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* —

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* —

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From "The Osterman Weekend" (1983) —

Counting symmetries of the R. T. Curtis Omega:

An Illustration from Shakespeare's birthday —

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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 four-dimensional

affine space over the two-element Galois

field, the appropriate **Ω **is the 4×4 grid above.

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The previous post displayed a set of

24 unit-square “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.

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The webpage Rosenhain and Göpel Tetrads in PG(3,2)

has been updated to include more material on symplectic structure.

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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.

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"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).

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Shown below is the matrix Omega from notes of Richard Evan Schwartz.

See also earlier versions (1976-1979) by Steven H. Cullinane.

Backstory: The Schwartz Notes (June 1, 2011), and Schwartz on

the American Mathematical Society's current home page:

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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 signed-out 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 12-14 of the document, which is untitled, undated, and

unsigned, discuss the finite-geometry 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.

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**See as well a search for Gray Space in this journal.**

**Related material: The Schwartz Omega .**

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

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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 Zeta-related theory —

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“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.

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Comments Off on A Graveyard Smash: *Galois Geometry Meets Nordic Aliens*

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 finite-geometry properties of the 4×4 square as

a finite affine 4-space — properties that are of use in studying the Mathieu

group M_{24 }with the aid of the MOG.

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“… 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"

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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.

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**Continued from last night**

From the American Mathematical Society, a news item

dated Thursday, March 9, 2017 —

Remarks by Schwartz quoted here on March 7—

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"The Bitter End’s signature stage backdrop —

a bare 150-year-old 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:

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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 many-faceted 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 4-space over

the two-element 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**

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Slowness is sometimes in the eye of the beholder.

See this journal on Slow Art Day 2015.

**Related material: Epistemic States in this journal.**

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See the previous post, "Space," as well as…

SymOmega in this journal and a suggested motto

for The University of Western Australia.

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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 …

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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.

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"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*

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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
— |

See also the tag Points Omega.

(Scroll down to January 11-12, 2015.)

Related material:

"Now, for example, in how far are

the six sides of a symmetric die

'equally possible' upon throwing?"

— From "The Natural-Range Conception

of Probability," by Dr. Jacob Rosenthal,

page 73 in *Time, Chance, and
Reduction: Philosophical Aspects of
Statistical *

Gerhard Ernst and Andreas Hüttemann,

Cambridge U. Press, 2010, pp. 71-90

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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 24-set * Ω *of R. T. Curtis*

Click on the images for related material.

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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.

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"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 *.

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**The Oslo Version and The Lyche Omega**

Those who prefer more traditional art

may consult The Portal Project.

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"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 (8-adult)

is to be the first to form a line

consisting of 4 different

colored chips.

**Imagist Poem**

(Recall the Go-chip

in *Wild Palms.)*

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