Version from “The Avengers” (2012) —
Version from Josefine Lyche (2009) —
See also this journal on the date that the above Avengers video was uploaded.
Version from “The Avengers” (2012) —
Version from Josefine Lyche (2009) —
See also this journal on the date that the above Avengers video was uploaded.
From "The Osterman Weekend" (1983) —
Counting symmetries of the R. T. Curtis Omega:
An Illustration from Shakespeare's birthday —
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.
The previous post displayed a set of
24 unitsquare “points” within a rectangular array.
These are the points of the
Miracle Octad Generator of R. T. Curtis.
The array was labeled Ω
because that is the usual designation for
a set acted upon by a group:
* The title is an allusion to Point Omega , a novel by
Don DeLillo published on Groundhog Day 2010.
See “Point Omega” in this journal.
The webpage Rosenhain and Göpel Tetrads in PG(3,2)
has been updated to include more material on symplectic structure.
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.
"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).
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:
Today's news from Oslo suggests a review—
The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."
Some related philosophical remarks—
These news items suggest a review —
The above “Pynchon’s Paranoid History” page number appeared
in this journal on Groundhog Day, 2015 —
David Justice on a Zetarelated theory —
“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.
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.
“… 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"
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.
"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:
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
Slowness is sometimes in the eye of the beholder.
See this journal on Slow Art Day 2015.
Related material: Epistemic States in this journal.
See the previous post, "Space," as well as…
SymOmega in this journal and a suggested motto
for The University of Western Australia.
Notes on space for day 13 of May, 2015 —
The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."
Related poetic material:
The ninefold square and Apollo, as well as …
See Stevens + New Haven.
* The above figure may be viewed as
the Chinese “Holy Field” or as the
Chinese character for “Well”
inscribed in a square.
"William Zinsser, a writer, editor and teacher
whose book ‘On Writing Well’ sold more than
1.5 million copies by employing his own literary
craftsmanship to urge clarity, simplicity, brevity
and humanity, died on Tuesday [May 12, 2015]
at his home in Manhattan. He was 92."
— Douglas Martin in the online New York Times
From "Origins of the Logical Theory of Probability: von Kries, Wittgenstein, Waismann," by Michael Heidelberger — "Von Kries calls a range of objective possibilities of a hypothesis or event (under given laws) its Spielraum (literally: play space), which can mean ‘room to move’, ‘leeway’, ‘latitude of choice’, ‘degree of freedom’ or ‘free play’ and ‘clearance’ – or even ‘scope’. John Maynard Keynes translated it as ‘field’, but the term ‘range’ has generally been adopted in English. Von Kries now holds that if numerical probability were to make any sense at all it must be through this concept of the Spielraum . Von Kries’s theory is therefore called a ‘Spielraum theory’ or ‘range theory of probability’." — International Studies in the Philosophy of Science , Volume 15, Issue 2, 2001, pp. 177188 
See also the tag Points Omega.
(Scroll down to January 1112, 2015.)
Related material:
"Now, for example, in how far are
the six sides of a symmetric die
'equally possible' upon throwing?"
— From "The NaturalRange Conception
of Probability," by Dr. Jacob Rosenthal,
page 73 in Time, Chance, and
Reduction: Philosophical Aspects of
Statistical Mechanics , ed. by
Gerhard Ernst and Andreas Hüttemann,
Cambridge U. Press, 2010, pp. 7190
A professor at Harvard has written about
“the urge to seize and display something
real beyond artifice.”
He reportedly died on January 3, 2015.
An image from this journal on that date:
Another Gitterkrieg image:
The 24set Ω of R. T. Curtis
Click on the images for related material.
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.
"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 .
The Oslo Version and The Lyche Omega
Those who prefer more traditional art
may consult The Portal Project.
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.
See also Harvard's Memorial Church in "Ready when you are, C. B."—
HARVARD CRIMSON/ ALEX R. LEVIN
Sharon Stone lectures at
Harvard's Memorial Church
on March 14, 2005…
From Under the Volcano , Chapter II—
Hotel Bella Vista
Gran Baile Noviembre 1938
a Beneficio de la Cruz Roja.
Los Mejores Artistas del radio en accion.
No falte Vd.
From Shining Forth—
"What he sees is something real."
— Charles Williams, The Figure of Beatrice
See "Nine is a Vine" and "Hereafter" in this journal.
As quoted here last October 23—
Margaret Atwood on Lewis Hyde's Trickster Makes This World: Mischief, Myth, and Art—
"Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159)
What is "the next world"? It might be the Underworld….
The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the conartist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.
From Galleri MGM in Oslo —
A connection to today's earlier post, Sunday School— The Oslo Version, from Friday, May 21, 2010.
Lyche's "Omega Point" portal, together with her last name, suggested three posts from the following Saturday morning— which later proved to be the date of Martin Gardner's death—
Art Space, Through the Lyche Gate and The Lyche Gate Asterisk.
For some further religious remarks, see November 9th, 2010— A Theory of Pure Design.
"What exactly was Point Omega?"
This is Robert Wright in Nonzero: The Logic of Human Destiny.
Wright is discussing not the novel Point Omega by Don DeLillo,
but rather a (related) concept of the Jesuit philosopher Pierre Teilhard de Chardin.
My own idiosyncratic version of a personal "point omega"—
The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."
For some other sorts of points, see the drawings
on the wall and Geometry Simplified—
The two points of the trivial affine space are represented by squares,
and the one point of the trivial projective space is represented by
a line segment separating the affinespace squares.
For related darkness at noon, see Derrida on différance
as a version of Plato's khôra—
The above excerpts are from a work on and by Derrida
published in 1997 by Fordham University,
a Jesuit institution— Deconstruction in a Nutshell—
For an alternative to the Villanova view of Derrida,
see Angels in the Architecture.
Back to the Real
Colum McCann on yesterday’s history:
“Fiction gives us access to a very real history.”
The Associated Press thought for today:
“Journalism allows its readers to witness history; fiction gives its readers an opportunity to live it.”
— John Hersey, American author (born on this date in 1914, died 1993).
From John Hersey’s The Child Buyer (1960):
“I was wondering about that this morning… About forgetting. I’ve always had an idea that each memory was a kind of picture, an insubstantial picture. I’ve thought of it as suddenly coming into your mind when you need it, something you’ve seen, something you’ve heard, then it may stay awhile, or else it flies out, then maybe it comes back another time…. If all the pictures went out, if I forgot everything, where would they go? Just out into the air? Into the sky? Back home around my bed, where my dreams stay?”
“We keep coming back and coming back
To the real: to the hotel instead of the hymns….”
— Wallace Stevens
Postcard from eBay
From Under the Volcano, by Malcolm Lowry, 1947, Chapter I:

"Everything that has a beginning
has an end."
— The Matrix Revolutions
Matrix, by Knots, Inc., 1979.
"Easy to master — A lifetime to enjoy!"
The object for 2 players (8adult)
is to be the first to form a line
consisting of 4 different
colored chips.
Imagist Poem
(Recall the Gochip
in Wild Palms.)
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