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

The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”

Related structures —

Greg Egan’s animated image of the Klein quartic —

For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen

Steiner quadruple system in eightfold cube

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 — Tags: , — 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 — Tags: — 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)

"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).

Tuesday, June 19, 2018

Ici vient M. Jordan

Filed under: General,Geometry — Tags: , — m759 @ 2:13 AM

NY Times correction, online June 16, about 'Here Comes Mr. Jordan' and 'Heaven Can Wait'

See also this  journal on Saturday morning, June 16.

Saturday, June 16, 2018

Kummer’s (16, 6) (on 6/16)

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

"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 166 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."

For June 16

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

"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

Monday, June 11, 2018

Arty Fact

Filed under: General,Geometry — Tags: , , — m759 @ 10:35 PM

The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.

The Eye of ARTI —

See also a post of May 19, "Uh-Oh" —

— and a post of June 6, "Geometry for Goyim" — 

Mystery box  merchandise from the 2011  J. J. Abrams film  Super 8 

An arty fact I prefer, suggested by the triangular computer-eye forms above —

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

This is from the July 29, 2012, post The Galois Tesseract.

See as well . . .


Filed under: General,Geometry — Tags: — m759 @ 8:32 PM

A Scientific American  headline today —

Glittering Diamond Dust in Space
Might Solve a 20-Year-Old Mystery

Related art —

"Never underestimate the power of glitter."

Glitter by Josefine Lyche, as of diamond dust

Background:  "Diamond Dust" + Glitter in this journal.

Finite Fields in 1956

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

The "more complicated" material mentioned by James Propp
in the previous post was notably described by A. A. Albert in 1956:

Sunday, June 10, 2018

Number Concept

Filed under: General,Geometry — m759 @ 2:45 PM

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() 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  is a prime
raised to the 2nd power or higher, describing the
elements of GF() 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.

Saturday, June 9, 2018

SASTRA paper

Filed under: General,Geometry — Tags: — m759 @ 11:14 PM

Now out from behind a paywall . . .

The diamond theorem at SASTRA —

Thursday, June 7, 2018

For Dan Brown

Filed under: General,Geometry — Tags: , — m759 @ 1:09 PM

See also Eightfold Trinity in this  journal.

Symbologist Robert Langdon views a corner of Solomon's Cube

Wednesday, June 6, 2018

Geometry for Goyim

Filed under: General,Geometry — Tags: , — m759 @ 12:45 PM

Mystery box  merchandise from the 2011  J. J. Abrams film  Super 8  —

A mystery box that I prefer —

Box containing Froebel's Third Gift-- The Eightfold Cube

Click image for some background.

See also Nicht Spielerei .

Monday, June 4, 2018

The Trinity Stone Defined

Filed under: General,Geometry — Tags: , — m759 @ 8:56 PM

“Unsheathe your dagger definitions.” — James Joyce, Ulysses

The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

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

Some small Galois spaces (the Cullinane models)

Sunday, June 3, 2018


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

<meta name="description"
content="Identidade generativa para o Diamonds Studio 

Desenvolvido em conjunto com


Baseado na Diamond Theory by Steven H Cullinane, 1977">

Thursday, May 31, 2018

Eightfold Suffering:

Filed under: General,Geometry — Tags: — m759 @ 9:23 PM

A New, Improved Version of  Quantum Suffering !

Background for group actions on the eightfold cube

See also other posts now tagged Quantum Suffering 
as well as — related to the image above of the Great Wall

Myspace China.

Monday, May 28, 2018


Filed under: General,Geometry — m759 @ 1:16 PM

A piece co-written by Ivanov, the author noted in the previous post, was cited
in my "Geometry of the 4×4 Square."

Also cited there —  A paper by Pasini and Van Maldeghem that mentions
the Klein quadric.

Those sources suggested a search —

The link is to some geometry recently described by Tabachnikov
that seems rather elegant:

For another, more direct, connection to the geometry of the 4×4 square,
see Richard Evan Schwartz in this  journal.

This same Schwartz appears also in the above Tabachnikov paper:

Sunday, May 27, 2018

Heptagons, Quilts, Mathieu Groups

Filed under: General,Geometry — m759 @ 1:02 PM

The phrase "ghostly heptagons" appears in A Piece of Justice , a 1995 novel 
by Jill Paton Walsh that features some fictional politically correct mathematics.
(See the previous post.)

Related material from a Google search today —

The Ivanov book is new:

“What Interests Me”

Filed under: General,Geometry — m759 @ 12:08 PM

See also a Log24 search for A Piece of Justice .

Sunday, May 20, 2018

Sometimes Function Follows Form

Filed under: General,Geometry — Tags: — m759 @ 9:45 PM

Function —

Follows —

Form —

Related material: Eight years ago today, Eliza Doolittle Day 2010.

Not So Cryptic

Filed under: General,Geometry — m759 @ 2:56 PM

From the date of the New York Times  James Bond video
referenced in the previous post, "A Cryptic Message" —

Some Style

Filed under: General,Geometry — Tags: — m759 @ 1:03 AM

Dialogue from the 1984 fourth draft of the script, as found on the Web,
for "Back to the Future" (1985) (apparently some changes were made
in the filming) —

A sort of "flux capacitor" (see previous post) —

The Rolls-Royce Cullinan

 plus "e" for Einstein 

Saturday, May 19, 2018

Flux Capacitor

Filed under: General,Geometry — Tags: , — m759 @ 4:13 PM

For Tom Hanks and Dan Brown —

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."

See as well a Google search for flux philosophy

Monday, May 14, 2018


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

"A generation lost in space" — Don McLean

See as well Varignon in the previous post.

Logos at Harvard

Filed under: General,Geometry — Tags: , — m759 @ 3:01 PM

In 2013, Harvard University Press changed its logo to an abstract "H."

Harvard University Press Logo, Before and After

Both logos now accompany a Harvard video first published in 2012,
"The World of Mathematical Reality." 

In the video, author Paul Lockhart discusses Varignon's theorem
without naming Varignon (1654-1722) . . .

Paul Lockhart on geometry

A related view of "mathematical reality" —

Note the resemblance to Plato's Diamond.

Sunday, May 13, 2018


Filed under: General,Geometry — m759 @ 9:48 PM


"Robert Noel Hall (December 25, 1919 – November 7, 2016)
was an American engineer and applied physicist."

The New York Times  on May 10, 2018

"A product of his inventive labor can also be found
in most kitchens nowadays: the microwave oven.

Yet for all the widespread familiarity of what Dr. Hall wrought
as a remarkably ingenious physicist, his death, at 96,
on Nov. 7, 2016, gained little notice."

A fictional kitchen —

170703-The_Forger-Christopher_Plummer-2015-500w.jpg (500×336)

In memoriam:  Kindergarten Relativity .


Friday, May 11, 2018

A Pure Geometry

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

From posts tagged Modernism

Sunday, December 10, 2006

m759 @ 9:00 PM

A Miniature Rosetta Stone:

The 3x3 grid

“Function defined form, expressed in a pure geometry
that the eye could easily grasp in its entirety.”

– J. G. Ballard on Modernism
(The Guardian , March 20, 2006)

“The greatest obstacle to discovery is not ignorance –
it is the illusion of knowledge.”

— Daniel J. Boorstin,
Librarian of Congress, quoted in Beyond Geometry

On this date 19 years ago…

Filed under: General,Geometry — Tags: — m759 @ 2:35 PM

See as well other posts tagged Modernism.

Wednesday, May 9, 2018

Once Upon a . . .

Filed under: General,Geometry — Tags: — m759 @ 10:13 AM

Click the text below for a slideshow.

Monday, January 8, 2018


Filed under: Uncategorized — m759 @ 12:00 AM 

Tuesday, May 8, 2018


Filed under: General,Geometry — Tags: — m759 @ 1:29 PM

The glitter-ball-like image discussed in the previous post
is of an artwork by Olafur Eliasson.

See the kaleidoscopic  section of his website.

From that section —

Eliasson, 'When Love Is Not Enough' wall, 2007

Related art in keeping with the theme of last night's Met Gala —

See also my 2005 webpage Kaleidoscope Puzzle.

Monday, May 7, 2018

Fish Babel

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

Stanley Fish in the online New York Times  today —

". . . Because it is an article of their faith that politics are bad
and the unmediated encounter with data is good,
internet prophets will fail to see the political implications
of what they are trying to do, for in their eyes political implications
are what they are doing away with.

Indeed, their deepest claim — so deep that they are largely
unaware of it — is that politics can be eliminated. They don’t
regard politics as an unavoidable feature of mortal life but as
an unhappy consequence of the secular equivalent of the
Tower of Babel: too many languages, too many points of view.
Politics (faction and difference) will just wither away when
the defect that generates it (distorted communication) has
been eliminated by unmodified data circulated freely among
free and equal consumers; everyone will be on the same page,
reading from the same script and apprehending the same
universal meanings. Back to Eden!"

The final page, 759, of the Harry Potter saga —

"Talk about magical thinking!" — Fish, ibidem .

See also the above Harry Potter page 
in this  journal Sunday morning.

Sunday, May 6, 2018

The Osterman Omega

Filed under: General,Geometry — Tags: , — m759 @ 5:01 PM

From "The Osterman Weekend" (1983) —

Counting symmetries of the R. T. Curtis Omega:

An Illustration from Shakespeare's birthday

Counting symmetries with the orbit-stabilizer theorem

Saturday, May 5, 2018

Galois Imaginary

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

" 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

A related passage —

From The French Mathematician
by Tom Petsinis (Nov. 30, 1998) —


I had foreseen it all in precise detail.
One step led inevitably to the next,
like the proof of a shining theorem,
down to the conclusive shot that still echoes
through time and space. 
Facedown in the damp pine needles,
I embraced that fatal sphere
with my whole body. Dreams, memories,
even the mathematics I had cherished
and set down in my last will and testament–
all receded. I am reduced to
a singular point; in an instant
I am transformed to .

i = an imaginary being

Here, on this complex space,
i  am no longer the impetuous youth
who wanted to change the world
first with a formula and then with a flame.
Having learned the meaning of infinite patience,
i  now rise to the text whenever anyone reads 
about Evariste Galois, preferring to remain 
just below the surface, 
like a goldfish nibbling the fringe of a floating leaf.
Ink is more mythical than blood
(unless some ancient poet slit his 
vein and wrote an epic in red):
The text is a two-way mirror 
that allows me to look into
the life and times of the reader. 
Who knows, someday i  may rise
to a text that will compel me 
to push through to the other side.
Do you want proof that i  exist? Where am ?
Beneath every word, behind each letter, 
on the side of a period that will never see the light.

Friday, May 4, 2018

Art & Design

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


A star figure and the Galois quaternion.

The square root of the former is the latter.

See also a passage quoted here a year ago today
(May the Fourth, "Star Wars Day") —

Cube symmetry subgroup of order 8 from 'Geometry and Symmetry,' Paul B. Yale, 1968, p.21

The Tuchman Radical*

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

Two excerpts from today's Art & Design section of
The New York Times  —

For the deplorables of France —

For further remarks on l'ordre ,
see posts tagged Galois's Space
( tag=galoiss-space).

* The radical of the title is Évariste Galois (1811-1832).


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

A more serious note in memory of Anatole Katok:

"Entropy measures the unpredictability
of a system that evolves over time."

Alex Wright, BULLETIN (New Series)

Volume 53, Number 1, January 2016, Pages 41–56


Article electronically published on September 8, 2015:



"This short expository note gives an elementary
introduction to the study of dynamics on certain
moduli spaces and, in particular, the recent 
breakthrough result of Eskin, Mirzakhani,
and Mohammadi. We also discuss the context
and applications of this result, and its connections
to other areas of mathematics, such as algebraic
geometry, Teichmüller theory, and ergodic theory
on homogeneous spaces."

See also the lives of Ratner and Mirzakhani.

Wednesday, May 2, 2018

Galois’s Space

Filed under: General,Geometry — Tags: , , — m759 @ 2:20 PM

(A sequel to Foster's Space and Sawyer's Space)

See posts now tagged Galois's Space.

Sunday, April 29, 2018


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

From the online New York Times  this afternoon:

Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.

From this  journal this morning:

"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to 
make a fancy amusement-park ride
There had to be more—"

Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168

"The sensation of folding space . . . ."

Or unfolding:

Click the above unfolded space for some background.

Wednesday, April 25, 2018

A Deathly Triangle

Filed under: General,Geometry — Tags: , , , — m759 @ 11:27 PM

'Imprisoned in a Tesseract,' a study of novelist James Blish

An Idea

Filed under: General,Geometry — m759 @ 11:45 AM

"There was an idea . . ." — Nick Fury in 2012

". . . a calm and objective work that has no special
dance excitement and whips up no vehement
audience reaction. Its beauty, however, is extraordinary.
It’s possible to trace in it terms of arithmetic, geometry,
dualism, epistemology and ontology, and it acts as
a demonstration of art and as a reflection of
life, philosophy and death."

New York Times  dance critic Alastair Macaulay,
    quoted here in a post of August 20, 2011.

Illustration from that post —

A 2x4 array of squares

See also Macaulay in
last night's 10 PM post.

Tuesday, April 24, 2018

Dance, Music, Space

Filed under: General,Geometry — m759 @ 10:00 PM

". . . dance, fueled by music, opens up space."

—  Alastair Macaulay in the online New York Times  today

Putting aside the unfortunate fuel metaphor, this suggests a review —

A video published on the above date —

The video has six-plus-two dancers, a more concise arrangement
than the eight-plus-two discussed by Macaulay.

Another approach to six plus two:  the diamond-theorem correlation.

Illustrators of the Word

Filed under: General,Geometry — m759 @ 1:30 AM

Tom Wolfe in The Painted Word  (1975) 

“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”

The above group of 322,560 permutations appears also in a 2011 book —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

— and in 2013-2015 papers by Anne Taormina and Katrin Wendland:

Monday, April 23, 2018

Blockbuster Exhibition

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

Mike Hale in The New York Times  online today —

Review ‘Genius’ Paints Picasso by the Numbers

"… the production’s tinselly soul.

For instance, it’s on the record that Picasso’s lovers
Dora Maar and Marie-Thérèse Walter had
a wrestling match in his studio while he was
painting 'Guernica.'  'Genius' includes that
scene, naturally, but adds its own detail:
The altercation helps Picasso overcome a creative block
and gleefully set to work on the gigantic painting.
It may be news to scholars that one of art’s
greatest testaments to the horror of war was
inspired, in part, by the excitement of being
fought over by a pair of jealous women."

Related Art


A Creative Block

Sunday, April 22, 2018

The Omega Matrix

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


Angels and Demons cross within a diamond (page 306), and Finite Geometry logo

Tuesday, April 17, 2018

A Necessary Possibility*

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

"Without the possibility that an origin can be lost, forgotten, or
alienated into what springs forth from it, an origin could not be
an origin. The possibility of inscription is thus a necessary possibility,
one that must always be possible."

— Rodolphe Gasché, The Tain of the Mirror ,
     Harvard University Press, 1986

IMAGE- Harvard University Press, 1986 - A page on Derrida's 'inscription'

An inscription from 2010 —

An inscription from 1984 —

American Mathematical Monthly, June-July 1984, p. 382


Triangles are square

"Every triangle consists of  n congruent copies of itself"
is true if and only if  n is a square. (The proof is trivial.) 
— Steven H. Cullinane

* See also other Log24 posts mentioning this phrase.

Sunday, April 15, 2018

Colorado Olympiad

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

Or:  Personalities Before Principles

Personalities —

Principles —

This  journal on April 28, 2004 at 7:00 AM.

Backstory —

Square Triangles in this journal.

Monday, April 9, 2018

The Long Hello

Filed under: General,Geometry — Tags: , — m759 @ 9:17 PM

Transportes Ometochtli   >>

Es la línea de transporte más antigua que va de Cuernavaca a Tepoztlán . . . .

Image from the 1973 Elliott Gould film "The Long Goodbye" —

Some backstory . . . .

  1. https://hidden-films.com/2014/11/09/the-little-movie-that-couldnt-

  2. https://en.wikipedia.org/wiki/A_Glimpse_of_Tiger
  3. http://m759.net/wordpress/?p=71956 —

Monday, January 8, 2018


Filed under: Uncategorized — m759 @ 12:00 AM 

Sunday, April 8, 2018


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

From a Log24 post of Feb. 5, 2009 —

Design Cube 2x2x2 for demonstrating Galois geometry

An online logo today —

See also Harry Potter and the Lightning Bolt.


Saturday, April 7, 2018

Word and Image: Backstory for James Spader

Filed under: General,Geometry — m759 @ 2:14 PM

Actor James Spader in a 2014 interview

". . . my father taught English. My mother taught art . . . ."

Detail of part of a text by Magritte (1929) that appeared  
without attribution in the online New York Times  today —

See also, from a search for the phrase "Word and Image"
in this  journal —

The Philosophers' Stone as originally
illustrated in The New York Times 

http://www.log24.com/log/pix10A/100518-TheStoneNYT.jpg .

Related images —

Juliette Binoche in 'Blue'  The
 24 2x2 Cullinane Kaleidoscope animated images

See as well a Log24 search for "Philosophers' Stone"
and remarks related to the Magritte pictures above 
in the post Story of March 13, 2014.


Filed under: General,Geometry — Tags: , — m759 @ 11:47 AM

The FBI holding cube in "The Blacklist" —

" 'The Front' is not the whole story . . . ."

— Vincent Canby, New York Times  film review, 1976,
     as quoted in Wikipedia.

See also Solomon's Cube in this  journal.

IMAGE- 'Solomon's Cube'

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

Thursday, April 5, 2018


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


Geometry of the I Ching (Box Style)

Box-style I Ching , January 6, 1989

Wednesday, April 4, 2018

Gold Bug Variations (Continued)

Filed under: General,Geometry — Tags: , , — m759 @ 1:32 PM

See as well a search for "Gold Bug"  in this  journal.

From that search —

Richard Powers, The Gold Bug Variations , first published in 1991—

Botkin, whatever her gifts as a conversationist, is almost as old
as the rediscovery of Mendel. The other extreme in age, 
Joe Lovering, beat a time-honored path out of pure math 
into muddy population statistics. Ressler has seen the guy 
potting about in the lab, although exactly what the excitable kid 
does is anybody's guess. He looks decidedly gumfooted holding
any equipment more corporeal than a chi-square. Stuart takes
him to the Y for lunch, part of a court-your-resources campaign.
He has the sub, Lovering the congealed mac and cheese. 
Hardly are they seated when Joe whips out a napkin and begins
sketching proofs. He argues that the genetic code, as an 
algorithmic formal system, is subject to Gödel's Incompleteness
Theorem. "That would mean the symbolic language of the code 
can't be both consistent and complete. Wouldn't that be a kick 
in the head?"

Kid talk, competitive showing off, intellectual fantasy. 
But Ressler knows what Joe is driving at. He's toyed with similar 
ideas, cast in less abstruse terms. We are the by-product of the 
mechanism in there. So it must be more ingenious than us. 
Anything complex enough to create consciousness may be too 
complex for consciousness to understand. Yet the ultimate paradox
is Lovering, crouched over his table napkin, using proofs to 
demonstrate proof's limits. Lovering laughs off recursion and takes
up another tack: the key is to find some formal symmetry folded
in this four-base chaos
. Stuart distrusts this approach even more.
He picks up the tab for their two untouched lunches, thanking 
Lovering politely for the insight.

"The key is to find some formal symmetry…."

IMAGE- Valéry on ornament in 'Method of Leonardo,' with Valéry's serpent-and-key emblem

The Key

Filed under: General,Geometry — Tags: — m759 @ 12:32 PM

"The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, 'heptads' …."

— George M. Conwell, "The 3-space PG (3, 2) and Its Group," 
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in 
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference 
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.

For those who, like the author of The Eight  (a novel in which today's
date figures prominently), prefer fiction —

See as well . . .

Literary theorists may, if they wish, connect
cabalistically the Insidious  address "414" 
with the date  4/14 of the above post, and
the word Appletree with the biblical Garden.

Tuesday, April 3, 2018

Easter at Cambridge

Filed under: General,Geometry — m759 @ 11:33 PM

Three geometric readings, in chronological order —

Monday, April 2, 2018

Three Mother Cubes

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

From a Toronto Star video pictured here on April 1 three years ago:

The three connected cubes are labeled "Harmonic Analysis," 'Number Theory,"
and "Geometry."

Related cultural commentary from a review of the recent film "Justice League" —

"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."

The nineteenth-century German mathematician Felix Christian Klein
as Steppenwolf —

Volume I of a treatise by Klein is subtitled
"Arithmetic, Algebra, Analysis." This covers
two of the above three Toronto Star cubes.

Klein's Volume II is subtitled "Geometry."

An excerpt from that volume —

Further cultural commentary:  "Glitch" in this journal.

Sunday, April 1, 2018


Filed under: General,Geometry — Tags: — m759 @ 11:07 AM

The Eightfold Cube

Quantum logo

Business logo

Happy April 1.

Saturday, March 31, 2018


Filed under: General,Geometry — m759 @ 2:56 PM

“The greatest obstacle to discovery
is not ignorance —
it is the illusion of knowledge.”

— Daniel J. Boorstin,
Librarian of Congress,
quoted here in 2006.

Related material —

Remarks on Rubik's Cube from June 13, 2014 and . . .

See as well a different Gresham, author of Nightmare Alley 
and Log24 posts on that book and the film of the same name .

Thursday, March 29, 2018

To Imagine (or, Better, to Construct)

Filed under: General,Geometry — m759 @ 8:10 PM

The title reverses a phrase of Fano —
costruire (o, dirò meglio immaginare).

Illustrations of imagining (the Fano plane) and of constructing (the eightfold cube) —


The Fano plane and the eightfold cube

“Before Creation Itself . . .”

Filed under: General,Geometry — Tags: , , , — m759 @ 10:13 AM

From the Diamond Theorem Facebook page —

A question three hours ago at that page

“Is this Time Cube?”

Notes toward an answer —

And from Six-Set Geometry in this journal . . .

Tuesday, March 27, 2018

Compare and Contrast

Filed under: General,Geometry — Tags: — m759 @ 4:28 PM

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Related material on automorphism groups —

The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture 
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.

Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure  above
than they are an eight-cube mechanical puzzle  like the one below.

Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."

Saturday, March 24, 2018


Filed under: General,Geometry — Tags: , — m759 @ 12:30 PM

Sure, Whatever.

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

The search for Langlands in the previous post
yields the following Toronto Star  illustration —

From a review of the recent film "Justice League" —

"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."

For other cubic adventures, see yesterday's post on A Piece of Justice 
and the block patterns in posts tagged Design Cube.

Friday, March 23, 2018


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

Copy editing — From Wikipedia

"Copy editing (also copy-editing or copyediting, sometimes abbreviated ce)
is the process of reviewing and correcting written material to improve accuracy,
readability, and fitness for its purpose, and to ensure that it is free of error,
omission, inconsistency, and repetition. . . ."

An example of the need for copy editing:

Related material:  Langlands and Reciprocity in this  journal.

Piece Prize

Filed under: General,Geometry — Tags: , — m759 @ 6:15 PM

The Waymark Prize from 'A Piece of Justice' (1995) by Jill Paton Walsh

The Waymark Prize Mystery - 'A Piece of Justice' (1995) p. 138

From the Personal to the Platonic

Filed under: General,Geometry — Tags: , , — m759 @ 11:01 AM

On the Oslo artist Josefine Lyche —

"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."

Ann Cathrin Andersen

Personal —

The Rushkoff Logo

— From a 2016 graphic novel by Douglas Rushkoff.

See also Rushkoff and Talisman in this journal.

Platonic —

The Diamond Cube.

Compare and contrast the shifting hexagon logo in the Rushkoff novel above 
with the hexagon-inside-a-cube in my "Diamonds and Whirls" note (1984).

Thursday, March 22, 2018

In Memoriam

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

Also on March 18, 2015 . . .

The Diamond Cube

Filed under: General,Geometry — Tags: , — m759 @ 11:32 AM

The Java applets at the webpage "Diamonds and Whirls"
that illustrate Cullinane cubes may be difficult to display.

Here instead is an animated GIF that shows the basic unit
for the "design cube" pages at finitegeometry.org.

Wednesday, March 21, 2018


Filed under: General,Geometry — Tags: — m759 @ 2:15 PM

WISC = Wechsler Intelligence Scale for Children

RISCReduced Instruction Set Computer   or
             Rust Inventory of Schizotypal Cognitions

See related material in earlier WISC RISC posts.

See also . . .

"Many parents ask us about the Block Design section
on the WISC and hope to purchase blocks and exercises
like those used on the WISC test. We explain that doing that
has the potential to invalidate their child's test results.
These Froebel Color Cubes will give you a tool to work with
your child on the skills tested for in the Block Design section
of the WISC in an ethical and appropriate way. These same
skills are applicable to any test of non-verbal reasoning like  
the NNAT, Raven's or non-verbal sections of the CogAT or OLSAT. "

An online marketing webpage

For a webpage that is perhaps un ethical and in appropriate,
see Block Designs in Art and Mathematics.

Monday, March 12, 2018


Filed under: General,Geometry — m759 @ 9:55 PM

Stein reportedly died at 100 last Friday (March 9).

Related material —

Textiles by Stein arranged on the six faces of a cube —

Ethel Stein, "Circus & Slapstick," 1996

See also a less amusing approach to
patterns on the faces of a cube.

“Quantum Tesseract Theorem?”

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

Remarks related to a recent film and a not-so-recent film.

For some historical background, see Dirac and Geometry in this journal.

Also (as Thas mentions) after Saniga and Planat —

The Saniga-Planat paper was submitted on December 21, 2006.

Excerpts from this  journal on that date —

A Halmos tombstone and the tale of HAL and the pod bay doors

     "Open the pod bay doors, HAL."

Thursday, March 8, 2018

Thanking the Academy…

Filed under: General,Geometry — m759 @ 9:11 PM


Wednesday, March 7, 2018

Unite the Seven.

Filed under: General,Geometry — Tags: , , — m759 @ 10:31 AM

Related material —

The seven points of the Fano plane within 

The Eightfold Cube.

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

"Before time began . . . ."

  — Optimus Prime

Sunday, March 4, 2018

The Square Inch Space: A Brief History

Filed under: General,Geometry — Tags: , — m759 @ 11:21 AM

1955  ("Blackboard Jungle") —

1976 —

2009 —

2016 —

 Some small Galois spaces (the Cullinane models)

Wednesday, February 28, 2018

Mathematics and Narrative

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


Excerpts from a post of May 25, 2005 —

Above is an example I like of mathematics….

Here is an example I like of narrative:

Kate felt quite dizzy. She didn't know exactly what it was
that had just happened, but she felt pretty damn  certain  that
it  was  the  sort of experience that her mother would not have
approved of on a first date.
     "Is this all part of what we have to do to go to  Asgard?"
she said. "Or are you just fooling around?"
     "We will go to Asgard...now," he said.
     At that moment he raised his hand as if to pluck an apple,
but instead of plucking he made a tiny, sharp turning movement.
The effect  was as if he had twisted the entire world through a
billionth part of a billionth  part  of  a  degree.  Everything
shifted,  was  for  a  moment  minutely  out of focus, and then
snapped back again as a suddenly different world.

— Douglas Adams, The Long Dark Tea-Time of the Soul

Image from a different  different world —

Hat-tip to a related Feb. 26 weblog post
at the American Mathematical Society.

Tuesday, February 27, 2018

Raiders of the Lost Images

Filed under: General,Geometry — m759 @ 11:28 AM

On the recent film "Justice League" —

From DC Extended Universe Wiki, "Mother Box" —

"However, during World War I, the British rediscovered
mankind's lost Mother Box. They conducted numerous studies
but were unable to date it due to its age. The Box was then
shelved in an archive, up until the night Superman died,
where it was then sent to Doctor Silas Stone, who
recognized it as a perpetual energy matrix. . . ." [Link added.]

The cube shape of the lost Mother Box, also known as the
Change Engine, is shared by the Stone in a novel by Charles Williams,
Many Dimensions . See the Solomon's Cube webpage.

See too the matrix of Claude Lévi-Strauss in posts tagged
Verwandlungslehre .

Some literary background:

Who speaks in primordial images speaks to us
as with a thousand trumpets, he grips and overpowers,
and at the same time he elevates that which he treats
out of the individual and transitory into the sphere of
the eternal. 
— C. G. JUNG

"In the conscious use of primordial images—
the archetypes of thought—
one modern novelist stands out as adept and
grand master: Charles Williams.
In The Place of the Lion  he incarnates Plato’s
celestial archetypes with hair-raising plausibility.
In Many Dimensions  he brings a flock of ordinary
mortals face to face with the stone bearing
the Tetragrammaton, the Divine Name, the sign of Four.
Whether we understand every line of a Williams novel
or not, we feel something deep inside us quicken
as Williams tells the tale.

Here, in The Greater Trumps , he has turned to
one of the prime mysteries of earth . . . ."

— William Lindsay Gresham, Preface (1950) to
Charles Williams's The Greater Trumps  (1932)

For fans of what the recent series Westworld  called "bulk apperception" —

Monday, February 26, 2018

The Unfolding

Filed under: General,Geometry — Tags: — m759 @ 9:20 AM

From the website of Richard P. Gabriel

" As part of my studies, I came up with a 'theory of poetry' 
based loosely on Christopher Alexander’s 'Nature of Order.' "
[The Alexander link is mine, not Gabriel’s.]

A phrase from this  journal a year ago today — "poetic order" —
links to the theory of Gabriel —

From Gabriel's "The Nature of Poetic Order" —

Positive Space

• Positive space is the characteristic of a center
that moves outward from itself, seemingly oozing life
rather than collapsing on itself
• An image that resonates is showing positive space
• A word that has many connotations that fit with the
other centers in the poem is showing positive space
• It is an expansion outward rather than a contraction
inward, and it shows that the poem is unfolding
in front of us and not dying

Related material —

From a post of April 26, 2017

Saturday, February 24, 2018

The Ugly Duck

Filed under: General,Geometry — Tags: — m759 @ 7:11 PM

"What of the night
That lights and dims the stars?
Do you know, Hans Christian,
Now that you see the night?"

— The concluding lines of
"Sonatina to Hans Christian,"
by Wallace Stevens
(in Harmonium  (second edition, 1931))

From "Mathmagic Land" (May 22, 2015)

Donald Duck with Pythagorean pentagram on hand

Donald in Mathmagic Land

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

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

Pentagon with pentagram    

Friday, February 23, 2018

The Dark, Seductive Art of Phillips Exeter

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

From two pedagogues in Montana —


Related material from author Dan Brown's father,
a pedagogue who taught at Phillips Exeter Academy :

Click the above image for some background.

Related material from Log24 —


Compare and contrast with the above
Transformational Geometry  cover:

Related material from Vienna — The previous post and
Wittgenstein on Bewitchment.

See as well . . .

Click to enlarge.

Thursday, February 22, 2018

Black Mountain Meets Blue Ridge

Filed under: General,Geometry — Tags: — m759 @ 10:48 AM

Click to enlarge

Related material —

Tuesday, February 20, 2018

The System

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

"It's the system  that matters. 
How the data arrange themselves inside it."

— Gravity's Rainbow  

"Examples are the stained-glass windows of knowledge."

— Vladimir Nabokov   

Map Systems (decomposition of functions over a finite field)

Saturday, February 17, 2018

The Binary Revolution

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

  • “The digital revolution based on the 2 symbols (0,1)”
  • “The algebra of George Boole”
  • “Binary codes”
  • “Dirac’s spinors, with their up/down dichotomy”

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

  • The 2×2 square
  • The 2x2x2 cube
  • The 4×4 square
  • The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

God’s Dice

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

On a Trinity classmate of Ian Macdonald (see previous post)—

Atiyah's eulogy of Shaw in Trinity Annual Record 2017 
is on pages 137 through 146.  The conclusion —


Friday, February 16, 2018

Two Kinds of Symmetry

Filed under: General,Geometry — m759 @ 11:29 PM

The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter 
revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below. 

In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald 
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)

The adjective "modular"  might aptly be applied to . . .

The adjective "affine"  might aptly be applied to . . .

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.

Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but 
did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For Macdonald's own  use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms," 
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.

Tuesday, February 13, 2018

A Titan of the Field

Filed under: General,Geometry — Tags: — m759 @ 9:45 AM

On the late Cambridge astronomer Donald Lynden-Bell —

"As an academic at a time when students listened and lecturers lectured, he had the disconcerting habit of instead picking on a random undergraduate and testing them on the topic. One former student, now a professor, remembered how he would 'ask on-the-spot questions while announcing that his daughter would solve these problems at the breakfast table'.

He got away with it because he was genuinely interested in the work of his colleagues and students, and came to be viewed with great affection by them. He also got away with it because he was well established as a titan of the field."

The London Times  on Feb. 8, 2018, at 5 PM (British time)

Related material —

Two Log24 posts from yesteday, Art Wars and The Void.

See as well the field GF(9)


and the 3×3 grid as a symbol of Apollo
    (an Olympian rather than a Titan) —


Wednesday, February 7, 2018

Conceptual Minimalism

Filed under: General,Geometry — m759 @ 10:08 AM


"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

See also AS IS.

Monday, February 5, 2018

Stranger Things than Pulp Fiction

Filed under: General,Geometry — m759 @ 12:30 PM

Diamond Theory cover, said to resemble Proginoskes in 'A Wind in the Door'

Click on the image for a
relevant Wallace Stevens poem.

A new Facebook page will describe
some background for the above image.

Sunday, February 4, 2018

Logos for Sunday, February 4

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

"The walls in the back of the room show geometric shapes
that remind us of the logos on a space shuttle. "

Web page on an Oslo art installation by Josefine Lyche.

See also Subway Art posts.

The translation above was obtained via Google.

The Norwegian original —

"På veggene bakerst i rommer vises geometriske former
som kan minne om logoene på en romferge."

Related logos — Modal Diamond Box in this journal:

Nietzsche, 'law in becoming' and 'play in necessity'

Logos for Philosophers
(Suggested by Modal Logic) —

Nietzsche, 'law in becoming' and 'play in necessity'

Friday, February 2, 2018

For Plato’s Cave

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

"Plato's allegory of the cave describes prisoners,
inhabiting the cave since childhood, immobile,
facing an interior wall. A large fire burns behind
the prisoners, and as people pass this fire their
shadows are cast upon the cave's wall, and
these shadows of the activity being played out
behind the prisoner become the only version of
reality that the prisoner knows."

— From the Occupy Space gallery in Ireland

IMAGE- Patrick McGoohan as 'The Prisoner,' with lapel button that says '6.'

Thursday, January 25, 2018

Beware of Analogical Extension

Filed under: General,Geometry — Tags: — m759 @ 11:29 AM

"By an archetype  I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."

— Max Black in Models and Metaphors 
    (Cornell, 1962, p. 241)

"Others … spoke of 'ultimate frames of reference' …."

A "frame of reference" for the concept  four quartets

A less reputable analogical extension  of the same
frame of reference

Madeleine L'Engle in A Swiftly Tilting Planet :

"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."

See also the phrase Galois tesseract .

Wednesday, January 24, 2018

The Pentagram Papers

Filed under: General,Geometry — Tags: — m759 @ 12:40 PM


From a Log24 post of March 4, 2008 —

"Are Children the Ultimate Literary Critics?"
— Top of the News 29 (Nov. 1972): 32-36.

"Sets forth his own aims in writing for children and laments
'slice of life' and chaos in children's literature. Maintains that
children like good plots, logic, and clarity, and that they
have a concern for 'so-called eternal questions.'"

— An Annotated Listing of Criticism
by Linnea Hendrickson

"She returned the smile, then looked across the room to
her youngest brother, Charles Wallace, and to their father,
who were deep in concentration, bent over the model
they were building of a tesseract: the square squared,
and squared again: a construction of the dimension of time."

— A Swiftly Tilting Planet,
by Madeleine L'Engle

Cover of 'A Swiftly Tilting Planet' and picture of tesseract

For "the dimension of time," see A Fold in TimeTime Fold,
and Diamond Theory in 1937

A Swiftly Tilting Planet  is a fantasy for children 
set partly in Vespugia, a fictional country bordered by
Chile and Argentina.


The pen's point:

Wm. F. Buckley as Archimedes, moving the world with a giant pen as lever. The pen's point is applied to southern South America.
John Trever, Albuquerque Journal, 2/29/08

Note the figure on the cover of National Review  above —

A related figure from Pentagram Design

See, more generally,  Isaac Singer  in this  journal.

Tuesday, January 23, 2018


Filed under: General,Geometry — Tags: , — m759 @ 1:27 PM

For the original Davos icon by Wink-Minneapolis,
see the previous post.  

For related geometry, see posts tagged Barth Art.

Monday, January 22, 2018

Hollywood Moment

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

Matt B. Roscoe and Joe Zephyrs, both of Missoula, Montana, authors of article on quilt block symmetries

A death on the date of the above symmetry chat,
Wednesday, August 17, 2016

'Love Story' director dies

An Hispanic Hollywood moment:

Ojo de Dios —

Click for related material.

For further Hispanic entertainment,
see Ben Affleck sing 
"Aquellos Ojos Verdes "
in "Hollywoodland."

Saturday, January 20, 2018

The Chaos Symbol of Dan Brown

Filed under: General,Geometry — m759 @ 1:30 AM

In the following passage, Dan Brown claims that an eight-ray star
with arrowheads at the rays' ends is "the mathematical symbol for
entropy."  Brown may have first encountered this symbol at a 
questionable "Sacred Science" website.  Wikipedia discusses
some even less  respectable uses of the symbol.

Related news —

Related symbolism —


A star figure and the Galois quaternion.

The square root of the former is the latter.

Wednesday, January 17, 2018

“Before Time Began, There Was the Cube”

Filed under: General,Geometry — m759 @ 8:00 AM

See Eightfold Froebel.

The Paradise of Childhood'-- Froebel's Third Gift

Tuesday, January 16, 2018

An Orison for Ha-Why

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

'Cloud Atlas' book cover illustrating the film

Lines from characters played in the film by Tom Hanks and Halle Berry —

— Cloud Atlas , by David Mitchell (2004).

An orison of sorts from a post on Martin Scorsese's
birthday, Sunday, Nov. 18, 2007 —

BlackBerry with pictures from Log24

Displayed on the BlackBerry are parts
of Log24 posts from October 25, 2007,
and October 24, 2007.

Related pattern geometry 

From a Log24 search for Angleton + Brotherhood:
A photo of Angleton in a post from 12/9/5

The image “http://www.log24.com/log/pix05B/051209-Angleton.jpg” cannot be displayed, because it contains errors.

From a post of 11/7/8


A cryptic note for Dan Brown:

The above dates 11/7/8 and 12/9/5 correspond to the corner-labels
(read clockwise and counter-clockwise) of the two large triangles
in the Finkelstein Talisman

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points

Above: More symbology for Tom Hanks from
this morning's post The Pentagram Papers.

The above symbology is perhaps better suited to Hanks in his
role as Forrest Gump than in his current role as Ben Bradlee.

The image “http://www.log24.com/log/pix05B/051211-gump.jpg” cannot be displayed, because it contains errors.

For Hanks as Dan Brown's Harvard symbologist 
Robert Langdon, see the interpretation 12/5/9, rather
than 12/9/5, of the above triangle/cube-corner label.

The Pentagram Papers

Filed under: General,Geometry — Tags: — m759 @ 9:16 AM

Other intersection-points-counting material —

The Finkelstein Talisman:

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points

See also Hanks + Cube in this journal —

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube.

Monday, January 8, 2018

Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: — m759 @ 11:15 PM

The Quantum Tesseract Theorem 



Raiders —

A Wrinkle in Time
starring Storm Reid,
Reese Witherspoon,
Oprah Winfrey &
Mindy Kaling


Time Magazine  December 25, 2017 – January 1, 2018


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

Sunday, January 7, 2018

Space Program

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

Or:  Interality Illustrated

See also Seven Seals.

Saturday, January 6, 2018

Report from Red Mountain

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

Tom Wolfe in The Painted Word  (1975):

“It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde  understood, they
were not only theories but … hot news,
straight from the studios, from the scene.”

Harold Rosenberg in The New Yorker  (click to enlarge)

See also Interality  and the Eightfold Cube .

Yale News

Filed under: General,Geometry — Tags: — m759 @ 5:24 AM

The Yale of the title is not the university, but rather the
mathematician Paul B. Yale. Yale's illustration of the Fano
plane is below.

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

A different illustration from a mathematician named Greenberg —

This illustration of the ominous phrase "line at infinity"
may serve as a sort of Deathly Hallows  for Greenberg.
According to the AMS website yesterday, he died on
December 12, 2017:

A search of this  journal for Greenberg yields no mention of
the dead mathematician, but does yield some remarks
on art that are pehaps less bleak than the above illustration.

For instance —

Art adapted from the Google search screen. Discuss.

Friday, January 5, 2018

Types of Ambiguity

Filed under: General,Geometry — Tags: , — m759 @ 2:56 AM

From "The Principle of Sufficient Reason," by George David Birkhoff
in "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940 —

From the same lecture —

Up to the present point my aim has been to consider a variety of applications of the Principle of Sufficient Reason, without attempting any precise formulation of the Principle itself. With these applications in mind I will venture to formulate the Principle and a related Heuristic Conjecture in quasi-mathematical form as follows:

PRINCIPLE OF SUFFICIENT REASON. If there appears in any theory T a set of ambiguously  determined ( i e . symmetrically entering) variables, then these variables can themselves be determined only to the extent allowed by the corresponding group G. Consequently any problem concerning these variables which has a uniquely determined solution, must itself be formulated so as to be unchanged by the operations of the group G ( i e . must involve the variables symmetrically).

HEURISTIC CONJECTURE. The final form of any scientific theory T is: (1) based on a few simple postulates; and (2) contains an extensive ambiguity, associated symmetry, and underlying group G, in such wise that, if the language and laws of the theory of groups be taken for granted, the whole theory T appears as nearly self-evident in virtue of the above Principle.

The Principle of Sufficient Reason and the Heuristic Conjecture, as just formulated, have the advantage of not involving excessively subjective ideas, while at the same time retaining the essential kernel of the matter.

In my opinion it is essentially this principle and this conjecture which are destined always to operate as the basic criteria for the scientist in extending our knowledge and understanding of the world.

It is also my belief that, in so far as there is anything definite in the realm of Metaphysics, it will consist in further applications of the same general type. This general conclusion may be given the following suggestive symbolic form:

Image-- Birkhoff diagram relating Galois's theory of ambiguity to metaphysics

While the skillful metaphysical use of the Principle must always be regarded as of dubious logical status, nevertheless I believe it will remain the most important weapon of the philosopher.

Related remarks by a founding member of the Metaphysical Club:

See also the previous post, "Seven Types of Interality."

Seven Types of Interality*

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

'Paradise of CHildhood'— on Froebel's Third Gift

* See the term interality  in this journal.
For many synonyms, see
The Human Seriousness of Interality,”
by Peter Zhang, Grand Valley State University,
China Media Research  11(2), 2015, 93-103.

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