Log24

Saturday, May 8, 2021

A Tale of Two Omegas

Filed under: General — m759 @ 5:00 AM

The Greek capital letter Omega, 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.

See the Cullinane diamond theorem .

If the group is the large Mathieu group  of
244,823,040 permutations of 24 things,
the appropriate Ω  is the 4×6 grid below.

See the Miracle Octad Generator  of R. T. Curtis.

Sunday, May 6, 2018

The Osterman Omega

Filed under: General,Geometry — Tags: , — m759 @ 5:01 PM

From "The Osterman Weekend" (1983) —

Counting symmetries of the R. T. Curtis Omega:

An Illustration from Shakespeare's birthday

Counting symmetries with the orbit-stabilizer theorem

Sunday, March 5, 2017

The Omega Matrix

Filed under: General,Geometry — Tags: — m759 @ 5:00 PM

Richard Evan Schwartz on
the mathematics of the 4×4 square

See also Priority in this journal.

Monday, June 15, 2015

Omega Matrix

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

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 &Omegais the 4×4 grid above.

See the Cullinane diamond theorem.

Monday, January 12, 2015

Points Omega*

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

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.

Friday, August 15, 2014

The Omega Matrix

Filed under: General,Geometry — Tags: — m759 @ 11:00 PM

(Continued)

The webpage Rosenhain and Göpel Tetrads in PG(3,2)
has been updated to include more material on symplectic structure.

Thursday, August 7, 2014

The Omega Mystery

Filed under: General — Tags: — m759 @ 11:00 PM

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.”

— Richard Evan Schwartz

Some context — "Mathematics, Magic, and Mystery" —
See posts tagged April Awareness 2014.

Tuesday, August 5, 2014

The Omega Story

Filed under: General — Tags: — m759 @ 1:00 AM
 

"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."

Joan Didion

See also a post from May 4, 2011 (the date, according to a Google
search, of untitled notes regarding a matrix called Omega).

Sunday, August 3, 2014

The Omega Matrix

Filed under: General,Geometry — Tags: , , — m759 @ 10:31 PM

Shown below is the matrix Omega from notes of Richard Evan Schwartz.
See also earlier versions (1976-1979) by Steven H. Cullinane.

IMAGE- The matrix Omega from notes of Richard Evan Schwartz. See also earlier versions (1977-1979) by Steven H. Cullinane.

Backstory:  The Schwartz Notes (June 1, 2011), and Schwartz on
the American Mathematical Society's current home page:

(Click to enlarge.)

Sunday, September 18, 2011

Alpha and Omega

Filed under: General,Geometry — Tags: — m759 @ 2:22 AM

http://www.log24.com/log/pix11B/110918-AlphaAndOmega.jpg

A transcription—

"Now suppose that α  is an element of order 23 in M 24 ; we number the points of Ω
as the projective line , 0, 1, 2, … , 22 so that α : i i  + 1 (modulo 23) and fixes . In
fact there is a full L 2 (23) acting on this line and preserving the octads…."

— R. T. Curtis, "A New Combinatorial Approach to M 24 ,"
Mathematical Proceedings of the Cambridge Philosophical Society  (1976), 79: 25-42

Saturday, September 11, 2021

Reality for Academia

Filed under: General — Tags: — m759 @ 4:16 PM

The title of the previous post, "Ground Omega," suggests a related  nightmare . . .

A writer of fiction in the previous post

"When we say a thing is unreal, we mean it is too real…."

Old joke —

"What you mean 'we,' paleface?"

At Ground Omega  in the above My Hero Academia  site —

"The twenty-four students are split into six groups of four…."

I prefer the similar splittings of  the Curtis  Omega

Monday, August 16, 2021

The Space of Possibilities

Filed under: General — Tags: , — m759 @ 3:57 AM

The title is from "Federico Ardila on Math, Music and
the Space of Possibilities
," a podcast from Steven Strogatz's
Quanta Magazine  series. The transcript is dated March 29, 2021.

Ardila: … in a nutshell, what combinatorics is about is just
the study of possibilities and how do you organize them,
given that there’s too many of them to list them.

Strogatz:  So, I love it. Combinatorics is not just
the art of the possible, but the enumeration of the possible,
the counting of the possible and the organizing of the possible.

Strogatz:  It’s such a poetic image, actually: the space of possibilities.

This  journal on the podcast date, March 29, 2021 —

A more precise approach to the space of possibilities:

Wednesday, May 26, 2021

Heavy Metaphor

Filed under: General — Tags: , — m759 @ 9:51 PM

Earlier posts that are also tagged "Points Omega" suggest some 
context for a May 19 New Republic  illustration.

      x -1/x
 

  See as well
"Flowers and Brown."

Monday, October 26, 2020

Theory

Filed under: General — Tags: , — m759 @ 11:37 AM

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 —

Monday, June 1, 2020

The Gefter Boundary

Filed under: General — Tags: , — m759 @ 5:09 PM

“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.

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

See as well the previous post.

A Graveyard Smash: Galois Geometry Meets Nordic Aliens

Filed under: General — Tags: — m759 @ 4:02 PM

See also Vril Chick.

Wednesday, May 20, 2020

Cue the Violins

Filed under: General — Tags: , — m759 @ 11:36 PM

In memory of a music editor.

Blangsted reportedly died on May 1.
See also that date in this journal, among
other posts tagged The Next Level.

Mathematics as a Black Art

Filed under: General — Tags: , , — m759 @ 11:34 AM

Sunday, July 28, 2019

Review

Filed under: General — m759 @ 11:18 AM

From a news article featured on the American Mathematical Society
home page today

A joint Vietnam-USA mathematical meeting in Vietnam on
June 10-13, 2019:

This  journal on June 12, 2019:

Wednesday, June 12, 2019

The Osterman Haiku

Filed under: General — m759 @ 11:32 PM Edit This

Click on the book cover below for posts tagged "Haiku."

'Point Omega' by DeLillo

See also the Twentieth of May, 2008 —

Welcome to the Garden Club, Pilgrim.

Wednesday, June 12, 2019

The Osterman Haiku

Filed under: General — m759 @ 11:32 PM

Click on the book cover below for posts tagged "Haiku."

IMAGE- 'Point Omega' by DeLillo

Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

Filed under: General — Tags: , — m759 @ 12:00 AM

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 M24  with the aid of the MOG.

Wednesday, October 24, 2018

The Cracked Potter

Filed under: General — Tags: , — m759 @ 1:52 PM

See also an embedded ad in The Atlantic  magazine, Sept. 2017.

Tuesday, October 23, 2018

Simplicity Versus Complexity

Filed under: General — m759 @ 7:13 PM

Simplicity  (Click for some complexity.)

Complexity  (Click for some simplicity.)

A passage from the 2011 book Idea Man  that was suggested by
a recent New Yorker  article on the book's author, the late Paul Allen —

Left-click image to enlarge.

Sunday, October 21, 2018

Simplicity

Filed under: General — m759 @ 11:00 AM


 

Thursday, July 12, 2018

Kummerhenge Illustrated

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

      

“… 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, May 7, 2018

Fish Babel

Filed under: General,Geometry — Tags: , , — m759 @ 10:00 AM

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.

Friday, May 26, 2017

Night at the Museum

Filed under: General — Tags: — m759 @ 10:05 PM

In memory of an author who reportedly 
died on Wednesday, May 24.

Tuesday, March 7, 2017

Signature Backdrop

Filed under: General — Tags: — m759 @ 9:00 PM

"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

Commentary

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

— Richard Evan Schwartz

See also Schwartz in "The Omega Matrix," a post of 5 PM ET Sunday:

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 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)

IMAGE- Desargues's theorem in light of Galois geometry

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Anticommuting Dirac matrices as spreads of projective lines

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

Monday, June 15, 2015

Slow Art

Filed under: General — Tags: , , — m759 @ 2:03 PM

Slowness is sometimes in the eye of the beholder.

See this journal on Slow Art Day 2015.

Related material: Epistemic States in this journal.

Wednesday, May 13, 2015

Motto

Filed under: General — Tags: — m759 @ 9:48 PM

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

SymOmega in this journal and a suggested motto
for The University of Western Australia.

Space

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

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 

http://www.log24.com/log/pix11A/110426-ApolloAndDionysus.jpg

Tuesday, May 12, 2015

Writing Well*

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM

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.

Zinsser Obituary

Filed under: General — Tags: — m759 @ 10:48 PM

"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

Monday, February 2, 2015

Spielraum as Ω

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM
 

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. 177-188

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 
Mechanics , ed. by 
     Gerhard Ernst and Andreas Hüttemann, 
     Cambridge U. Press, 2010, pp. 71-90

Sunday, January 11, 2015

Real Beyond Artifice

Filed under: General,Geometry — Tags: , , , — m759 @ 7:20 PM

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.

Tuesday, August 26, 2014

Lux et Veritas

Filed under: General,Geometry — m759 @ 7:59 AM

Omega by Lux:

Omega by Curtis:

Wednesday, August 13, 2014

Stranger than Dreams*

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM

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.

Monday, August 4, 2014

A Wrinkle in Space

Filed under: General,Geometry — Tags: , — m759 @ 10:30 AM

"There is  such a thing as a tesseract." — Madeleine L'Engle

An approach via the Omega Matrix:

http://www.log24.com/log/pix10A/100619-TesseractAnd4x4.gif

See, too, Rosenhain and Göpel as The Shadow Guests .

Friday, April 26, 2013

High White

Filed under: General — Tags: , — m759 @ 12:00 PM

(Continued)

For Times Square Church
Click image for a video.

Review

Filed under: General — Tags: — m759 @ 11:00 AM

The Oslo Version and The Lyche Omega

Those who prefer more traditional art 
may consult The Portal Project.

Wednesday, June 1, 2011

The Schwartz Notes

Filed under: General,Geometry — Tags: , , — m759 @ 2:00 PM

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—

http://www.log24.com/log/pix11A/110601-Search.jpg

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.

Sunday, July 4, 2010

Brightness at Noon (continued)

Filed under: General,Geometry — m759 @ 12:00 PM

Today's sermon mentioned the phrase "Omega number."

Other sorts of Omega numbers— 24 and 759— occur
in connection with the set named Ω by R. T. Curtis in 1976—

Image-- In a 1976 paper, R.T. Curtis names the 24-set of his Miracle Octad Generator 'Omega.'

— R. T. Curtis, "A New Combinatorial Approach to M24,"
Math. Proc. Camb. Phil. Soc. (1976), 79, 25-42

Sunday, May 23, 2010

Annals of Conceptual Art

Filed under: General — Tags: , — m759 @ 2:02 AM

Josefine Lyche's
  "Theme and Variations" (Oslo, 2009)

http://www.log24.com/log/pix10A/100523-LycheTandV.jpg

Some images in reply—

  Frame Tale

Image by R. T. Curtis from 'Further Elementary Techniques...'

Click on images for further details.

"In the name of the former
and of the latter
and of their holocaust.
  Allmen."

Finnegans Wake

Wednesday, November 5, 2003

Wednesday November 5, 2003

Filed under: General — Tags: , — m759 @ 2:23 PM

Game Over

 "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

Digital 'tears in the rain'

Image suggesting the 'Go chip' in 'Wild Palms'

(Recall the Go-chip
in Wild Palms.)

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