Log24

Sunday, April 22, 2018

The Omega Matrix

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

(Continued)

Angels and Demons cross within a diamond (page 306), and Finite Geometry logo

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.

Friday, August 15, 2014

The Omega Matrix

Filed under: General,Geometry — 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.

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.)

Thursday, June 21, 2018

Cavell’s Matrix

Filed under: General — Tags: , — m759 @ 3:00 AM

From an obituary for Stanley Cavell, Harvard philosopher
who reportedly died at 91 on Tuesday,  June 19:

The London Review of Books  weblog yesterday —

"Michael Wood reviewed [Cavell’s] 
Philosophy the Day after Tomorrow  in 2005:

'The ordinary slips away from us. If we ignore it, we lose it.
If we look at it closely, it becomes extraordinary, the way
words or names become strange if we keep staring at them.
The very notion turns into a baffling riddle.' "

See also, in this  journal, Tuesday morning's Ici vient M. Jordan  and
this  morning's previous post.

Update of 3:24 AM from my RSS feed —

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

Friday, November 24, 2017

The Matrix Meets the Grid

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

The Matrix

  The Grid —

  Picturing the Witt Construction

     "Read something that means something." — New Yorker  ad

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.

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).

Saturday, February 23, 2019

Code Rain

Filed under: General — Tags: — m759 @ 11:30 PM

See 031105-code in this journal, the Wachowskis
at Wikipedia, and The Omega Matrix in this journal.

Thursday, June 21, 2018

Kummerhenge

Filed under: General,Geometry — Tags: , — m759 @ 2:19 AM

See also the Omega Matrix in this  journal.

Tuesday, May 29, 2018

The Schwartz Meme

Filed under: General — m759 @ 1:33 PM

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

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 —

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:

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 .

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.

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"

Tuesday, June 19, 2018

Ici vient M. Jordan

Filed under: General,Geometry — Tags: , — m759 @ 2:13 AM

NY Times correction, online June 16, about 'Here Comes Mr. Jordan' and 'Heaven Can Wait'

See also this  journal on Saturday morning, June 16.

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.

Monday, November 27, 2017

The Golay Code via Witt’s Construction

Filed under: General,Geometry — Tags: — m759 @ 2:08 PM

Reference

Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053. 
http://scholarworks.sjsu.edu/etd_theses/4053.

See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG in this journal.

Tuesday, March 21, 2017

Requiescat in Pace

Filed under: General — Tags: — m759 @ 7:45 PM

The Chicago Tribune  today —

H. Wilbert Norton, college president
and Christian missionary, dies

Norton reportedly died at 102 on Feb. 20, 2017.

This evening's previous post linked the death dates of two
academics to two Log24 posts that both contained the
following image —

For some backstory, see the Log24 posts from the date
of Norton's reported death, February 20.

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.

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.

Monday, October 10, 2011

The Aleph

Filed under: General — m759 @ 12:00 AM

COLLEGE OF THE DESERT
Minutes — Organization Meeting
11:00 a.m., Saturday, July 1, 1961—

15. Preparation of College Seal:

By unanimous consent preparation of a College
Seal to contain the following features was
authorized: A likeness of the Library building
set in a matrix of date palms, backed by
a mountain skyline and rising sun; before
the Library an open book, the Greek symbol
Alpha on one page and Omega on the other;
the Latin Lux et Veritas, College of the
Desert, and 1958 to be imprinted within or
around the periphery of the seal.

From the website http://geofhagopian.net/ of
Geoff Hagopian, Professor of Mathematics,
College of the Desert—

http://www.log24.com/log/pix11C/111010-CollegeOfTheDesert-Seal.gif

Note that this version of the seal contains
an Aleph  and Omega instead of Alpha and Omega.

From another Hagopian website, another seal.

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

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

(Recall the Go-chip
in Wild Palms.)

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