Wednesday, April 1, 2020
Adventures in the Matrix
Thursday, June 21, 2018
Cavell’s Matrix
From an obituary for Stanley Cavell, Harvard philosopher
who reportedly died at 91 on Tuesday, June 19:
The London Review of Books weblog yesterday —
"Michael Wood reviewed [Cavell’s]
Philosophy the Day after Tomorrow in 2005:
'The ordinary slips away from us. If we ignore it, we lose it.
If we look at it closely, it becomes extraordinary, the way
words or names become strange if we keep staring at them.
The very notion turns into a baffling riddle.' "
See also, in this journal, Tuesday morning's Ici vient M. Jordan and
this morning's previous post.
Update of 3:24 AM from my RSS feed —
Sunday, April 22, 2018
Friday, April 20, 2018
Friday, December 1, 2017
The Architect and the Matrix
In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —
Update from the morning of December 2 —
The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:
.
See as well Tsimtsum in this journal.
Thursday, November 30, 2017
The Matrix for Quantum Mystics
Scholia on the title — See Quantum + Mystic in this journal.
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude LéviStrauss, 1976
I prefer the earlier, betterknown, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —
From a Log24 post of June 2627, 2017:
A work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
Friday, November 24, 2017
The Matrix Meets the Grid
The Matrix —
The Grid —
^{ Picturing the Witt Construction —}
"Read something that means something." — New Yorker ad
Thursday, November 23, 2017
The Matrix
David Brooks in The New York Times today —
"We once had a unifying national story, celebrated each Thanksgiving.
It was an Exodus story. Americans are the people who escaped oppression,
crossed a wilderness and are building a promised land. The Puritans brought
this story with them. Each wave of immigrants saw themselves in this story.
The civil rights movement embraced this story.
But we have to admit that many today do not resonate with this story. . . .
Today, we have no common national narrative, no shared way
of interpreting the flow of events. Without a common story,
we don’t know what our national purpose is. We have no
common set of goals or ideals.
We need a new national narrative."
From a post of August 15, 2010 —
For some background, see Java Jive and Today's Theology.
Related readings —
From 1928:
From the previous post:
"Thus, instead of Propp's chronological scheme,
in which the order of succession of events
is a feature of the structure . . .
another scheme should be adopted, which would present
a structural model defined as the group of transformations
of a small number of elements. This scheme would appear
as a matrix . . . ."
— Claude LéviStrauss, 1960
Wednesday, March 8, 2017
The Matrix Hypothesis
"And so both of these bizarre events put one in mind of
a simple but arresting thesis: that we are living in the Matrix,
and something has gone wrong with the controllers. . . .
The people or machines or aliens who are supposed to be
running our lives are having some kind of breakdown.
There’s a glitch, and we are in it.
Once this insight is offered, it must be said, everything else
begins to fall in order."
— Adam Gopnik in The New Yorker , Feb. 27, 2017
More recently …
Sunday, March 5, 2017
The Omega Matrix
Saturday, October 1, 2016
“A Matrix of Four”
The title is from a book quoted in the previous post.
A related illustration from 7:31 AM Tuesday, September 27 —
"The matrix at left below represents the feminine yin principle
and the diamond at right represents the masculine yang ."
Friday, November 27, 2015
Once Upon a Matrix
Monday, June 15, 2015
Omega Matrix
See that phrase in this journal.
See also last night's post.
The Greek letter Ω is customarily used to
denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the fourdimensional
affine space over the twoelement Galois
field, the appropriate Ω is the 4×4 grid above.
Saturday, April 25, 2015
A Matrix for Corliss
See Richard Corliss's 1999 review of "The Matrix," "Popular Metaphysics."
See as well the previous Log 24 post and the following publisher's
book description —
Harvard Heart of Gold by Dustin Aguilar
You may be a storybook character after all!!!
This philosophic, fantastical journey is a newfangled fairytale
where fun and unusual happenings are all too common, and
you—the reader—become a character just like Harvard or Kansas
and are subject to the allknowing, allpowerful, author of the story.
This daring piece tests the bounds of reality and subtly suggests
that you should question everything you know —
While most people in this story believe they are reallife, walking
talking humans, a small, somewhat violent sect of society has
realized they are actually part of a book. They lash out and
demand that the story have a happy ending, and they'll whomever
they have to. [Sic] An enormous battle erupts catching Harvard
and Kansas trapped in the middle forced to rely on their cunning
and a little help from an extralarge talking tarantula to save the day.
Thursday, November 13, 2014
Progressive Matrix
Yesterday's post and recent Hollywood news suggest
a meditation on a Progressive Matrix —
Click to enlarge.
"My card."
Structurally related images —
A sample Raven's Progressive Matrices test item
(such items share the 3×3 structure of the hash symbol above):
Structural background —
Tuesday, September 23, 2014
Matrix
Monday, September 22, 2014
The Matrix continues…
From the opening credits of the 2013 film “Paranoia,”
a hint that you may have taken the red pill:
(Note the reflection.)
Friday, August 15, 2014
The Omega Matrix
The webpage Rosenhain and Göpel Tetrads in PG(3,2)
has been updated to include more material on symplectic structure.
Sunday, August 3, 2014
The Omega Matrix
Shown below is the matrix Omega from notes of Richard Evan Schwartz.
See also earlier versions (19761979) by Steven H. Cullinane.
Backstory: The Schwartz Notes (June 1, 2011), and Schwartz on
the American Mathematical Society's current home page:
Friday, May 30, 2014
Combinatorial Matrix Classes
A book by this title, Richard A. Brualdi’s Combinatorial Matrix Classes ,
was published by Cambridge University Press in 2006:
For some related remarks, see The Counter (March 13, 2011).
My own work deals with combinatorial properties of matrices
of 0’s and 1’s, but in the context of Galois (i.e., finite) fields,
not the real or complex fields. Despite the generality of
their titles, Combinatorial Matrix Theory and Combinatorial
Matrix Classes do not deal with Galois matrices.
Saturday, June 2, 2012
Matrix Problem Revolutions
(The sequel to yesterday's Matrix Problem Reloaded)
Wikipedia on the scifi weblog io9.com—
Newitz explained the significance of the name "io9":
"Well, io9s are inputoutput devices that let you see into the future.
They're brain implants that were outlawed because they drove
anyone who used one insane. We totally made that (device) up
to name the blog."
— Jenna Wortham at wired.com, Jan. 2, 2008
From io9.com itself—
"Science fiction writer Ken MacLeod has another term for io9ers.
He calls them rapture fuckers.*"
For the relevance of the term "revolutions" in this post's title, see
Wikipedia on Ken MacLeod.
I prefer to associate the number 9 with The Holy Field.
* MacLeod used this phrase in one of his novels, Newton's Wake.
Friday, June 1, 2012
Matrix Problem Reloaded
For Frigg's Day—
"The newest badass fantasy flick to hit theaters is Snow White and the Huntsman , out today, and I believe it proves my theory that Kristen Stewart is the Keanu Reeves of her generation…. With few lines but a lot of convincingly heroic facial expressions, Kristen Stewart does a great job channeling Keanu Reeves in The Matrix (one of the dwarfs even calls her 'the one'!)."
— Annalee Newitz this morning, "The Awesome Terribleness of Snow White and the Huntsman "
See also yesterday's Matrix Problem.
Thursday, May 31, 2012
Matrix Problem
Poster from Walpurgisnacht 2012
Raven’s Progressive Matrices problem:
Click the problem for a related story.
For some related geometry, see Elements Diamond.
See also a post (Dream Time, May 3, 2010)
about geometry and an earlier Walpurgisnacht.
Sunday, May 8, 2011
Matrix
From Thomas Mann, "Schopenhauer," 1938, in Essays of Three Decades , translated by H. T. LowePorter, Alfred A. Knopf, 1947, pp. 372410—
Page 372: THE PLEASURE we take in a metaphysical system, the gratification purveyed by the intellectual organization of the world into a closely reasoned, complete, and balanced structure of thought, is always of a preeminently aesthetic kind. It flows from the same source as the joy, the high and ever happy satisfaction we get from art, with its power to shape and order its material, to sort out life's manifold confusions so as to give us a clear and general view.
Truth and beauty must always be referred the one to the other. Each by itself, without the support given by the other, remains a very fluctuating value. Beauty that has not truth on its side and cannot have reference to it, does not live in it and through it, would be an empty chimera— and "What is truth?"
Saturday, August 7, 2010
The Matrix Reloaded
For aficionados of mathematics and narrative —
Illustration from
"The Galois Quaternion— A Story"
This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—
The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean plane, but rather with unit squares
representing points in a finite Galois affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.
See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.
Thursday, August 5, 2010
The Matrix
The Presence of God in the Group: A Neglected Dimension
Ronald Sandison, Ledbury, Herefordshire, HR8 2EY, UK
Group Analysis March 1993, vol. 26 no. 1, 5565
Abstract
When people gather in small groups they are often aware of an indefinable suprapersonal dimension. Anyone who has taken part in a campfire singsong will have experienced this. The group setting at sundown was the time and place when folktales and fairy stories were woven. It was the time when the elders met to discuss the events of the day, when evensong or compline was sung or said in churches. There is something "special" about this conjunction of the group and the "going down of the sun." In this article I develop the idea that to take part in a small group can be, and often is, a spiritual experience. In giving a name to this experience I consider the proposition that God, or Spirit, is ever present in the group. I view the matrix as not only nurturing and holding but also as a dynamic container, and perhaps even generator, of spiritual force.
Related material
Dr. Sandison's obituary in this evening's Telegraph and this journal on the date of his death.
Monday, June 1, 2020
The Gefter Boundary
“The message was clear: having a finite frame of reference
creates the illusion of a world, but even the reference frame itself
is an illusion. Observers create reality, but observers aren’t real.
There is nothing ontologically distinct about an observer, because
you can always find a frame in which that observer disappears:
the frame of the frame itself, the boundary of the boundary.”
— Amanda Gefter in 2014, quoted here on Mayday 2020.
See as well the previous post.
Wednesday, May 20, 2020
Friday, November 15, 2019
Operators
Easy E
Not So Easy: EOperators
"A great many other properties of Eoperators
have been found, which I have not space
to examine in detail."
— Sir Arthur Eddington, New Pathways in Science ,
Cambridge University Press, 1935, page 271.
(This book also presents Eddington's unfortunate
speculations on the finestructure constant.)
Saturday, September 21, 2019
Annals of Random Fandom
For Dan Brown fans …
… and, for fans of The Matrix, another tale
from the above death date: May 16, 2019 —
An illustration from the above
Miracle Octad Generator post:
Related mathematics — Tetrahedron vs. Square.
Tuesday, August 13, 2019
Putting the Structure in Structuralism
(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2 (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée , No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)
The structure of the matrix of LéviStrauss —
Illustration from Diamond Theory , by Steven H. Cullinane (1976).
The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.
The Central Myth
Wikipedia on film producer Lorenzo di Bonaventura
(Harvard AB (1980) in Intellectual History*) —
"His tenure at Warner Bros. included discovering and
shepherding The Matrix into production, and the
purchase of the rights to the Harry Potter books by
J. K. Rowling."
From the previous post —
"But he enters into the central myth of
this book at another, higher level as well;
for he is an artist, a potter . . . ."
— Wendy Doniger O'Flaherty on
Claude LéviStrauss, author of
The Jealous Potter
* See as well "What is Intellectual History?" and
Magic for Liars .
Wednesday, July 24, 2019
The Batty Farewell
Adam Rogers today on "Rutger Hauer in Blade Runner , playing
the artificial* person Roy Batty in his death scene."
* See the word "Artifice" in this journal,
as well as Tears in Rain . . .
Friday, May 3, 2019
The Structure of Story Space
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A Permanent Order of Wondertale Elements
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). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 9798. p. 108). If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns). 
… Or by congruent quartersections.
Sunday, April 7, 2019
Block Talk
David Brooks in The New York Times Sunday Review today —
" 'In the deeps are the violence and terror of which psychology
has warned us,' Annie Dillard writes in 'Teaching a Stone to Talk.'
'But if you ride these monsters deeper down, if you drop with them
farther over the world’s rim, you find what our sciences cannot locate
or name, the substrate, the ocean or matrix or ether which buoys
the rest, which gives goodness its power for good, and evil its power
for evil, the unified field: our complex and inexplicable caring for
each other.' "
Annie Dillard on the legendary philosopher's stone —
“… if Holy the Firm is matter at its dullest, Aristotle’s materia prima ,
absolute zero, and since Holy the Firm is in touch with the Absolute
at base, then the circle is unbroken. And it is…. Holy the Firm is
in short the philosopher’s stone.”
See also "The Thing and I."
Monday, March 25, 2019
Espacement
(Continued from the previous post.)
InBetween "Spacing" and the "Chôra " (Ch. 2 in Henk Oosterling & Ewa Plonowska Ziarek (Eds.), Intermedialities: Philosophy, Arts, Politics , Lexington Books, October 14, 2010) "The term 'spacing' ('espacement ') is absolutely central to Derrida's entire corpus, where it is indissociable from those of différance (characterized, in the text from 1968 bearing this name, as '[at once] spacing [and] temporizing' ^{1}), writing (of which 'spacing' is said to be 'the fundamental property' ^{2}) and deconstruction (with one of Derrida's last major texts, Le Toucher: JeanLuc Nancy , specifying 'spacing ' to be 'the first word of any deconstruction' ^{3})." 1 Jacques Derrida, “La Différance,” in Marges – de la philosophie (Paris: Minuit, 1972), p. 14. Henceforth cited as D . 2 Jacques Derrida, “Freud and the Scene of Writing,” trans. A. Bass, in Writing and Difference (Chicago: University of Chicago Press, 1978), p. 217. Henceforth cited as FSW . 3 Jacques Derrida, Le Toucher, JeanLuc Nancy (Paris: Galilée, 2000), p. 207. . . . . "… a particularly interesting point is made in this respect by the French philosopher, Michel Haar. After remarking that the force Derrida attributes to différance consists simply of the series of its effects, and is, for this reason, 'an indefinite process of substitutions or permutations,' Haar specifies that, for this process to be something other than a simple 'actualisation' lacking any real power of effectivity, it would need “a soubassement porteur ' – let’s say a 'conducting underlay' or 'conducting medium' which would not, however, be an absolute base, nor an 'origin' or 'cause.' If then, as Haar concludes, différance and spacing show themselves to belong to 'a pure Apollonism' 'haunted by the groundless ground,' which they lack and deprive themselves of,^{16} we can better understand both the threat posed by the 'figures' of space and the mother in the Timaeus and, as a result, Derrida’s insistent attempts to disqualify them. So great, it would seem, is the menace to différance that Derrida must, in a 'properly' apotropaic gesture, ward off these 'figures' of an archaic, chthonic, spatial matrix in any and all ways possible…." 16 Michel Haar, “Le jeu de Nietzsche dans Derrida,” Revue philosophique de la France et de l’Etranger 2 (1990): 207227. . . . . … "The conclusion to be drawn from Democritus' conception of rhuthmos , as well as from Plato's conception of the chôra , is not, therefore, as Derrida would have it, that a differential field understood as an originary site of inscription would 'produce' the spatiality of space but, on the contrary, that 'differentiation in general' depends upon a certain 'spatial milieu' – what Haar would name a 'groundless ground' – revealed as such to be an 'inbetween' more 'originary' than the play of differences it informs. As such, this conclusion obviously extends beyond Derrida's conception of 'spacing,' encompassing contemporary philosophy's continual privileging of temporization in its elaboration of a preontological 'opening' – or, shall we say, 'inbetween.' 
For permutations and a possible "groundless ground," see
the eightfold cube and group actions both on a set of eight
building blocks arranged in a cube (a "conducting base") and
on the set of seven natural interstices (espacements ) between
the blocks. Such group actions provide an elementary picture of
the isomorphism between the groups PSL(2,7) (acting on the
eight blocks) and GL(3,2) (acting on the seven interstices).
Espacements
For the Church of Synchronology —
See also, from the reported publication date of the above book
Intermedialities , the Log24 post Synchronicity.
Saturday, February 23, 2019
Code Rain
See 031105code in this journal, the Wachowskis
at Wikipedia, and The Omega Matrix in this journal.
Friday, February 15, 2019
A Simple Interlacing
Paul Valéry, "Introduction to the Method of Leonardo da Vinci,"
La Nouvelle Revue , Paris, Vol. 95 (1895)—
"Regarded thus, the ornamental conception is to the individual arts
what mathematics is to the other sciences. …the objects chosen
and arranged with a view to a particular effect seem as if disengaged
from most of their properties and only reassume them in the effect,
in, that is to say, the mind of the detached spectator. It is thus
by means of an abstraction that the work of art can be constructed,
and is more or less easy to define according as the elements borrowed
from reality for it are more or less complex. Inversely it is by a sort of
induction, by the production of mental images, that all works of art are
appreciated, and this production must equally be more or less active,
more or less tiring, according as it is set in motion by a simple interlacing
on a vase or a broken phrase by Pascal."
— Translated by Thomas McGreevy (Valéry's Selected Writings,
New Directions, 1950)
Related art —

Departure dates in "Where Gravity Makes You Float,"
by Oslo artist Josefine Lyche  Apollo and Niobe at the Meadow
 "Enter the Matrix" game
Thursday, February 7, 2019
Geometry of the 4×4 Square: The Kummer Configuration
From the series of posts tagged Kummerhenge —
A Wikipedia article relating the above 4×4 square to the work of Kummer —
A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis. Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finitegeometry properties of the 4×4 square as
a finite affine 4space — properties that are of use in studying the Mathieu
group M_{24 }with the aid of the MOG.
Sunday, January 13, 2019
The Clifford Narrative
See also Clifford in this journal, in particular
The Matrix for Quantum Mystics
(Log24, St. Andrew's Day, 2017).
Thursday, July 12, 2018
Kummerhenge Illustrated
“… 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"
Monday, July 9, 2018
Beaty Confidential
Thursday, July 5, 2018
Paz:
Some context for what Heidegger called
das SpiegelSpiel des Gevierts
From Helen Lane's translation of El Mono Gramático ,
a book by Nobel winner Octavio Paz first published
in Barcelona by Seix Barral in 1974 —
Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oftrepeated and everdifferent metaphor of identity.
— Paz, Octavio. The Monkey Grammarian 
A related 1960 meditation from Claude LéviStrauss taken from a
Log24 post of St. Andrew's Day 2017, "The Matrix for Quantum Mystics":
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude LéviStrauss
I prefer the earlier, betterknown, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets
(from The Matrix Meets the Grid) —
Tuesday, July 3, 2018
Lost in Quantum Space
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, June 25, 2018
The Trials of Device
"A blank underlies the trials of device."
"Designing with just a blank piece of paper is very quiet."
Related material —
An image posted at 12 AM ET December 25, 2014:
The image stands for the
phrase "five by five,"
meaning "loud and clear."
Other posts featuring the above 5×5 square with some added structure:
Thursday, June 21, 2018
Tuesday, June 19, 2018
Ici vient M. Jordan
Tuesday, May 29, 2018
The Schwartz Meme
Aficionados of the preposterous joke
(see yesterday's post Epstein on Art)
may consult a Google Image Search for
Schwartz Meme.
I prefer Schwartz même —
Saturday, April 14, 2018
Immanentizing the Transcendence
The title refers to the previous two posts.
Related literature —
Plato's Ghost: The Modernist Transformation of Mathematics
(Princeton University Press, 2008) and . . .
Plato's diamondinamatrix:
Tuesday, February 27, 2018
Raiders of the Lost Images
On the recent film "Justice League" —
From DC Extended Universe Wiki, "Mother Box" —
"However, during World War I, the British rediscovered
mankind's lost Mother Box. They conducted numerous studies
but were unable to date it due to its age. The Box was then
shelved in an archive, up until the night Superman died,
where it was then sent to Doctor Silas Stone, who
recognized it as a perpetual energy matrix. . . ." [Link added.]
The cube shape of the lost Mother Box, also known as the
Change Engine, is shared by the Stone in a novel by Charles Williams,
Many Dimensions . See the Solomon's Cube webpage.
See too the matrix of Claude LéviStrauss 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 hairraising 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, November 27, 2017
The Golay Code via Witt’s Construction
Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053.
http://scholarworks.sjsu.edu/etd_theses/4053.
See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG in this journal.
Saturday, November 25, 2017
Night at the Museum
The previous post suggests a review of remarks by Adam Gopnik
in The New Yorker on February 27, 2017 on "The Matrix" hypothesis—
"The thesis that we are in a simulation is, as people who
track such things know—my own collegeage son has
explained it to me—far from a joke, or a mere conceit.
The argument, actually debated at length at the
American Museum of Natural History just last year, is that
the odds are overwhelming that ours is a simulated universe.
The argument is elegant."
No, it is not.
See as well my own remarks on the date of the above museum debate —
Tuesday, April 5, 2016.
From those remarks, a Halloween 2014 image that provides a
companionpiece to the "Easy E" of today's previous post —
EElements
German mathematician Wolf Barth reportedly died
on December 30, 2016.
Flashback to this journal on that date * —
From "The Man Who Tried to Redeem the World with Logic" —
"The following June, 1945, von Neumann penned Image from von Neumann's report —
Version converted to text —

* And, of course, to the later post Easy E for Cullinan (Feb. 28, 2017).
Cullinan, second from left below, is the nowfamous Oscars accountant.
Thursday, November 23, 2017
LéviStrauss vs. Propp
Claude LéviStrauss
From his “Structure and Form: To maintain. as I have done. that the permutability of contents is not arbitrary amounts to saying that, if the analysis is carried to a sufficiently deep level, behind diversity we will discover constancy. And, of course. the avowed constancy of form must not hide from us that functions are also permutable. The structure of the folktale as it is illustrated by Propp presents a chronological succession of qualitatively distinct functions. each constituting an independent genre. One can wonder whether—as with dramatis personae and their attributes— Propp does not stop too soon, seeking the form too close to the level of empirical observation. Among the thirtyone functions that he distinguishes, several are reducible to the same function reappearing at different moments of the narrative but after undergoing one or a number of transformations . I have already suggested that this could be true of the false hero (a transformation of the villain), of assigning a difficult task (a transformation of the test), etc. (see p. 181 above), and that in this case the two parties constituting the fundamental tale would themselves be transformations of each other. Nothing prevents pushing this reduction even further and analyzing each separate partie into a small number of recurrent functions, so that several of Propp’s functions would constitute groups of transformations of one and the same function. We could treat the “violation” as the reverse of the “prohibition” and the latter as a negative transformation of the “injunction.” The “departure” of the hero and his “return” would appear as the negative and positive expressions of the same disjunctive function. The “quest” of the hero (hero pursues someone or something) would become the opposite of “pursuit” (hero is pursued by something or someone), etc. 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). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 9798. p. 108) If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns). These critical remarks are certainly valid for the method used by Propp and for his conclusions. However. it cannot be stressed enough that Propp envisioned them and in several places formulated with perfect clarity the solutions I have just suggested. Let us take up again from this viewpoint the two essential themes of our discussion: constancy of the content (in spite of its permutability) and permutability of functions (in spite of their constancy). * Translated from a 1960 work in French. It appeared in English as Chapter VIII 
See also “LéviStrauss” + Formula in this journal.
Some background related to the previous post —
Sunday, August 13, 2017
The Red Pill
"We have passed through the looking glass
and down the rabbit hole."
— Kurt Andersen in the Sept. 2017 Atlantic
See as well "the desert of the real" and,
for comparison, College of the Desert in this journal.
Tuesday, March 21, 2017
Requiescat in Pace
The Chicago Tribune today —
H. Wilbert Norton, college president
and Christian missionary, dies
Norton reportedly died at 102 on Feb. 20, 2017.
This evening's previous post linked the death dates of two
academics to two Log24 posts that both contained the
following image —
For some backstory, see the Log24 posts from the date
of Norton's reported death, February 20.
Wednesday, March 8, 2017
Inscapes
"The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."
— Kent Johnson in a 1993 essay
The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214): 1. , , , , , 2. , , , , , 3. , , , , , 4. , , , , , 5. , , , , , 6. , , , , . SEE ALSO: Pauli Matrices REFERENCES: Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211217, 1985. Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed. Oxford, England: Pergamon Press, pp. 8084, 1982. Bethe, H. A. and Salpeter, E. Quantum Mechanics of One and TwoElectron Atoms. New York: Plenum, pp. 4748, 1977. Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics. New York: McGrawHill, 1964. Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, 1982. Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: AddisonWesley, p. 580, 1980. Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187211, 1955. Referenced on WolframAlpha: Dirac Matrices CITE THIS AS: Weisstein, Eric W. "Dirac Matrices."
From MathWorld— A Wolfram Web Resource. 
Tuesday, March 7, 2017
Signature Backdrop
"The Bitter End’s signature stage backdrop —
a bare 150yearold brick wall — helped distinguish it from
other popular bohemian hangouts like the Village Gate
and the Village Vanguard. It appeared on the cover of
Peter, Paul and Mary’s first album."
— The New York Times this evening on a Sunday death
“Looking carefully at Golay’s code is like staring into the sun.”
See also Schwartz in "The Omega Matrix," a post of 5 PM ET Sunday:
Monday, February 20, 2017
Mathematics and Narrative
Mathematics —
Hudson's parametrization of the
4×4 square, published in 1905:
A later parametrization, from this date in 1986:
A note from later in 1986 shows the equivalence of these
two parametrizations:
Narrative —
Posts tagged MemoryHistoryGeometry.
The mathematically challenged may prefer the narrative of the
Creation Matrix from the religion of the Transformers:
"According to religious legend, the core of the Matrix
was created from Solomus, the god of wisdom,
trapped in the form of a crystal by Mortilus, the god
of death. Following the defeat of Mortilus, Solomus
managed to transform his crystal prison into the Matrix—
a conduit for the energies of Primus, who had himself
transformed into the lifegiving computer Vector Sigma."
Monday, December 12, 2016
Tuesday, November 22, 2016
Jargon
See "sacerdotal jargon" in this journal.
For those who prefer scientific jargon —
"… open its reading to
combinational possibilities
outside its larger narrative flow.
The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."
— Kent Johnson in a 1993 essay
For some science that is not just jargon, see …
and, also from posts tagged Dirac and Geometry …
The above line complex also illustrates an outer automorphism
of the symmetric group S_{6}. See last Thursday's post "Rotman and
the Outer Automorphism."
Saturday, October 1, 2016
Doris and Oscar
An image from last night's post Brand Name —
"Squared into a matrix of four"
YouTube data suggested by the above passage —
Related literary remarks —
— A Heart for the Gods of Mexico , Conrad Aiken, 1939
Tuesday, September 27, 2016
Chomsky and LéviStrauss in China
Or: Philosophy for Jews
From a New Yorker weblog post dated Dec. 6, 2012 —
"Happy Birthday, Noam Chomsky" by Gary Marcus—
"… two titans facing off, with Chomsky, as ever,
defining the contest"
"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."
Socrates and the slave boy discussed a rather elementary "truth
about geometry" — A diamond inscribed in a square has area 2
(and side the square root of 2) if the square itself has area 4
(and side 2).
Consider that notparticularlydeep structure from the Meno dialogue
in the light of the following…
The following analysis of the Meno diagram from yesterday's
post "The Embedding" contradicts the LéviStrauss dictum on
the impossibility of going beyond a simple binary opposition.
(The Chinese word taiji denotes the fundamental concept in
Chinese philosophy that such a goingbeyond is both useful
and possible.)
The matrix at left below represents the feminine yin principle
and the diamond at right represents the masculine yang .
From a post of Sept. 22,
"Binary Opposition Illustrated" —
A symbol of the unity of yin and yang —
Related material:
A much more sophisticated approach to the "deep structure" of the
Meno diagram —
Wednesday, May 25, 2016
Framework
"Studies of spin½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng and H. van Dam,
February 20, 2009
For one such framework,* see posts from that same date
four years earlier — February 20, 2005.
* A 4×4 array. See the 1977, 1978, and 1986 versions by
Steven H. Cullinane, the 1987 version by R. T. Curtis, and
the 1988 ConwaySloane version illustrated below —
Cullinane, 1977
Cullinane, 1978
Cullinane, 1986
Curtis, 1987
Update of 10:42 PM ET on Sunday, June 19, 2016 —
The above images are precursors to …
Conway and Sloane, 1988
Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.
Tuesday, May 24, 2016
Rosenhain and Göpel Revisited
The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .
"This famous book is a prototype for the possibility
of explaining and exploring a manyfaceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."
— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com
Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4space over
the twoelement Galois field GF(2).
Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .
Some related work of my own (click images for related posts)—
Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)
Göpel tetrads as 15 of the 35 projective lines in PG(3,2)
Related terminology describing the Göpel tetrads above
Sunday, May 1, 2016
Sunday Appetizer from 1984
Judith Shulevitz in The New York Times
on Sunday, July 18, 2010
(quoted here Aug. 15, 2010) —
“What would an organic Christian Sabbath look like today?”
The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —
Mathematische Appetithäppchen: Autor: Erickson, Martin —
"Weitere Informationen zu diesem Themenkreis finden sich 
That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."
The title of the document, "Binary Coordinate Systems,"
is highly relevant to figure 11.16c on page 312 of a book
published four years after the document was written: the
1988 first edition of Sphere Packings, Lattices and Groups ,
by J. H. Conway and N. J. A. Sloane —
A passage from the 1984 document —
Sunday, March 6, 2016
Friday, December 11, 2015
Street View
Continued from Once Upon a Matrix (November 27, 2015).
Click image below to enlarge.
“… Which makes it a giltedged priority that one of us
gets into that Krell lab and takes that brain boost.”
— American adaptation of Shakespeare's Tempest , 1956
Midrash —
"Remember me to Herald Square."
Thursday, August 27, 2015
Friday, July 24, 2015
Field of Manifestation
This post was suggested by a book title in
the previous post: "Pieces of the Action."
A group action is a mathematical concept.
Related meditation:
"The number 9, that is to say, relates traditionally
to the Great Goddess of Many Names (Devi,
Inanna, Ishtar, Astarte, Artemis, Venus, etc.)
as matrix of the cosmic process, whether in the
macrocosm or in a microcosmic field of manifestation."
— Joseph Campbell,
The Inner Reaches of Outer Space
Examples:
Monday, July 13, 2015
Weaving World
(The title was suggested by the novel Weaveworld .)
Recent public selfie by Oslo artist Josefine Lyche —
Lyche's shirt honors the late Kurt Cobain.
Notsorecent image of Hugo Weaving as
Agent Smith in "The Matrix" —
"Smells like teen spirit."
See also Weaving in the new film "Strangerland."
Monday, June 15, 2015
Slow Art
Slowness is sometimes in the eye of the beholder.
See this journal on Slow Art Day 2015.
Related material: Epistemic States in this journal.
Saturday, June 13, 2015
Plan 9 Continues…
A version of the song in the previous post that I prefer:
A related meditation —
For a more abstract version of the
"matrix of the cosmic process,"
see "3×3 Grid" in this journal.
Saturday, April 25, 2015
Tuesday, December 30, 2014
An Education
AP Today in History
Thought for the Day:
“I respect faith, but doubt is what
gives you an education.”
— Wilson Mizner,
American playwright (18761933)*
From this journal on the (wide) release date
of "XMen: First Class" —
A minimalist 3×3 matrix favicon—
This may, if one likes, be viewed as the "nothing"
present at the Creation. See Jim Holt on physics.
* A source —
Wednesday, December 17, 2014
Launched from Cuber
Continued from Nobel Note (Jan. 29, 2014).
From Tradition in Action , "The Missal Crisis of '62,"
remarks on the revision of the Catholic missal in that year—
"Neither can the claim that none of these changes
is heretical in content be used as an argument
in favor of its use, for neither is the employment of
hula girls, fireworks, and mariachis strictly speaking
heretical in itself, but they belong to that class of novel
and profane things that do not belong in the Mass."
— Fr. Patrick Perez, posted Sept. 11, 2007
See also this journal on November 22, 2014…
… and on Bruce Springsteen's birthday this year —
Tuesday, September 23, 2014

Thursday, December 4, 2014
The Pony Argument
The title was suggested by this morning's post "Follow This."
From a previous incarnation of my home website, m759.com —
The second of the site's three pages mentions authors
Alfred Bester and Zenna Henderson :
"Bester and Henderson are particularly good at
fictional accounts of telepathy. The noted Harvard
philosopher W. V. Quine doubts such a thing exists,
but I prefer the 'There must be a pony' argument."
Related material: The date Nov. 27, 2014, in a web search today …
… in The Washington Post …
… and in this journal …
Thursday, November 27, 2014
In the Details
For the late novelist P. D. James, author of, among other things,
The Children of Men .
Michael Caine in the film of that book —
Detail —
Raven's Progressive Matrices example —
A quote for tellers of tales —
“There have long been rumors of a mythical Ninth Element
that grants ultimate power to the Wizard who masters it.
The Order of Magick says there is no such thing. But….”
— Website of Magicka: The Ninth Element Novel
Tuesday, November 18, 2014
The Abacus Conundrum…
Prequel from 1961 (click image for context):
Detail that may be interpreted as the Chinese
3×3 "Holy Field" and a Chinese temple bell—
"Tingaling." — Kurt Vonnegut.
Our Most Important Product
Hexagram 35:
Progress
"Then came a 'Robot Psychologist,' known as a Psychological Matrix Rotator,
developed for the Department of Defense. It is still used to literally 'see' that
the right man gets the right Army job."
— Ronald Reagan, 1961 GE Sales Meeting
"Always with a little humor." — Yen Lo
In memory of Dr. Irving Peress,
who reportedly died on Thursday,
November 13, 2014.
Monday, September 22, 2014
A Holy Cross for Brooklyn
“Once again, Harvard defeats Holy Cross.”
That post quoted Joseph Campbell describing a
“… matrix of the cosmic process, whether in the macrocosm
or in a microcosmic field of manifestation.”
Related material —
Macrocosm:
Microcosmic field of manifestation:
Wednesday, August 13, 2014
Stranger than Dreams*
Illustration from a discussion of a symplectic structure
in a 4×4 array quoted here on January 17, 2014 —
See symplectic structure in this journal.
* The final words of Point Omega , a 2010 novel by Don DeLillo.
See also Omega Matrix in this journal.
Thursday, August 7, 2014
The Omega Mystery
See a post, The Omega Matrix, from the date of her death.
Related material:
"When Death tells a story, you really have to listen."
— Cover of The Book Thief
A scene from the film of the above book —
“Looking carefully at Golay’s code is like staring into the sun.”
Some context — "Mathematics, Magic, and Mystery" —
See posts tagged April Awareness 2014.
Tuesday, August 5, 2014
The Omega Story
"We tell ourselves stories in order to live…. We interpret what we see, select the most workable of multiple choices. We live entirely, especially if we are writers, by the imposition of a narrative line upon disparate images, by the 'ideas' with which we have learned to freeze the shifting phantasmagoria which is our actual experience." 
See also a post from May 4, 2011 (the date, according to a Google
search, of untitled notes regarding a matrix called Omega).
Monday, August 4, 2014
A Wrinkle in Space
"There is such a thing as a tesseract." — Madeleine L'Engle
An approach via the Omega Matrix:
See, too, Rosenhain and Göpel as The Shadow Guests .
Tuesday, June 3, 2014
Galois Matrices
The webpage Galois.us, on Galois matrices , has been created as
a starting point for remarks on the algebra (as opposed to the geometry)
underlying the rings of matrices mentioned in AMS abstract 79TA37,
“Symmetry invariance in a diamond ring.”
See also related historical remarks by Weyl and by Atiyah.
Thursday, May 29, 2014
Lost in Translation
Translation by Barbara Johnson:
"The minimum number of rows— lines or columns—
that contain all the zeros in a matrix is equal to
the maximum number of zeros
located in any individual line or column ."
In the original:
"situés sur des lignes ou des colonnes distinctes "
Update of 11:30 PM ET May 29, 2014:
Derrida in 1972 was quoting Philippe Sollers, Nombres
(Paris: Éditions du Seuil , 1968). Sollers in turn was
perhaps quoting A. Kaufmann, Méthodes et Modèles
de la Recherche Opérationnelle , Paris, Dunod , 1964,
L'Économie d'Entreprise 10 , vol. 2, page 305:
"Le nombre minimal de rangées
(lignes et/ou colonnes) contenant
tous les zéros d'une matrice, est égal
au nombre maximal de zéros
situés sur des lignes et des colonnes distinctes."
Wednesday, May 7, 2014
Meanwhile, in São Paulo…
This post was suggested by last night’s posts on conceptual art
(in São Paulo) and on a quartertothree story.
From this journal on May 31, 2012:
Matrix Problem:
Click image for some related material.
Meanwhile…
A game released on the above date:
Friday, January 17, 2014
The 4×4 Relativity Problem
The sixteendot square array in yesterday’s noon post suggests
the following remarks.
“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
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16point affine geometry over the twoelement Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79TA37.)
See also a 2011 publication of the Mathematical Association of America —
Wednesday, December 25, 2013
Sunday, December 22, 2013
The Bronfman Catechism
Meanwhile…
Log24 on Sunday, October 5, 2008— “In a game of chess, the knight’s move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a ‘knight’s move’ in intelligence.” Related material: Terence McKenna: “Schizophrenia is not a psychological disorder peculiar to human beings. Schizophrenia is not a disease at all but rather a localized traveling discontinuity of the space time matrix itself. It is like a travelling whirlwind of radical understanding that haunts time. It haunts time in the same way that Alfred North Whitehead said that the color dove grey ‘haunts time like a ghost.’” Anonymous author: “‘Knight’s move thinking’ is a psychiatric term describing a thought disorder where in speech the usual logical sequence of ideas is lost, the sufferer jumping from one idea to another with no apparent connection. It is most commonly found in schizophrenia.” 
Related journalism—
"What's the 'S' stand for?" — Amy Adams
Wednesday, December 11, 2013
The WellTempered Monolith
Tuesday, December 10, 2013
Blue Note
For the late Jim Hall—
Backstory: Icon, 1:44 PM ET today.
Update of 11 PM ET Dec. 10, 2013 —
For all the notes, see Da Capo (11 AM today)
and the Cullinane frequency matrix (12×12).
Matrix used to illustrate the welltempered
scale. The integer frequencyratio values
are only approximate in such a scale.
Friday, September 6, 2013
Space
"A vast space that travels down to the bedrock
upon which the towers were built, the museum
winds its way deeper and deeper underground,
taking visitors on a journey to the very bottom."
— The Associated Press in
this evening's Washington Post
This suggests a review of a different sort of
bedrock:—
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
Monday, August 12, 2013
Form
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The Galois tesseract is the basis for a representation of the smallest
projective 3space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday’s post.
The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—
As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator (MOG) of
R. T. Curtis.
Tuesday, July 9, 2013
Vril Chick
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) 
Clearly most of this (the nonhighlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
Tuesday, May 28, 2013
Codes
The hypercube model of the 4space over the 2element Galois field GF(2):
The phrase Galois tesseract may be used to denote a different model
of the above 4space: the 4×4 square.
MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galoistesseract model of the 4space over GF(2).
The thirtyfive 4×4 structures within the MOG:
Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:
A later book coauthored by Sloane, first published in 1988,
did recognize the 4×4 MOG patterns as based on the 4×4
Galoistesseract model.
Between the 1977 and 1988 Sloane books came the diamond theorem.
Update of May 29, 2013:
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliamsSloane book was first published):
Monday, April 15, 2013
M Theory
This morning's previous post pictured the cover
of a book titled "The Mystery of the Quantum World."
That title, together with Peter Woit's post on Hawking
yesterday, suggests a review of the phrase "M theory."
See remarks on that topic in the October 1998
Notices of the American Mathematical Society :
"The richer theory, which has as limiting cases
the five string theories studied in the last generation,
has come to be called M theory, where M stands for
magic, mystery, or matrix, according to taste."
— "Magic, Mystery, and Matrix," by Edward Witten
See also, in this journal, a post mentioning Hawking's
vulgarized M theory book The Grand Design , as well
as a post of January 9, 2012, titled "M Theory."
Of Witten's three alternative meanings for M , I prefer "matrix."
Tuesday, March 26, 2013
Nothing Revisited
"… the primal ground of this 'one' is 'nothing.'
The elucidation of the generative matrix of the
mythwork is thus completed…." — Jadran Mimica
Related material— The Nothing That Is.
Wednesday, March 6, 2013
The Red Pill
"A couple of years later, I asked him why,
so late in the day, he had decided to
adopt socialism. He acknowledged that he
had come to it late, long after most of
the world had abandoned it, but said that it
had clicked for him after he had read
Victor Hugo’s epic novel Les Misérables .
That, and listening to Fidel."
— The New Yorker 's Jon Lee Anderson
"As you can see, we've had our eye on you
for some time now, Mr. Anderson."
Tuesday, January 22, 2013
Raven Light
"…a fundamental cognitive ability known as 'fluid' intelligence: the capacity to solve novel problems, to learn, to reason, to see connections and to get to the bottom of
…matrices are considered the gold standard of fluidintelligence tests. Anyone who has taken an intelligence test has seen matrices like those used in the Raven’s: three rows, with three graphic items in each row, made up of squares, circles, dots or the like. Do the squares get larger as they move from left to right? Do the circles inside the squares fill in, changing from white to gray to black, as they go downward? One of the nine items is missing from the matrix, and the challenge is to find the underlying patterns— up, down and across— from six possible choices. Initially the solutions are readily apparent to most people, but they get progressively harder to discern. By the end of the test, most test takers are baffled."
— Dan Hurley, "Can You Make Yourself Smarter?," NY Times , April 18, 2012
See also "Raven Steals the Light" in this journal.
Related material:
Plan 9 from MIT and, perhaps exemplifying crystallized rather than fluid intelligence, Black Diamond.
Monday, December 24, 2012
Eternal Recreation
Memories, Dreams, Reflections
by C. G. Jung
Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989
From pages 195196:
"Only gradually did I discover what the mandala really is:
'Formation, Transformation, Eternal Mind's eternal recreation.'*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate selfdeceptions."
* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:
… Gestaltung, Umgestaltung,
Des ewigen Sinnes ewige Unterhaltung….
Jung's "Formation, Transformation" quote is from the realm of
the Mothers (Faust , Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.
See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:
"In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust's next encounter with
the feminine, but it's obviously of a very different kind than his
relationship with Gretchen."
The phrase "Gestaltung, Umgestaltung " suggests a more mathematical
approach to the Unterhaltung . Hence…
Part I: Mothers
"The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time."
— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity ,
Johns Hopkins Press, 1969, page 37
Part II: Matrices
Part III: Spaces and Hypercubes
Click image for some background.
Part IV: Forms
Forms from the I Ching :
Click image for some background.
Forms from Diamond Theory :
Click image for some background.
Sunday, December 23, 2012
In a Nutshell…
The Kernel of the Concept of the Object…
according to the New York Lottery yesterday—
From 4/27
From 11/24
A page numbered 176
A page numbered 187
Sunday, December 2, 2012
Knight’s Labyrinth
A magic— indeed, diabolic— square:
For the construction, see a book
by W. W. Rouse Ball, founding president
of a Cambridge University magic society.
For some related religious remarks,
see Raiders of the Lost Matrix.
Saturday, November 24, 2012
Deep Agents
New today…
John Markoff in this morning's New York Times—
"…what is new in recent months is the growing speed and accuracy
of deeplearning programs, often called artificial neural networks
or just 'neural nets' for their resemblance to the neural connections
in the brain."
Not so new…
From the early days of Mike Lynch's Autonomy Inc.—
"Autonomy 's intelligent agents use neural networking
to search for patterns of information, rather than
specific words or phrases, thereby distinguishing relevant
from irrelevant information." (Business Wire , October 8, 1996)
Tuesday, October 30, 2012
Black October
From AntiChristmas 2010—
Art theorist Rosalind Krauss and The Ninefold Square
Krauss is a cofounder of the art journal October .
For some backgound, see the overlapping searches
Krauss October and Matrix Architect.
Thursday, October 25, 2012
Bunker Bingo*
Mathematics and Narrative continues…
Steiner's version of "classical functional analysis"—
"Mein Führer… Steiner…"
* See the story by Kilgore Trout. See also On Linguistic Creation,
The Matrix of Abraham, and The Thoreau Foundation.
Monday, October 1, 2012
Foundation
» more
This, together with this morning's post involving the squares 16 and 9,
suggests a review of Conceit at Harvard (October 25, 2006), which
contains the following figure involving the squares 16, 9, and 25—
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
Thursday, September 6, 2012
Glory Road (continued)
"In ancient Greece, 9 was the number of the Muses,
patron goddesses of the arts. They were the daughters
of Mnemosyne ('memory'), the source
of imagination, which in turn is the carrier of archetypal,
elementary ideas to artistic realization in the field
of spacetime. The number 9, that is to say, relates
traditionally to the Great Goddess of Many Names
(Devi, Inanna, Ishtar, Astarte, Artemis, Venus, etc.),
as matrix of the cosmic process, whether in the
macrocosm or in a microcosmic field of manifestation."
— Joseph Campbell in The Inner Reaches of Outer Space ,
first published in 1986
From Robert A. Heinlein’s Glory Road (1963): Her face turned thoughtful. “Would you like to call me ‘Ettarre’?” “Is that one of your names?” “It is much like one of them, allowing for different spelling and accent. Or it could be ‘Esther’ just as closely. Or ‘Aster.’ Or even ‘Estrellita.’ “ ” ‘Aster,’ ” I repeated. “Star. Lucky Star!” 
Saturday, October 22, 2011
The Meter is Running
Suggested by this morning's previous post (Araby) as well as
by Thursday's posts Jack and Jill and The Thing Itself …
an item from Google News—
Note the beauty of the headline's meter.
A midrash for Bloomberg—
"Let us return to the insertions." —André Topia, "The Matrix and the Echo."
Monday, October 10, 2011
The Aleph
COLLEGE OF THE DESERT
Minutes — Organization Meeting
11:00 a.m., Saturday, July 1, 1961—
15. Preparation of College Seal:
By unanimous consent preparation of a College
Seal to contain the following features was
authorized: A likeness of the Library building
set in a matrix of date palms, backed by
a mountain skyline and rising sun; before
the Library an open book, the Greek symbol
Alpha on one page and Omega on the other;
the Latin Lux et Veritas, College of the
Desert, and 1958 to be imprinted within or
around the periphery of the seal.
From the website http://geofhagopian.net/ of
Geoff Hagopian, Professor of Mathematics,
College of the Desert—
Note that this version of the seal contains
an Aleph and Omega instead of Alpha and Omega.
From another Hagopian website, another seal.
Sunday, September 11, 2011
First Lady
Betty Skelton, "the First Lady of Firsts," died on the last day of August.
From this journal on August thirtyfirst—
"The Tesseract was the jewel of Odin's treasure room."
Hugo Weaving also played Agent Smith
in The Matrix Trilogy .
For Cynthia Zarin, biographer of Madeleine L'Engle—
"There is such a thing as a tesseract."
— A Wrinkle in Time
Saturday, September 3, 2011
The Galois Tesseract (continued)
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
twothirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79TA37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG’s
4×4 square as the affine 4space over the 2element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four “special tetrads” within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 “special tetrads” rather by the parity
of their intersections with the square’s rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The “35 structures” of the abstract were listed, with an application to
Latinsquare orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M_{24}, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was misnamed as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Wednesday, August 31, 2011
Odin’s Day
Today is Wednesday.
O.E. Wodnesdæg "Woden's day," a Gmc. loantranslation of L. dies Mercurii "day of Mercury" (cf. O.N. Oðinsdagr , Swed. Onsdag , O.Fris. Wonsdei , M.Du. Wudensdach ). For Woden , see Odin . — Online Etymology Dictionary
Above: Anthony Hopkins as Odin in the 2011 film "Thor"
Hugo Weaving as Johann Schmidt in the related 2011 film "Captain America"—
"The Tesseract* was the jewel of Odin's treasure room."
Weaving also played Agent Smith in The Matrix Trilogy.
The figure at the top in the circle of 13** "Thor" characters above is Agent Coulson.
"I think I'm lucky that they found out they need somebody who's connected to the real world to help bring these characters all together."
— Clark Gregg, who plays Agent Coulson in "Thor," at UGO.com
For another circle of 13, see the Crystal Skull film implicitly referenced in the Bright Star link from Abel Prize (Friday, Aug. 26, 2011)—
Today's New York Times has a quote about a former mathematician who died on that day (Friday, Aug. 26, 2011)—
"He treated it like a puzzle."
Sometimes that's the best you can do.
* See also tesseract in this journal.
** For a different arrangement of 13 things, see the cube's 13 axes in this journal.
Friday, June 3, 2011
MIT Day
Today is Commencement Day at MIT.
“To measure the changes — ShingTung Yau, To measure the changes: The smartest are nothing: 
Well, perhaps not quite nothing.
The above pictures were posted here on the day the following book was published—
The lives of the nine Jews in the above book amount to more than Yau's "nothing."
Note, however, that claims by Jews (see Jill Abramson yesterday)
that their secular publications constitute a substitute for religion
and contain only "absolute truth" should be viewed with at least one
raised eyebrow.
Abramson's remark yesterday that her promotion to New York Times executive editor
was like "ascending to Valhalla" had a religious flavor worthy of yesterday's
Feast of the Ascension.
In related news from yesterday's Times—
See also a symbol related to Apollo, to nine, and to "nothing"—
A minimalist 3×3 matrix favicon—
This may, if one likes, be viewed as the "nothing"
present at the Creation. See Jim Holt on physics.
Wednesday, June 1, 2011
The Schwartz Notes
A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signedout search activity was disabled.)
The same search limited to results from only the past month yielded,
in addition, the following—
This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.
Pages 1214 of the document, which is untitled, undated, and
unsigned, discuss the finitegeometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.
See Schwartz’s page 12, page 13, and page 14.
Compare to the web pages from today’s earlier search.
There are no references at the end of the Schwartz document,
but there is this at the beginning—
These are some notes on error correcting codes. Two good sources for
this material are
• From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
• Sphere Packings, Lattices, and Simple Groups by J. H. Conway and N.
Sloane
Planet Math (on the internet) also some information.
It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.
Wednesday, May 18, 2011
Minimalist Icon
The source of the mysterious generic
3×3 favicon with one green cell —
— has been identified.
For minimalists, here is a purer 3×3 matrix favicon—
This may, if one likes, be viewed as the "nothing"
present at the Creation. See Jim Holt on physics.
See also Visualizing GL(2,p), Coxeter and the Aleph, and Ayn Sof.
Sunday, March 13, 2011
The Counter
"…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—
Cover Illustration Number 1 (1976):
Cover Illustration Number 2 (1991):
The Shrikhande Graph
______________________________________________________________________________
This post was prompted by two remarks…
1. In a different weblog, also on August 26, 2010—
The Accidental Mathematician— "The Girl Who Played with Fermat's Theorem."
"The worst thing about the series is the mathematical interludes in The Girl Who Played With Fire….
Salander is fascinated by a theorem on perfect numbers—
one can verify it for as many numbers as one wishes, and it never fails!—
and then advances through 'Archimedes, Newton, Martin Gardner,*
and a dozen other classical mathematicians,' all the way to Fermat’s last theorem."
2. "The fact that the pattern retains its symmetry when you permute the rows and columns
is very well known to combinatorial theorists who work with matrices."
[My italics; note resemblance to the BrualdiRyser title above.]
–Martin Gardner in 1976 on the diamond theorem
* Compare Eric Temple Bell (as quoted at the MacTutor history of mathematics site)—
"Archimedes, Newton, and Gauss, these three, are in a class by themselves
among the great mathematicians, and it is not for ordinary mortals
to attempt to range them in order of merit."
This is from the chapter on Gauss in Men of Mathematics .
Saturday, January 8, 2011
True Grid (continued)
"Rosetta Stone" as a Metaphor
in Mathematical Narratives
For some backgound, see Mathematics and Narrative from 2005.
Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).
For some other properties, see (at the collegeundergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."
His conclusion:
When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….
This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!
5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
American Mathematical Society, 1997.
Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)
Investigation of this word yields a related essay at the graduatestudent, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."
From the DolgachevArtebani introduction–
In this paper we discuss some old and new results about the widely known Hesse
configuration of 9 points and 12 lines in the projective plane P^{2}(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (12_{3}, 9_{4}).
PlanetMath.org on the Hesse configuration—
A picture of the Hesse configuration–
(See Visualizing GL(2,p), a note from 1985).
Related notes from this journal —
From last November —
From the December 2010 American Mathematical Society Notices—
Related material from this journal— Consolation Prize (August 19, 2010) 
From 2006 —
Sunday December 10, 2006
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, 
Also from 2006 —
Sunday November 26, 2006
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The NonObjective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more uptodate example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal*– or, according to Krauss,
"This is the garden of Apollo, the field of Reason…."
For more on the field of reason, see
A reasonable set of "strange correspondences" Unreason is, of course, more popular. * The ninefold square is perhaps a "concrete universal" in the sense of Hegel: "Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."
– Hegel, Lectures on the History of Philosophy ,
"For every kind of vampire, 
And from last October —
Friday, October 8, 2010
Starting Out in the Evening This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .
"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."
– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
* The Web version's title has a misprint— 