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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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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Version from “The Avengers” (2012) —

Version from Josefine Lyche (2009) —

See also this journal on the date that the above *Avengers* video was uploaded.

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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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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—

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**And I'd like to thank the heroine of Finale —**

* The title refers to a 2009 sculpture by Lyche —

For a related shape, see today's noon post.

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The "compact key to universal wisdom" passage in the previous post seemed

too well written to be the work of an anonymous webforum author.

Here is a slightly expanded version—

Throughout history mystics and philosophers have sought

a compact key to universal wisdom, a finite formula or text

that would provide the answer to every question. The use of

the Bible, the Koran and the I Ching for divination and the

tradition of the secret books of Hermes Trismegistus and the

medieval Jewish Cabala exemplify this belief or hope. Such

sources of universal wisdom are traditionally protected from

casual use by being difficult to find as well as difficult to un-

derstand and dangerous to use, tending to answer more quest-

ions and deeper ones than the searcher wishes to ask. The

esoteric book is, like God, simple yet undescribable. It is om-

niscient, and it transforms all who know it. The use of clas-

sical texts to foretell mundane events is considered supersti-

tious nowadays, yet in another sense science is in quest of its

own Cabala, a concise set of natural laws that would explain

all phenomena. In mathematics, where no set of axioms can

hope to prove all true statements, the goal might be a concise

axiomatization of all "interesting" true statements.

Ω is in many senses a Cabalistic number. It can be known

of through human reason, but not known. To know it in detail

one must accept its uncomputable sequence of digits on faith,

like words of a sacred text.

This is Martin Gardner's* and Charles H. Bennett's

revised version of a passage from Bennett's paper

"On Random and Hard-to-Describe Numbers," 1979.

The original passage from Bennett's paper—

Throughout history mystics and philosophers have sought a compact key to

universal wisdom, a finite formula or text which, when known and understood,

would provide the answer to every question. The Bible, the Koran, the mythical

secret books of Hermes Trismegistus, and the medieval Jewish Cabala have

been so regarded. Sources of universal wisdom are traditionally protected from

casual use by being hard to find, hard to understand when found, and dangerous

to use, tending to answer more and deeper questions than the user wishes to

ask. Like God the esoteric book is simple yet undescribable, omniscient, and

transforms all who know It. The use of classical texts to fortell [sic] mundane events

is considered superstitious nowadays, yet, in another sense, science is in quest of

its own Cabala, a concise set of natural laws which would explain all phenomena.

In mathematics, where no set of axioms can hope to prove all true statements,

the goal might be a concise axiomatization of all "interesting" true statements.

Ω is in many senses a Cabalistic number. It can be known of, but not known,

through human reason. To know it in detail, one would have to accept its un-

computable digit sequence on faith, like words of a sacred text.

The Bennett paper deals with Gregory Chaitin's concept of an "Omega Number."

I prefer the Omega of Josefine Lyche—

See also All Hallows' Eve, 2002.

* Martin Gardner's *Mathematical Games* column

"The Random Number Omega Bids Fair to Hold the Mysteries of the Universe,"

*Scientific American*, November 1979, 241(5), pp. 20–34.

The column is reprinted as "Chaitin's Omega," Ch. 21, pp. 307-319 in the

collection of Gardner's columns titled *Fractal Music, Hypercards and More*,

W.H. Freeman & Co., 1991

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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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“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.

The title was suggested by the "Crystal Cult" installations

of Oslo artist Josefine Lyche and by a post of May 30 —

Filed under: General — m759 @ 8:02 PM Edit This
Jeff Nichols, director of
"When asked about the film's similarities to See also Jung's four-diamond figure and the previous post. |

Writers of fiction are, of course, also dabblers in the collective unconscious.

For instance . . .

A 1971 British paperback edition of ** The Dreaming Jewels**,

a story by Theodore Sturgeon (first version published in 1950):

The above book cover, together with the Death Valley location

Zabriskie Point, suggests . . .

Those less enchanted by the collective unconscious may prefer a

different weblog's remarks on the same date as the above Borax post . . .

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**Foursquare, Inscape, Subway **

**Foursquare —**

**Inscape —**

**Subway —**

Art installation, "Crystal Cult" by Josefine Lyche, at an Oslo subway station —

See also today's previous post.

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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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See also "Romancing the Omega" —

Related mathematics — Guitart in this journal —

See also Weyl + Palermo in this journal —

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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

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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**In Like Flynn**

From the *Wall Street Journal* site Friday evening—

ESSAY September 21, 2012, 9:10 p.m. ET Americans' IQ scores have risen steadily over the past century. |

No, thank you. I prefer the ninth configuration as is—

Why? See Josefine Lyche's art installation "Grids, you say?"

Her reference there to "High White Noon" is perhaps

related to the use of that phrase in this journal.

The phrase is from a 2010 novel by Don DeLillo.

See "Point Omega," as well as Lyche's "Omega Point,"

in this journal.

The *Wall Street Journal* author above, James R. Flynn (born in 1934),

"is famous for his discovery of the Flynn effect, the continued

year-after-year increase of IQ scores in all parts of the world."

—Wikipedia

His son Eugene Victor Flynn is a mathematician, co-author

of the following chapter on the Kummer surface—

For use of the Kummer surface in Buddhist metaphysics, see last night's

post "Occupy Space (continued)" and the letters of Nanavira Thera from the

late 1950s at nanavira.blogspot.com.

These letters, together with Lyche's use of the phrase "high white noon,"

suggest a further quotation—

*You know that it would be untrue
You know that I would be a liar
If I was to say to you
Girl, we couldn't get much higher*

See also the Kummer surface at the web page Configurations and Squares.

Comments Off on Plan 9 (continued)–

*we are just like a couple of tots…*

*— Sinatra*

Born 1973 in Bergen. Lives and works in Oslo.

Education

2000 – 2004 National Academy of Fine Arts, Oslo

1998 – 2000 Strykejernet Art School, Oslo, NO

1995 – 1998 Philosophy, University of Bergen

**University of Bergen—**

It might therefore seem that the idea of digital and analogical systems as rival fundaments to human experience is a new suggestion and, like digital technology, very modern. In fact, however, the idea is as old as philosophy itself (and may be much older). In his
What we shall see is something like a battle of gods and giants going on between them over their quarrel about reality [γιγαντομαχία περì της ουσίας] ….One party is trying to drag everything down to earth out of heaven and the unseen, literally grasping rocks and trees in their hands, for they lay hold upon every stock and stone and strenuously affirm that real existence belongs only to that which can be handled and offers resistance to the touch. They define reality as the same thing as body, and as soon as one of the opposite party asserts that anything without a body is real, they are utterly contemptuous and will not listen to another word. (…) Their adversaries are very wary in defending their position somewhere in the heights of the unseen, maintaining with all their force that true reality [την αληθινήν ουσίαν] consists in certain intelligible and bodiless forms. In the clash of argument they shatter and pulverize those bodies which their opponents wield, and what those others allege to be true reality they call, not real being, but a sort of moving process of becoming. On this issue an interminable battle is always going on between the two camps [εν μέσω δε περι ταυτα απλετος αμφοτέρων μάχη τις (…) αει συνέστηκεν]. (…) It seems that only one course is open to the philosopher who values knowledge and truth above all else. He must refuse to accept from the champions of the forms the doctrine that all reality is changeless [and exclusively immaterial], and he must turn a deaf ear to the other party who represent reality as everywhere changing [and as only material].
The gods and the giants in Plato’s battle present two varieties of the analog position. Each believes that ‘true reality’ is singular, that "real existence belongs only to" one side or other of competing possibilities. For them, difference and complexity are secondary and, as secondary, deficient in respect to truth, reality and being (την αληθινήν ουσίαν, το όν τε και το παν). Difference and complexity are therefore matters of "interminable battle" whose intended end for each is, and
— Cameron McEwen, "The Digital Wittgenstein," |

* See that phrase in this journal.

Comments Off on Child’s Play (continued*)

See Notes for a Haiku.

Related material—

A novel published on Groundhog Day, 2010—

— as well as Conceptual Art, Josefine Lyche's

"Grids, You Say?" and The Speed of Thought.

Comments Off on Grids

For Fischer's fellow database enthusiast Codd, see Cross-Referenced (April 24, 2003).

For Birtel's fellow pseudoscience enthusiast Frank Tipler, see the artist's comment linked to in Romancing the Omega (April 19, 2011)—

"Omega Point" is a term used by mathematical physicist Frank Tipler for what he maintains is the ultimate fate of the universe required by the laws of physics."

Comments Off on Endings

**For Norway's Niels Henrik Abel (1802-1829)
on his birthday, August Fifth
**

(6 PM Aug. 4, Eastern Time, is 12 AM Aug. 5 in Oslo.)

**Plato's Diamond
**

The above version by Peter Pesic is from Chapter I of his book *Abel's Proof* , titled "The Scandal of the Irrational." Plato's diamond also occurs in a much later mathematical story that might be called "The Scandal of the Noncontinuous." The story—

## Paradigms"These passages suggest that the Form is a character or set of characters common to a number of things, i.e. the feature in reality which corresponds to a general word. But Plato also uses language which suggests not only that the forms exist separately ( … Both in the – William and Martha Kneale, Plato's paradigm in the Changed paradigm in the diamond theorem (2×2 case) — Aspects of the paradigm change— Monochrome figures to Areas to Continuous transformations to Euclidean geometry to Euclidean quantities to The 24 patterns resulting from the paradigm change— Each pattern has some ordinary or color-interchange symmetry. This is the 2×2 case of a more general result. The patterns become more interesting in the 4×4 case. For their relationship to finite geometry and finite fields, see the diamond theorem. |

Related material: Plato's Diamond by Oslo artist Josefine Lyche.

“*Plato’s Ghost* evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”

— Princeton University Press on *Plato’s Ghost: The Modernist Transformation of Mathematics* (by Jeremy Gray, September 2008)

"Remember me to her."

— Closing words of the Algis Budrys novel *Rogue Moon* .

Background— Some posts in this journal related to Abel or to random thoughts from his birthday.

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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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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.

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"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 affine-space 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.

Comments Off on Brightness at Noon, continued

"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.)*

Comments Off on Wednesday November 5, 2003

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