From "The Osterman Weekend" (1983) —

Counting symmetries of the R. T. Curtis Omega:

An Illustration from Shakespeare's birthday —

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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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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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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"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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From a post of July 24, 2011 —

A review —

“The story, involving the Knights Templar, the Vatican, sunken treasure,

the fate of Christianity and a decoding device that looks as if it came out of

a really big box of medieval Cracker Jack, is the latest attempt to combine

Indiana Jones derring-do with ‘Da Vinci Code’ mysticism.”

A feeble attempt at a purely mathematical "decoding device"

from this journal earlier this month —

For some background, see a question by John Baez at Math Overflow

on Aug. 20, 2015.

The nonexistence of a 24-cycle in the large Mathieu group

might discourage anyone hoping for deep new insights from

the above figure.

See Marston Conder's "Symmetric Genus of the Mathieu Groups" —

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The latest *Visual Insight* post at the American Mathematical

Society website discusses group actions on the McGee graph,

pictured as 24 points arranged in a circle that are connected

by 36 symmetrically arranged edges.

Wikipedia remarks that …

"The automorphism group of the McGee graph

is of order 32 and doesn't act transitively upon

its vertices: there are two vertex orbits of lengths

8 and 16."

The partition into 8 and 16 points suggests, for those familiar

with the Miracle Octad Generator and the Mathieu group M_{24},

the following exercise:

Arrange the 24 points of the projective line

over GF(23) in a circle in the natural cyclic order

( ∞, 1, 2, 3, … , 22, 0 ). Can the McGee graph be

modeled by constructing edges in any natural way?

In other words, if the above set of edges has no

"natural" connection with the 24 points of the

projective line over GF(23), does some *other*

set of edges in an isomorphic McGee graph

have such a connection?

Update of 9:20 PM ET Sept. 20, 2015:

Backstory: A related question by John Baez

at Math Overflow on August 20.

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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

Comments Off on Spielraum as Ω

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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Anyone tackling the *Raumproblem* described here

on Feb. 21, 2014 should know the history of coordinatizations

of the 4×6 Miracle Octad Generator (MOG) array by R. T. Curtis

and J. H. Conway. Some documentation:

The above two images seem to contradict a statement by R. T. Curtis

in a 1989 paper. Curtis seemed in that paper to be saying, falsely, that

his original 1973 and 1976 MOG coordinates were those in array M below—

This seemingly false statement involved John H. Conway's supposedly

definitive and natural canonical coordinatization of the 4×6 MOG

array by the symbols for the 24 points of the projective line over GF(23)—

{**∞, 0, 1, 2, 3… , 21, 22}:**

An explanation of the apparent falsity in Curtis's 1989 paper:

By "two versions of the MOG" Curtis seems to have meant merely that the

*octads* , and not the projective-line *coordinates *, in his earlier papers were

mirror images of the *octads *that resulted later from the Conway coordinates,

as in the images below.

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Despite the blocking of Doodles on my Google Search

screen, some messages get through.

Today, for instance —

"Your idea just might change the world.

Enter Google Science Fair 2014"

Clicking the link yields a page with the following image—

Clearly there is a problem here analogous to

the square-triangle coordinatization problem,

but with the 4×6 rectangle of the R. T. Curtis

Miracle Octad Generator playing the role of

the square.

I once studied this 24-triangle-hexagon

coordinatization problem, but was unable to

obtain any results of interest. Perhaps

someone else will have better luck.

* For a rather different use of this word,

see Hermann Weyl in the Stanford

Encyclopedia of Philosophy.

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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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