AM

The title is a useful search phrase:

Saturday, November 17, 2018


Filed under: General,Geometry — Tags: — m759 @ 7:00 AM

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.

Friday, November 16, 2018

The Transposed Squares

Filed under: G-Notes,General,Geometry — m759 @ 9:12 PM

I.e. (click to enlarge) —


Thursday, November 15, 2018

Mathematics and Narrative

Filed under: General,Geometry — m759 @ 9:08 PM

Mathematics —

See (for instance) a research group at Ghent University.

For those who prefer narrative . . .

See also . . .

Wednesday, November 14, 2018

Phase Space

Filed under: General,Geometry — m759 @ 3:33 AM

"Open the pod bay doors, Bernard."

Tuesday, November 13, 2018

Blackboard Jungle Continues.

Filed under: G-Notes,General,Geometry — m759 @ 6:19 PM

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 —

Koen Thas, 'Unextendible Mututally Unbiased Bases' (2016)

See also posts tagged Dirac and Geometry. For more
background on finite  geometry, see a web page
at Thas's institution, Ghent University.

Thursday, November 8, 2018

Reality vs. Axiomatic Thinking

Filed under: G-Notes,General,Geometry — m759 @ 11:16 PM

From https://blogs.scientificamerican.com/…

A  Few  of  My  Favorite  Spaces:
The Fano Plane

The intuition-challenging 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.")

Geometry Lesson

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 12:00 AM

From "The Trials of Device" (April 24, 2017) —

Wittgenstein's pentagram and 4x4 'counting-pattern'

Pentagon with pentagram    

See also Wittgenstein in a search for "Ein Kampf " in this journal.

Tuesday, November 6, 2018

On Mathematical Beauty

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 2:18 AM

A phrase from the previous post —
"a size-eight dame in a size-six dress" —
suggests a review . . .

See as well the diamond-theorem correlation and . . .

Why PSL(2,7) is isomorphic to GL(3.2)

Saturday, November 3, 2018

Foundations of Geometry

Filed under: G-Notes,General,Geometry — m759 @ 1:40 PM

"costruire (o, dirò meglio immaginare) un ente" — Fano, 1892

"o, dirò meglio, costruire" — Cullinane, 2018

Thursday, November 1, 2018

Formation, Transformation . . . . Solution, Dissolution

Filed under: General,Geometry — m759 @ 8:40 PM

Tuesday, October 23, 2018

Plan 9 from Inner Space

Filed under: G-Notes,General,Geometry — m759 @ 9:57 AM

Click the image for some context.

Saturday, October 20, 2018


Filed under: G-Notes,General,Geometry — m759 @ 10:30 PM

See also Stella Octangula.

Study in Blue and Pink

Filed under: General,Geometry — Tags: — m759 @ 3:00 PM

Related Log24 posts — See Blade + Chalice.

Wednesday, October 17, 2018

Breakthrough Prize

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

181017-Breakthrough_Prize-news.jpg (500×212)

"…  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


Filed under: General,Geometry — Tags: — m759 @ 11:22 AM

From "The Phenomenology of Mathematical Beauty,"
by Gian-Carlo 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 one-shot idea of mathematical beauty saves us from having to deal with a concept that comes in degrees. Talk of mathematical beauty is a cop-out to avoid confronting enlightenment, a cop-out intended to keep our description of mathematics as close as possible to the description of a mechanism. This cop-out 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?

Monday, October 15, 2018

History at Bellevue

Filed under: General,Geometry — Tags: , — m759 @ 9:38 PM

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

For Zingari Shoolerim*

Filed under: General,Geometry — Tags: , — m759 @ 12:19 PM

IMAGE- Site with keywords 'Galois space, Galois geometry, finite geometry' at DiamondSpace.net

The structure at top right is that of the
in the previous post.

* "Zingari shoolerim" is from
    Finnegans Wake .

Friday, October 5, 2018

The “Ignotiles” of Paul Hertz

Filed under: General,Geometry — m759 @ 3:09 PM

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.

Wednesday, October 3, 2018

Adamantine Meditation

Filed under: General,Geometry — m759 @ 12:24 PM


A Catholic philosopher —

Related art —

Image result for mog miracle octad bricks

Sunday, September 30, 2018

Iconology of the Eightfold Cube

Filed under: General,Geometry — Tags: — m759 @ 11:13 AM

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 ) —

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.

Saturday, September 29, 2018

“Ikonologie des Zwischenraums”

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 9:29 AM

The title is from Warburg. The Zwischenraum  lines and shaded "cuts"
below are to be added together in characteristic two, i.e., via the
set-theoretic symmetric difference  operator.

Some small Galois spaces (the Cullinane models)

Sunday, September 23, 2018

Three Times Eight

Filed under: General,Geometry — Tags: — m759 @ 9:21 AM

The New York Times 's Sunday School today —

I prefer the three bricks of the Miracle Octad Generator —

Image result for mog miracle octad bricks

Saturday, September 22, 2018

Minimalist Configuration

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 11:03 PM

From the previous post

From Wikipedia

From Log24

The Venturi Manifesto

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 1:00 PM

Venturi reportedly died on Tuesday, September 18.*

See also this journal on that date.

* Fact check:

Symmetric Generation, by Curtis

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 10:15 AM

Norwegian artist Josefine Lyche —

Lyche's shirt honors the late Kurt Cobain.

"Here we are now, entertain us."

Friday, September 21, 2018


Filed under: G-Notes,General,Geometry — Tags: — m759 @ 4:36 AM

Monday, September 17, 2018

The 123 Configurations

Filed under: General,Geometry — m759 @ 8:35 AM



Lying at the Axis

Filed under: G-Notes,General,Geometry — m759 @ 12:00 AM

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

Saturday, September 15, 2018


Filed under: G-Notes,General,Geometry — m759 @ 9:50 AM

Tieszen— 'Kurt Godel and Phenomenology' — 1992

Update of 10:18 AM the same day —

See also Logicomix  in this  journal and, at Harvard,


  • 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.

Eidetic Reduction in Geometry

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 1:23 AM

"Husserl is not the greatest philosopher of all times.
He is the greatest philosopher since Leibniz."

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.

Terminology: Eidos

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.

Friday, September 14, 2018


Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:00 AM


I Ching Geometry search result

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.


Thursday, September 13, 2018

Iconology of the Interstice

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 6:25 AM

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.

Godard and Interality

Filed under: General,Geometry — Tags: , — m759 @ 12:14 AM

The previous post, "One Plus One," suggests some further
art-historical 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 in-between,’ 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 twenty-first century archaeologist of the moving image, constructing a meta-cinematic collage that, while consisting of an indexing of (almost exclusively) pre-existing 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 non-existent one (superimposition, co-presence 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. Jean-Luc 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. Jean-Luc Godard, "Introduction à une véritable histoire du cinéma," Albatros , Paris,1980, p. 145

* * * * P. 783 —

If it is in ‘the in-between’ 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 déjà-vu 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 Didi-Huberman, L’image survivante, p. 503. (Georges Didi-Huberman, L’image survivante. Histoire de l’art et temps des fantômes selon Aby Warburg , Minuit, Paris, 2002)

Sunday, September 9, 2018

Plan 9 Continues.

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 9:00 AM

"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 Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics
(May, 1997), pp. 183-196. 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

Saturday, September 8, 2018


Filed under: General,Geometry — Tags: — m759 @ 9:26 PM

'Space' in Chinese

For example —

'Projective space' in Chinese

See also Interality in this journal.


Filed under: General,Geometry — m759 @ 2:00 AM

Affine groups on small binary spaces

Wednesday, September 5, 2018

Multifaceted Narrative

Filed under: General,Geometry — m759 @ 8:19 PM



See also, in this  journal, 23-cycle.

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  —

Counting symmetries with the orbit-stabilizer theorem

Saturday, September 1, 2018

Ron Shaw — D. 21 June 2016

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 —


Friday, August 31, 2018

Perception of Number

Filed under: General,Geometry — Tags: — m759 @ 8:31 PM

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

Thursday, August 30, 2018

Perception* of Space

Filed under: General,Geometry — m759 @ 2:12 PM




* 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

Monday, August 27, 2018

Geometry and Simplicity

Filed under: General,Geometry — m759 @ 9:27 PM


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 705-710, 735). Kindle Edition.


For some much simpler spaces of various
dimensions, see Galois Space in this journal.

Some small Galois spaces (the Cullinane models)

Saturday, August 25, 2018

“Waugh, Orwell. Orwell, Waugh.”

Filed under: General,Geometry — m759 @ 4:00 PM

Suggested by a review of Curl on Modernism —


Related material —

Waugh + Orwell in this journal and

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Point at Infinity

Filed under: General,Geometry — Tags: — m759 @ 5:20 AM

In literature —


In film —


In mathematics —


Sunday, August 19, 2018

Possible Permutations

Filed under: General,Geometry — Tags: — m759 @ 9:18 PM

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


Thursday, August 16, 2018

Mathematics and Narrative

Filed under: General,Geometry — Tags: — m759 @ 11:00 PM



Alperin, Groups and Representations, p. 61


The Tale of the Flux Capacitor

Wednesday, August 15, 2018

An Illusion of Brilliance

Filed under: General,Geometry — Tags: , — m759 @ 5:25 PM

" . . . 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":


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.


An Exceptional Isomorphism

Filed under: General,Geometry — Tags: , , — m759 @ 5:40 AM

Why PSL(2,7) is isomorphic to GL(3.2)

From previous posts on this topic —



Tuesday, August 14, 2018


Filed under: General,Geometry — Tags: , , , — m759 @ 12:06 AM




Monday, August 13, 2018

Trojan Horsitude

Filed under: General,Geometry — Tags: — m759 @ 3:33 AM

"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



Friday, August 10, 2018

For Noah Cross in Chinatown

Filed under: General,Geometry — m759 @ 2:00 PM


Geometry of the I Ching (Box Style)

Thursday, August 9, 2018

True Grids

Filed under: General,Geometry — Tags: , — m759 @ 7:59 PM

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" —

IMAGE- Joaquin Phoenix, corridor scene in 'The Master'

A fanciful novel referring to the number 8,
and a not -so-fanciful 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.

Wednesday, August 8, 2018


Filed under: General,Geometry — Tags: , , , — m759 @ 6:00 AM

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

'Knight' octad labeling by the 8 points of the projective line over GF(7) .

Background — Relativity Problem in Log24.

Monday, August 6, 2018

The Girl with Kaleidoscope Eyes

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM


“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).
Penguin Publishing Group. Kindle Edition.

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.

Sunday, August 5, 2018

Board Structure

Filed under: General,Geometry — m759 @ 11:48 AM



Saturday, August 4, 2018

The Secret Life of Harry Albers

Filed under: General,Geometry — m759 @ 3:00 PM

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).

Manifestations of Exquisite Geometry

Filed under: General,Geometry — m759 @ 1:23 PM

An alleged manifestation in physics, from Scientific American  —


Manifestations in pure mathematics, from Plato and R. T. Curtis  —

Counting symmetries with the orbit-stabilizer theorem

For some entertaining literary  manifestations, see Wrinkle.

Tuesday, July 31, 2018

Easter Storytelling

Filed under: General,Geometry — m759 @ 11:00 AM

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).


Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

"A blank underlies the trials of device." — Wallace Stevens

IMAGE- The ninefold square .

Sunday, July 29, 2018

The Materialization

Filed under: General,Geometry — m759 @ 11:01 PM

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.

Saturday, July 28, 2018


Filed under: General,Geometry — m759 @ 11:22 AM

Floyd:  "You're trying to figure out this length.
            That's the hypotenuse. So you have to
             know this angle."

Wednesday, July 25, 2018

Bucharest Physics

Filed under: General,Geometry — m759 @ 3:52 PM

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 .

Tuesday, July 24, 2018

Artistic Style

Filed under: General,Geometry — m759 @ 10:45 AM

Monday, July 23, 2018

Eightfold Cube for Furey*

Filed under: General,Geometry — m759 @ 10:31 PM

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

Space 101

Filed under: General,Geometry — Tags: — m759 @ 1:01 AM

From the April 1st publication date of "Interality Shows Through,"
by Geling Shang —

See too yesterday's post  Space.

Sunday, July 22, 2018


Filed under: General,Geometry — Tags: , — m759 @ 10:29 AM

See also interality in the eightfold cube.

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

Saturday, July 21, 2018

Comic-Con 2018

Filed under: General,Geometry — m759 @ 9:31 PM

"For many of us, the geometry course sounded the death knell
for our progress — and interest — in mathematics."

— "Shape and Space in Geometry"

© 1997-2003 Annenberg/CPB. All rights reserved.
Legal Policy

Building-Block Theory

Filed under: General,Geometry — m759 @ 10:56 AM

(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.

Friday, July 20, 2018

Geometry for Jews

Filed under: General,Geometry — m759 @ 12:00 PM


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 Holden-Day in 1968.

Wednesday, July 18, 2018


Filed under: General,Geometry — m759 @ 12:00 PM

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 —

Counting symmetries with the orbit-stabilizer theorem.

Monday, July 16, 2018

Greatly Exaggerated Report

Filed under: General,Geometry — Tags: — m759 @ 1:21 PM

"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 —

IMAGE- Richard Kiley in 'Blackboard Jungle,' with grids and broken records

"Through the unknown, remembered gate . . . ."

— T. S. Eliot, Four Quartets

Sunday, July 15, 2018

Jewish Oases

Filed under: General,Geometry — m759 @ 10:06 PM

"… 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: 


From  the Oasis  in Steven Spielberg's "Ready Player One" (2018) —

I prefer, from a Log24 search for Flux Capacitor

Symbologist Robert Langdon views a corner of Solomon's Cube

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."

Saturday, July 14, 2018

Expanding the Spiel

Filed under: General,Geometry — m759 @ 1:15 PM


Cube Dance

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.

Friday, July 13, 2018

Segue for Harlan Ellison

Filed under: General,Geometry — m759 @ 12:00 PM

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"

© 1997-2003 Annenberg/CPB. All rights reserved.
Legal Policy


Filed under: General,Geometry — m759 @ 12:00 AM

Thursday, July 12, 2018

Kummerhenge Illustrated

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM


“… 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"

Wednesday, July 11, 2018

Clarity and Precision

Filed under: General,Geometry — m759 @ 9:13 AM

"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


— Page number in a 2016 Scribner edition
of Stephen King's IT

Sunday, July 8, 2018


Filed under: General,Geometry — m759 @ 12:00 PM

Eric Temple Bell, 'The Development of Mathematics'

See also Solomon's  cube.

* Title suggested by a 2011 dystopian novel.

Friday, July 6, 2018


Filed under: General,Geometry — Tags: — m759 @ 9:48 AM

"… 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 group-structure" —

See as well posts tagged Dirac and Geometry.

Thursday, July 5, 2018


Filed under: General,Geometry — Tags: — m759 @ 1:44 PM

Some context for what Heidegger called
das Spiegel-Spiel 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 oft-repeated and ever-different metaphor of identity.

— Paz, Octavio. The Monkey Grammarian 
(Kindle Locations 1185-1191). 
Arcade Publishing. Kindle Edition. 

A related 1960 meditation from Claude Lévi-Strauss taken from a 
Log24 post of St. Andrew's Day 2017,  "The Matrix for Quantum Mystics":

The Matrix of Lévi-Strauss —

"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 Lévi-Strauss

I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets
(from The Matrix Meets the Grid) —


Tuesday, July 3, 2018


Filed under: General,Geometry — Tags: — m759 @ 7:55 PM

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 —

From "Heidegger for Passover

"Propriation1 gathers the rift-design2 of the saying
and unfolds it3 in such a way that it becomes 
the well-joined structure4 of a manifold showing."

— p. 415 of Heidegger's Basic Writings ,
edited by David Farrell Krell,
HarperCollins paperback, 1993

"Das Ereignis versammelt den Aufriß der Sage
und entfaltet ihn zum Gefüge des vielfältigen Zeigens." 

— Heidegger, Weg zur Sprache

1. "Mirror-Play of the Fourfold"

2. "Christ descending into the abyss"

3. Barrancas of Cuernavaca

4. Combinatorics, Philosophy, Geometry

Lost in Quantum Space

Filed under: General,Geometry — Tags: — m759 @ 10:45 AM

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.

Monday, July 2, 2018

In Memoriam

Filed under: General,Geometry — m759 @ 9:10 PM

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.)

Sunday, July 1, 2018

Deutsche Ordnung

Filed under: General,Geometry — Tags: — m759 @ 8:22 PM

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 
09:00-09:40 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

Friday, June 29, 2018

Triangles in the Eightfold Cube

Filed under: General,Geometry — m759 @ 9:10 PM

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 eight-triangle 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 eight-triangle 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

For St. Stanley

Filed under: General,Geometry — m759 @ 1:26 PM

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

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

That  2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.

Wednesday, June 27, 2018

Taken In

Filed under: General,Geometry — Tags: — m759 @ 9:36 AM

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 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 .


"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."


Filed under: General,Geometry — m759 @ 9:27 AM

See Ballet Blanc 
and Still Point.

Monday, June 25, 2018

The Gateway Device

Filed under: General,Geometry — m759 @ 6:24 PM

<title data-rh="true">Frank Heart, Who Linked Computers Before the Internet, Dies at 89 – The New York Times</title>
. . . .
<meta data-rh="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 data-rh="true" property="article:published" itemprop="datePublished" content="2018-06-25T19:16:17.000Z"/>

See also yesterday's "For 6/24" and 

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

Sunday, June 24, 2018

For 6/24

Filed under: General,Geometry — Tags: — m759 @ 10:12 AM

A clue to the relationship between the Kummer (16, 6)
configuration and the large Mathieu group M24

Related material —

See too the diamond-theorem correlation.

Saturday, June 23, 2018

Meanwhile …

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

Backstory for fiction fans, from Log24 on June 11 —

Related non -fiction —

See as well the structure discussed in today's previous post.

Plan 9 from Inner Space

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

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  2-subsets of a 6-set —

Thursday, June 21, 2018

Models of Being

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

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.

Dirac and Geometry (continued)

Filed under: General,Geometry — Tags: , — m759 @ 10:04 AM

"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 "super-mathematician" 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.


Filed under: General,Geometry — Tags: , — m759 @ 2:19 AM

See also the Omega Matrix in this  journal.

Wednesday, June 20, 2018


Filed under: General,Geometry — m759 @ 5:29 AM

"… 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).

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