Log24

Monday, March 31, 2025

Endings

Filed under: General — Tags: — m759 @ 3:14 pm

LA Times obituary yesterday

"Glover performed in many plays, TV shows and movies,
including portraying Duffy in the 1974 drama "Chinatown,”
the villain Mr. Wint in the 1971 James Bond movie
“Diamonds Are Forever” and Feldman in the 2001 drama
Ghost World.”

Better Duffy and Walsh than Wint and Kidd.

Wednesday, March 26, 2025

For Patricia AnnetteThe Krazy Kat* Game

Filed under: General — Tags: , , — m759 @ 10:18 am

https://www.facebook.com/patricia.annette.31521301

* . . .

Thursday, February 29, 2024

Aaron

Filed under: General — Tags: , , ,
— m759 @ 10:01 am 

See earlier Log24 posts now tagged Aaron, about
the clash of Aaron Swartz and JSTOR, as well as
JSTOR in today's previous post, and an enthusiastic
post by @marrific yesterday on a different Aaron.

See also "A Tale of Two Intersections," about Venice,
CA, and St. Augustine's Church, New Orleans, LA. 
For a connection to the phrase "Where Madness Lies,"
used recently as a podcast title by Aaron Webman,
see the life of St. Augustine's parishioner George Herriman,
creator of "Krazy Kat."

Sunday, March 23, 2025

Annals of Brick Space . . .
Not-so-free Masonry

Filed under: General — Tags: — m759 @ 4:23 pm

A Google search today for "brick space" —

Related art apparently suggested by the phrase "building blocks"

A Mystery Popup

Filed under: General — Tags: — m759 @ 2:35 pm

The recent URL cubebrick.space forwards to . . .

http://m759.net/wordpress/?tag=brick-space.

The web posts so tagged are, as one would expect,
NOT in the Harvard Library system.  I was therefore
somewhat surprised to see the following popup today —

Clicking on the "Get article" link yields . . .

This metadata is actually quite helpful, as the cited article
does, in fact, give good references for what I have called,
using a term from the "Miracle Octad Generator" of R. T. Curtis,
"brick space" — the finite projective space PG(5,2).

Wednesday, March 5, 2025

“Old men ought to be explorers” — T. S. Eliot

Filed under: General — Tags: — m759 @ 2:22 am

See cubebrick.space.

Tuesday, March 4, 2025

Brick-Space News

Filed under: General — Tags: , , , , — m759 @ 7:38 pm

Earlier in this  journal, a more abstract approach . . .

Thursday, February 20, 2025

Coloring the Klein Correspondence

Filed under: General — Tags: , , , — m759 @ 11:43 pm

A Google search for "four color decomposition" yields an AI Overview

My "four-color decomposition" theorem supplies some background
for last New Year's Eve's post on the Klein Correspondence.

 

Tuesday, February 18, 2025

Grok 3 on the Log24 post of last New Year’s Eve

Filed under: General — Tags: — m759 @ 1:01 am

The Miracle Octad Generator (MOG) of R. T. Curtis —

The Miracle Octad Generator (MOG) of R. T. Curtis

Grok 3 beta, 12:56 AM ET Feb. 18, 2025 —

The diagram you've provided from the webpage titled "The Yellow Brick Road to the Miracle Octad Generator; with Conwell's Heptads" illustrates the concept of "brick space" within the context of projective geometry, specifically PG(5,2), which is the 5-dimensional projective space over the field with 2 elements (GF(2)). Here's a detailed breakdown based on the image:

Diagram Explanation:

    Components of the Diagram:
        Left Side:
            There are 28 points off the Klein quadric represented by grids with some cells shaded in blue.
        Middle Section:
            The 35 points of the Klein quadric are shown, which are added to the 28 points off the Klein quadric. This section shows grids with additional yellow shading.
        Right Side:
            The result of adding these points together to form the 63 points of PG(5,2), represented by grids with blue and yellow shading combined. This section also includes the 8 Conwell heptads (each heptad represented as a column).
    Mathematical Context:
        PG(5,2): As mentioned, this is the projective 5-space over GF(2), which has
        2^6 – 1 = 63
         points (since points in projective space are equivalence classes of non-zero vectors, excluding the zero vector).
        Klein Quadric: A specific geometric structure within PG(5,2) that can be visualized in this context as a subset of points with particular properties. Here, it's shown as contributing 35 points.
        Conwell Heptads: These are subsets of 7 points (heptads) in this geometry, named after Conwell, and are visually represented in columns on the right side.
    Visual Representation:
        The "brick space" metaphor likely comes from the way these points and subspaces are arranged in a grid-like or block-like fashion, similar to bricks in a wall. Each grid represents a configuration of points or subspaces within this projective space, and the combination of different colored shadings (blue, yellow) helps to distinguish different sets or properties of points.
    Purpose of the Diagram:
        This diagram is part of an exploration of the Miracle Octad Generator (MOG) by R. T. Curtis, where the Klein quadric serves as a background. The MOG is a structure used in the study of combinatorial designs and has applications in finite group theory, particularly with the Mathieu groups. The diagram visually aids in understanding how these points and heptads relate to each other within this geometric framework.
    Additional Notes:
        The note at the bottom mentions that 28 + 35 = 63, which matches the total number of points in PG(5,2).
        The mention of heptads being added on December 31, 2024, indicates updates or additions to the original concept or visualization.


This diagram, therefore, serves as a visual aid to understand the structure of PG(5,2) in terms of its points, subspaces, and specific combinatorial designs like the Conwell heptads, all within the context of the Miracle Octad Generator. It's an advanced mathematical visualization, combining geometric and combinatorial aspects in a way that's accessible to those familiar with these concepts.

For some remarks by Grok 3 beta made before it saw the
"yellow brick" diagram, see a PDF of prompts and responses.

Friday, February 14, 2025

Lyrics Game for Gretel —
“Lester, Lana … Lana, Lester.”

Filed under: General — Tags: , — m759 @ 12:50 am

Related reading:

Monday, February 10, 2025

Brick Space: Points with Parts

Filed under: General — Tags: , , , , — m759 @ 3:47 pm

This post's "Points with Parts" title may serve as an introduction to
what has been called "the most powerful diagram in mathematics" —
the "Miracle Octad Generator" (MOG) of Robert T. Curtis.

The Miracle Octad Generator (MOG) of R. T. Curtis

Curtis himself has apparently not written on the geometric background
of his diagram — the finite projective spaces PG(5,2) and PG(3,2), of 
five and of three dimensions over the two-element Galois field GF(2).

The component parts of the MOG diagram, the 2×4 Curtis "bricks,"
may be regarded* as forming both PG(5,2) and PG(3,2) . . .
Pace  Euclid, points with parts. For more on the MOG's geometric
background, see the Klein correspondence  in the previous post.

For a simpler example of "points with parts, see
http://m759.net/wordpress/?s=200229.

* Use the notions of Galois (XOR, or "symmetric-difference") addition
of even  subsets, and such addition "modulo complementation," to
decrease the number of dimensions of the spaces involved.

Sunday, February 9, 2025

An AI Overview of Klein Quadric and MOG

Filed under: General — Tags: , , — m759 @ 2:15 pm

See also a more accurate AI report from January 9, 2025 —

Grok 2 on the MOG and the Klein Correspondence:

HTML version, with corrections, of the above 9 January Grok 2 report —

Grok 2: Klein Correspondence and MOG, 9 Jan. 2025 . . .
______________________________________________

The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics.

Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between points and lines in a projective space.* Specifically, in the context of the MOG, it relates to the geometry of PG(3,2), the three-dimensional projective space over the field with two elements. The Klein correspondence, in this context, essentially connects the combinatorial aspects of partitions of an 8-set to the geometric structure of lines in PG(3,2).

Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system.

Relation Between Klein Correspondence and MOG:

  • Combinatorial and Geometric Link: The MOG uses the Klein correspondence to map partitions of an 8-set (into two 4-sets) to [the sets of] 4 parallel affine planes [that represent lines] within PG(3,2). This mapping is natural and symmetrical, which is crucial for the MOG's functionality. The 35 partitions of an 8-set correspond to the 35 lines in PG(3,2), and this correspondence helps in understanding how to generate and visualize octads within the MOG framework.
     
  • Structural Insight: The Klein correspondence provides a way to see how the MOG's method of generating octads can be understood geometrically. This insight is vital for exploring the symmetry and subgroup structure of M24, as the MOG's layout preserves these symmetries.

Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems.

* Correction: Should be "a correspondence between points in a five-dimensional projective space and lines in a three-dimensional projective space."

Update  of ca. 9 AM ET Monday, Feb. 10, 2024 —

Neither AI report above mentions the Cullinane model of the five-
dimensional projective space PG(5,2) as a brick space — a space
whose points are the 2×4 bricks  used in thte MOG. This is
understandable, as the notion of using bricks to model both  PG(5,2)
and PG(3,2) has appeared so far only in this journal. See an
illustration from New Year's Eve . . . Dec. 31, 2024 —

The Miracle Octad Generator (MOG) of R. T. Curtis

Friday, February 7, 2025

Apple Entertainment

Filed under: General — Tags: , — m759 @ 12:49 pm

The Miracle Octad Generator (MOG) of R. T. Curtis

Tuesday, February 4, 2025

Diagrams

Filed under: General — Tags: , , — m759 @ 11:30 am

"The most powerful diagram in mathematics" —

Anonymous YouTube lecturer

The YouTube lecturer is not referring to the Fano plane diagram cited
in the AI Overview below, but to a much more sophisticated figure,
the Miracle Octad Generator (MOG) of R. T. Curtis.

The Miracle Octad Generator of R. T. Curtis

Some context —

A rearrangement of the Miracle Octad Generator —

The Miracle Octad Generator (MOG) of R. T. Curtis

The diagram below may be less powerful , but it illustrates a result that,
although less miraculous , is perhaps more historically significant —

Pythagorean theorem proof by overlapping similar figures

Thursday, January 9, 2025

Grok 2 on the MOG and the Klein Correspondence

Filed under: General — Tags: , , — m759 @ 1:04 pm

Related illustration —

The Miracle Octad Generator (MOG) of R. T. Curtis

— Transcription —

Grok 2 on Klein correspondence and MOG — 9 Jan. 2025
______________________________________________________________

Prompt:

How is the Klein correspondence related to the Miracle Octad Generator?

Grok 2 response (with citations and links removed):

The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics.

Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between points and lines in a projective space. Specifically, in the context of the MOG, it relates to the geometry of PG(3,2), the three-dimensional projective space over the field with two elements. The Klein correspondence, in this context, essentially connects the combinatorial aspects of partitions of an 8-set to the geometric structure of lines in PG(3,2).

Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system.

Relation Between Klein Correspondence and MOG:

    Combinatorial and Geometric Link: The MOG uses the Klein correspondence to map partitions of an 8-set (into two 4-sets) to 4 parallel affine planes within PG(3,2). This mapping is natural and symmetrical, which is crucial for the MOG's functionality. The 35 partitions of an 8-set correspond to the 35 lines in PG(3,2), and this correspondence helps in understanding how to generate and visualize octads within the MOG framework.

    Structural Insight: The Klein correspondence provides a way to see how the MOG's method of generating octads can be understood geometrically. This insight is vital for exploring the symmetry and subgroup structure of M24, as the MOG's layout preserves these symmetries.

Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems.

Tuesday, January 7, 2025

The Yarrow Stalker

Filed under: General — Tags: , , — m759 @ 7:28 pm

John Huston and chessboard

And then there is Chinatown brick  space

Tuesday, December 31, 2024

The Yellow Brick Road to the
Miracle Octad Generator, with Conwell’s Heptads

Filed under: General — Tags: , , — m759 @ 2:42 am

The Klein quadric as background for the Miracle Octad Generator of R. T. Curtis —

The Klein quadric, PG(5,2), and the 'bricks' of the Miracle Octad Generator

See also Saniga on heptads in this journal.

The Miracle Octad Generator (MOG) of R. T. Curtis

Thursday, December 26, 2024

“Xanadu” . . . The Melody Lingers On

Filed under: General — Tags: , — m759 @ 12:20 am

Some earlier work of the above star, Radha Mitchell —

Wednesday, December 25, 2024

Coordinatizing Brick Space

Filed under: General — Tags: — m759 @ 12:07 am

For the assignment of zero-one coordinates (over GF(2)), the earlier
layout of the space posted here yesterday is less convenient than
the layout begun below (a work in progress with different basis vectors) —

Monday, December 23, 2024

A Projective-Space Home for the Miracle Octad Generator

Filed under: General — Tags: , — m759 @ 3:17 pm

The natural geometric setting for the "bricks" in the Miracle Octad Generator
(MOG) of Robert T. Curtis is PG(5,2), the projective 5-space over GF(2).

The Klein correspondence mirrors the 35 lines of PG(3,2) — and hence, via the 
graphic approach below, the 35 "heavy bricks" of the MOG that match those
lines — in PG(5,2), where the bricks may be studied with geometric methods,
as an alternative to Curtis's original MOG combinatorial construction methods.

The construction below of a PG(5,2) brick space  is analogous to the
"line diagrams"  construction of a PG(3,2) in Cullinane's diamond theorem.

Saturday, December 21, 2024

Coordinatizing Brick Space

Filed under: General — Tags: , — m759 @ 4:10 am

The Miracle Octad Generator (MOG) of R. T. Curtis

Exercise:  The eight-part diagrams in the graphic "brick space"
model of PG(5,2) below need to be suitably labeled with six-part
GF(2) coordinates to help illustrate the Klein correspondence that
underlies the large Mathieu group M24.

A possible approach:  The lines  separating dark squares from light
(i.e., blue from white or yellow) in the figure above may be added
in XOR fashion (as if they were diamond theorem  line diagrams)
to form a six  dimensional vector space, which, after a suitable basis
is chosen, may be represented by six-tuples of 0's and 1's.

Related reading —

log24.com/log24/241221-'Brick Space « Log24' – m759.net.pdf .

This is a large (15.1 MB) file.  The Foxit PDF reader is recommended.

The PDF is from a search for Brick Space  in this journal.

Some context:  http://m759.net/wordpress/?s=Weyl+Coordinatization.

Thursday, December 19, 2024

Different Angles

Filed under: General — Tags: , — m759 @ 7:26 pm

"Drawing the same face from different angles sounds fun,
but let me tell you – it’s not. It’s not fun at all. It’s HARD!!"

Loisvb on Instagram, Dec. 18, 2024

Likewise for PG(5,2).

Exercise:  The eight-part diagrams in the graphic "brick space"
model of PG(5,2) below need to be suitably labeled with six-part
GF(2) coordinates to help illustrate the Klein correspondence that
underlies the large Mathieu group M24.

Friday, October 25, 2024

The Space Structures Underlying M24

Filed under: General — Tags: , , — m759 @ 12:24 am

The structures of the title are the even subsets of a six-set and of
an eight-set, viewed modulo set complementation.

The "Brick Space" model of PG(5,2) —

Brick space: The 2x4 model of PG(5,2)

For the M24 relationship between these spaces, of 15 and of 63 points,
see G. M. Conwell's 1910 paper "The 3-Space PG (3,2) and Its Group,"
as well as Conwell heptads in this  journal.

The Miracle Octad Generator (MOG) of R. T. Curtis

Thursday, October 24, 2024

October Story: If It’s Day 24, This Must Be . . . BRICS?

Filed under: General — Tags: , , — m759 @ 12:12 pm

My own interests tend towards . . . not BRICS, but bricks —

The Miracle Octad Generator (MOG) of R. T. Curtis

I look forward to the November publication of . . .

Thursday, September 19, 2024

The Zen of Brick Space:
Embedding the Null Brick

Filed under: General — Tags: , — m759 @ 2:21 am


The "Brick Space" model of PG(5,2) —

Brick space: The 2x4 model of PG(5,2)

Related reading . . .

See also "Zero System."

Tuesday, September 17, 2024

Design for Pilot Fish

Filed under: General — Tags: , , — m759 @ 9:46 am

Geometry for Jews:
“Kickin’ Down the Cobblestones…” — Song lyric

Filed under: General — Tags: — m759 @ 2:58 am

Kirkenes at newyorker.com

Some less stressful material . . .

"a medley cobbled together" —

Brick Space Exercise

Filed under: General — Tags: , — m759 @ 1:23 am

Exercise: Assign coordinates over GF(2) to the graphic

Saturday, September 14, 2024

Notes on a Friday the 13th Death

Filed under: General — Tags: , , , — m759 @ 8:40 pm

A passage accessed via the new URL Starbrick.art*

Thursday, February 25, 2021

Compare and Contrast

Filed under: General — Tags: , ,
,
— m759 @ 12:31 pm

“… What is your dream—your ideal? 
What is your News from Nowhere, or, rather,
What is the result of the little shake your hand has given
to the old pasteboard toy with a dozen bits of colored glass
for contents? And, most important of all, can you present it
in a narrative or romance which will enable me to pass an
idle hour not disagreeably? How, for instance, does it compare
in this respect with other prophetic books on the shelf?”

— Hudson, W. H.. A Crystal Age , 1887.
Open Road Media, Kindle Edition, page 2.

A related cultural note suggested by the New York Times  obituary today
of fashion designer Mary McFadden, who reportedly died yesterday
(a Friday the Thirteenth) and is described by the Times  as a late-life
partner of "eightfold-way" physicist Murray Gell-Mann —

* A reference to the 2-column 4-row matrix (a "brick") that underlies
the patterns in the Miracle Octad Generator  of R. T. Curtis. The only
connection of this eight-part matrix to Gell-Mann's "Eightfold Way"
that I know of is simply the number 8 itself.

Friday, September 13, 2024

Starbrick Art

Filed under: General — Tags: , — m759 @ 4:47 am

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

— Richard Evan Schwartz

The monolith at the beginning of '2001'

Twelve basis vectors, in lexicographic order, for the binary Golay-code space

Tuesday, August 27, 2024

For Rubik Worshippers

Filed under: General — Tags: , , , — m759 @ 2:37 pm

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

The above is six-dimensional as an affine  space, but only five-dimensional
as a  projective  space . . . the space PG(5, 2).

As the domain of the smallest model of the Klein correspondence and the
Klein quadric, PG (5,2) is not without mathematical importance.

See Chess Bricks and Ovid.group.

This post was suggested by the date July 6, 2024 in a Warren, PA obituary
and by that date in this  journal.

Sunday, August 11, 2024

Double Duals

Filed under: General — Tags: — m759 @ 8:40 am

"That in which space itself is contained"
Wallace Stevens

In the ninefold square,
projective-perspectivity duality
corresponds to
projective-correlation duality.

Illustrations —

Saturday, August 10, 2024

The Yellow Brick Road Through PG(5,2)

Filed under: General — Tags: , , — m759 @ 8:20 pm

The Miracle Octad Generator (MOG) of R. T. Curtis

Thursday, August 1, 2024

For Coppola’s Megalopolis:
Brick Space in the Big Apple

Filed under: General — Tags: — m759 @ 12:00 am

See also Big Apple and Brick Space.

Thursday, July 25, 2024

Brick Space

Filed under: General — Tags: — m759 @ 6:09 am

http://m759.net/wordpress/?s=2×4

Related reading:

http://m759.net/wordpress/?tag=knight-move .

Thursday, July 18, 2024

Brick Space

Filed under: General — Tags: , , — m759 @ 1:45 am
 

Compare and Contrast

 

A rearranged illustration from . . .

R. T. Curtis, "A New Combinatorial Approach to M24 ,"
Mathematical Proceedings of the Cambridge Philosophical Society ,
Volume 79 , Issue 1 , January 1976 , pp. 25 – 42
DOI: https://doi.org/10.1017/S0305004100052075

The image “MOGCurtis03.gif” cannot be displayed, because it contains errors.


The "Brick Space" model of PG(5,2) —

Brick space: The 2x4 model of PG(5,2)

Background: See "Conwell heptads" on the Web.

See as well Nocciolo  in this journal and . . .

Friday, July 5, 2024

De Bruyn on the Klein Quadric

Filed under: General — Tags: , , , — m759 @ 10:31 pm

— De Bruyn, Bart. “Quadratic Sets on the Klein Quadric.”
JOURNAL OF COMBINATORIAL THEORY SERIES A,
vol. 190, 2022, doi:10.1016/j.jcta.2022.105635.

Related material —

Log24 on Wednesday, July 3, 2024: "The Nutshell Miracle" . . .

In particular, within that post, my own 2019 "nutshell" diagram of PG(5,2):

PG(5,2)

An Omega for Chinatown

Filed under: General — Tags: , — m759 @ 12:47 am

John Huston and chessboard

Wednesday, July 3, 2024

The Nutshell Miracle

Filed under: General — Tags: , — m759 @ 10:42 pm

'Then a miracle occurs' cartoon

Cartoon by S. Harris

From a search in this journal for nocciolo

From a search in this journal for PG(5,2)

From a search in this journal for Curtis MOG

IMAGE- The Miracle Octad Generator (MOG) of R.T. Curtis

Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M24,"
Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)

From a search in this journal for Klein Correspondence

Philippe Cara on the Klein correspondence

The picture of PG(5,2) above as an expanded nocciolo
shows that the Miracle Octad Generator illustrates
the Klein correspondence.

Update of 10:33 PM ET Friday, July 5, 2024 —

See the July 5 post "De Bruyn on the Klein Quadric."

Tuesday, May 14, 2024

Octad Space

Filed under: General — Tags: , , , — m759 @ 4:45 am

From "Geometry of 6 and 8," Dec. 8, 2019 —

Brick mosaic.

Compare and contrast —

Monday, April 22, 2024

Annals of American Beauty:
Sowing Confusion and Reaping …

Filed under: General — Tags: — m759 @ 11:11 am

Also reaping Kevin Spacey?

Monday, January 8, 2024

The Star Brick

Filed under: General — Tags: , , , — m759 @ 7:44 pm

From a post of January 3, 2024

Black monolith in death-and-rebirth sequence from '2001: A Space Odyssey'

"Hello  darkness,  my  old  friend.
I’ve  come  to  talk  with  you  again."

The above image was flipped to reverse left and right.
Related reading: Other posts tagged Darkness and

Related material: Other posts tagged Star Brick and . . .

"And we may see the meadow in December,
icy white and crystalline"

— Song lyric, "Midnight Sun"

Friday, September 22, 2023

Figurate Space

Filed under: General — Tags: , , — m759 @ 11:01 am

For the purpose of defining figurate geometry , a figurate space  might be
loosely described as any space consisting of finitely many congruent figures  —
subsets of Euclidean space such as points, line segments, squares, 
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them. 

Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate  space, considered as a collection of figures —  vertices, edges,
faces —
seen in the nineteenth century as acted upon by a group  of symmetries .

More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator 
of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M24 . That group may be viewed as acting
on various subsets of a 24-set for instance, the 759 octads  that are analogous
to the faces  of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.

Counting symmetries with the orbit-stabilizer theorem

Sunday, April 16, 2023

For Ashley Falls, Mass.

Filed under: General — Tags: , — m759 @ 10:30 pm

Honoring the Spaces, Minding the Gaps . . .

From this  journal on the above YouTube upload date, Sept. 9, 2022 —

Poetry enthusiasts might view the brick at left as
symbolizing the scepter'd isle  off the west coast
of Europe, and the gap between as the English 
Channel. Mind the gap.

Wednesday, November 30, 2022

Three Representations

Filed under: General — Tags: , , — m759 @ 12:05 pm

'Cube Bricks, 1984,' by Steven H. Cullinane

Cube Bricks, 1984

See also Impenetrability .

Sunday, September 11, 2022

Raiders of the Lost Space

Filed under: General — Tags: , — m759 @ 2:29 pm

From 1981 —

From today —

Update —

A Magma check of the motif-generated space shows that
its dimension is only 8, not 12 as with the MOG space.
Four more basis vectors can be added to the 24 motifs to
bring the generated space up to 12 dimensions: the left
brick, the middle brick, the top half (2×6), the left half (4×3).
I have not yet checked the minimum weight in the resulting
12-dimensional 4×6 bit-space.

— SHC 4 PM ET, Sept. 12, 2022.

Friday, March 18, 2022

Architectural Review

Filed under: General — Tags: , — m759 @ 12:30 pm
 

"Poincaré said that science is no more a collection of facts than a house is a collection of bricks. The facts have to be ordered or structured, they have to fit a theory, a construct (often mathematical) in the human mind.

… Mathematics may be art, but to the general public it is a black art, more akin to magic and mystery. This presents a constant challenge to the mathematical community: to explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that architecture is the best of the arts to compare with mathematics. The analogy between the two subjects is not hard to describe and enables abstract ideas to be exemplified by bricks and mortar, in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

Gottschalk Review —

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,
reviewed by Paul R. Halmos in Bulletin of the American Mathematical Society  61(6): 584-588 (November 1955).

The ending of the review —

The most striking virtue of the book is its organization. The authors' effort to arrange the exposition in an efficient order, and to group the results together around a few central topics, was completely successful; they deserve to be congratulated on a spectacular piece of workmanship. The results are stated at the level of greatest available generality, and the proofs are short and neat; there is no unnecessary verbiage. The authors have, also, a real flair for the "right" generalization; their definitions of periodicity and almost periodicity, for instance, are very elegant and even shed some light on the classical concepts of the same name. The same is true of their definition of a syndetic set, which specializes, in case the group is the real line, to Bohr's concept of a relatively dense set.

The chief fault of the book is its style. The presentation is in the brutal Landau manner, definition, theorem, proof, and remark following each other in relentless succession. The omission of unnecessary verbiage is carried to the extent that no motivation is given for the concepts and the theorems, and there is a paucity of illuminating examples. The striving for generality (which, for instance, has caused the authors to treat uniform spaces instead of metric spaces whenever possible) does not make for easy reading. The same is true of the striving for brevity; the shortest proof of a theorem is not always the most perspicuous one. There are too many definitions, especially in the first third of the book; the reader must at all times keep at his finger tips a disconcerting array of technical terminology. The learning of this terminology is made harder by the authors' frequent use of multiple statements, such as: "The term {asymptotic } {doubly asymptotic } means negatively {or} {and} positively asymptotic."

Conclusion: the book is a mine of information, but you sure have to dig for it.  — PAUL R. HALMOS

Friday, February 11, 2022

For Space Groupies

Filed under: General — Tags: , , — m759 @ 5:31 pm

A followup to Wednesday's post Deep Space

Related material from this journal on July 9, 2019

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

From "Tomorrowland" (2015) —

From other posts tagged 1984 Cubes

Sunday, October 11, 2020

Saniga on Einstein

Filed under: General — Tags: , — m759 @ 8:25 am

See Einstein on Acid” by Stephen Battersby
(New Scientist , Vol. 180, issue 2426 — 20 Dec. 2003, 40-43).

That 2003 article is about some speculations of Metod Saniga.

“Saniga is not a professional mystic or
a peddler of drugs, he is an astrophysicist
at the Slovak Academy of Sciences in Bratislava.
It seems unlikely that studying stars led him to
such a way-out view of space and time. Has he
undergone a drug-induced epiphany, or a period
of mental instability? ‘No, no, no,’ Saniga says,
‘I am a perfectly sane person.'”

Some more recent and much less speculative remarks by Saniga
are related to the Klein correspondence —

arXiv.org > math > arXiv:1409.5691:
Mathematics > Combinatorics
[Submitted on 17 Sep 2014]
The Complement of Binary Klein Quadric
as a Combinatorial Grassmannian

By Metod Saniga

“Given a hyperbolic quadric of PG(5,2), there are 28 points
off this quadric and 56 lines skew to it. It is shown that the
(286,563)-configuration formed by these points and lines
is isomorphic to the combinatorial Grassmannian of type
G2(8). It is also pointed out that a set of seven points of
G2(8) whose labels share a mark corresponds to a
Conwell heptad of PG(5,2). Gradual removal of Conwell
heptads from the (286,563)-configuration yields a nested
sequence of binomial configurations identical with part of
that found to be associated with Cayley-Dickson algebras
(arXiv:1405.6888).”

Related entertainment —

See Log24 on the date, 17 Sept. 2014, of Saniga’s Klein-quadric article:

Articulation Day.

Friday, August 7, 2020

Enormous Changes at 11:59

Filed under: General — Tags: , — m759 @ 11:59 pm

Carver reportedly died on Tuesday, August 4, 2020.

“This is Maggie the Cat” —

“This is not.” 

Thursday, April 23, 2020

Octads and Geometry

See the web pages octad.group and octad.us.

Related geometry (not the 759 octads, but closely related to them) —

The 4×6 rectangle of R. T. Curtis
illustrates the geometry of octads —

Counting symmetries with the orbit-stabilizer theorem

Curtis splits the 4×6 rectangle into three 4×2 "bricks" —

.

"In fact the construction enables us to describe the octads
in a very revealing manner. It shows that each octad,
other than Λ1, Λ2, Λ3, intersects at least one of these ' bricks' —
the 'heavy brick' – in just four points." . . . .

— R. T. Curtis (1976). "new combinatorial approach to M24,"
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42.

Sunday, December 29, 2019

Articulation Raid

Filed under: General — Tags: , , — m759 @ 7:45 am

“… And so each venture Is a new beginning,
a raid on the inarticulate….”

— T. S. Eliot, “East Coker V” in Four Quartets

arXiv:1409.5691v1 [math.CO]  17 Sep 2014

The Complement of Binary Klein Quadric as
a Combinatorial Grassmannian

Metod Saniga,
Institute for Discrete Mathematics and Geometry,
Vienna University of Technology,
Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria
(metod.saniga@tuwien.ac.at) and
Astronomical Institute, Slovak Academy of Sciences,
SK-05960 Tatransk ́a Lomnica, Slovak Republic
(msaniga@astro.sk)

Abstract

Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286,563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286,563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).

Keywords:

Combinatorial Grassmannian −
Binary Klein Quadric − Conwell Heptad

See also this  journal on the above date — 17 September 2014.

Sunday, December 8, 2019

Geometry of 6 and 8

Filed under: General — Tags: , , , , — m759 @ 4:03 am

Just as
the finite space PG(3,2) is
the geometry of the 6-set, so is
the finite space PG(5,2)
the geometry of the 8-set.*

Selah.

* Consider, for the 6-set, the 32
(16, modulo complementation)
0-, 2-, 4-, and 6-subsets,
and, for the 8-set, the 128
(64, modulo complementation)
0-, 2-, 4-, 6-, and 8-subsets.

Update of 11:02 AM ET the same day:

See also Eightfold Geometry, a note from 2010.

Tuesday, July 9, 2019

Schoolgirl Space: 1984 Revisited

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

From "Tomorrowland" (2015) —

From John Baez (2018) —

See also this morning's post Perception of Space 
and yesterday's Exploring Schoolgirl Space.

Friday, December 14, 2018

Small Space Odyssey

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 11:00 am

References in recent posts to physical space and 
to mathematical space suggest a comparison.

Physical space is well known, at least in the world
of mass entertainment.

Mathematical space, such as the 12-dimensional
finite space of the Golay code, is less well known.

A figure from each space —

The source of the Conway-Sloane brick —

Quote from a mathematics writer —

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

— Richard Evan Schwartz

The former practice yields reflections like those of Conway and Sloane.
The latter practice is not recommended.

Monday, November 12, 2018

Half Crazy

Filed under: General — Tags: , — m759 @ 10:42 pm

Douglas Rain, the voice of HAL in Kubrick's 2001 , reportedly
died at 90 on Sunday, Nov. 11, 2018. A piece from the Sunday,
April 1, 2018, print  edition of The New York Times  recalls that . . .

When HAL says, “I know I’ve made some very poor decisions recently, but I can give you my complete assurance that my work will be back to normal,” Mr. Rain somehow manages to sound both sincere and not reassuring. And his delivery of the line “I think you know what the problem is just as well as I do” has the sarcastic drip of a drawing-room melodrama and also carries the disinterested vibe of a polite sociopath.

Kubrick had Mr. Rain sing the 1892 love song “Daisy Bell” (“I’m half crazy, all for the love of you”) almost 50 times, in uneven tempos, in monotone, at different pitches and even just by humming it. In the end, he used the very first take. Sung as HAL’s brain is being disconnected, it’s from his early programming days, his computer childhood. It brings to an end the most affecting scene in the entire film.

— Gerry Flahive in the online New York Times 
 

"A version of this article appears in print on , on Page AR13 of the New York edition with the headline: HAL 9000 Wasn’t Always So Eerily Calm."

This  journal on the above online date, March 30, 2018 —

"Program or be programmed."
— A saying by Douglas Rushkoff

See as well the following link from this journal
on Armistice Day, the reported date of Mr. Rain's death —

"Halmos + Tombstone."

Monday, June 4, 2018

The Trinity Stone Defined

“Unsheathe your dagger definitions.” — James Joyce, Ulysses

The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

An Approach to Symmetric Generation of the Simple Group of Order 168

Some small Galois spaces (the Cullinane models)

Wednesday, July 5, 2017

Imaginarium of a Different Kind

Filed under: General,Geometry — Tags: , — m759 @ 9:00 pm

The title refers to that of the previous post, "The Imaginarium."

In memory of a translator who reportedly died on May  22, 2017,
a passage quoted here on that date —

Related material — A paragraph added on March 15, 2017,
to the Wikipedia article on Galois geometry

George Conwell gave an early demonstration of Galois geometry in 1910 when he characterized a solution of Kirkman's schoolgirl problem as a partition of sets of skew lines in PG(3,2), the three-dimensional projective geometry over the Galois field GF(2).[3] Similar to methods of line geometry in space over a field of characteristic 0, Conwell used Plücker coordinates in PG(5,2) and identified the points representing lines in PG(3,2) as those on the Klein quadric.

— User Rgdboer

Tuesday, March 28, 2017

Bit by Bit

Filed under: General,Geometry — Tags: , , , — m759 @ 11:45 am

From Log24, "Cube Bricks 1984" —

An Approach to Symmetric Generation of the Simple Group of Order 168

Also on March 9, 2017 —

For those who prefer graphic  art —

Broken Symmetries  in  Diamond Space  

Tuesday, April 19, 2016

The Folding

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

(Continued

A recent post about the eightfold cube  suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :

As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array from Turyn's
1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields the 2x2x2 eightfold cube .

Those who prefer an entertainment  approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space." 

Monday, April 4, 2016

Cube for Berlin

Foreword by Sir Michael Atiyah —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

 Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010

Judy Bass, Los Angeles Times , March 12, 1989 —

"Like Rubik's Cube, The Eight  demands to be pondered."

As does a figure from 1984, Cullinane's Cube —

The Eightfold Cube

For natural group actions on the Cullinane cube,
see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."

See also the recent post Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Related remark from the literature —

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

Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see "Cube Space, 1984-2003."

(From Anatomy of a Cube, Sept. 18, 2011.)

Sunday, March 13, 2016

Space Sermon

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

In memory of the late architect Patrick Hodgkinson

Harvey Court at Gonville & Caius College, Cambridge

For the architect, see yesterday's post "Brick-Perfect."

See as well a meditation on the numbers 9 and 13
in the post "Space" on day 13 of May, 2015.

Saturday, March 12, 2016

Masonic Melody

Filed under: General — Tags: , , , — m759 @ 1:31 am

"Button your lip baby
Button your coat
Let's go out dancing
Go for the throat"

Read more: Rolling Stones – Mixed Emotions Lyrics | MetroLyrics 

This melody was suggested by a post of February 25, 2016,
by tonight's previous post "Brick-Perfect," and by
the post "Cube Bricks 1984" of March 4, 2016.

"Only connect." — E. M. Forster.

“Brick-Perfect”

Filed under: General — Tags: , , — m759 @ 1:00 am

Patrick Hodgkinson, a British architect, reportedly died at 85 on 
February 21, 2016. From his March 4 obituary in the Telegraph

Before Brunswick, came Harvey Court for Gonville and Caius College, Cambridge. Colin St John Wilson, exLCC, his senior in the Martin studio, had done a scheme with four freestanding ranges in concrete. Hodgkinson radically transformed this at short notice into the final version presented to the College, a tight, connected square finished in local brick with a stepped section and impressive close-spaced brick columns on the exterior faces where the section overhung.

Never afflicted by modesty, Hodgkinson called it “designed to a brick-perfect, three-dimensional grid clear of ugly moments: the builders enjoyed making it”. It was attributed to Martin, Wilson and Hodgkinson jointly, but Hodgkinson felt that his contribution was under-appreciated, and again with the Law Library at Oxford, normally credited to Martin and Wilson. The theory of compact medium-rise courtyard forms derived from the Harvey Court design became central to Martin’s research programme at Cambridge in the 1960s; Hodgkinson felt that he deserved more credit for this too.

Thursday, October 1, 2015

Another Bad Song for Dave Barry

Filed under: General — Tags: — m759 @ 2:06 pm

"All work and no play…."

— Stanley Kubrick's film (1980) of The Shining  (1977)

"Each metaphor already modeled the modeler
that pasted it together. It seemed I might have
another fiction in me after all."

— Richard Powers, Galatea 2.2  (1995)

"In the space between what happens
And what gets left behind…."

— "Diamond Space" (2006), song by
      Michael Friedman and Sam Masich

Combining, as in a headline from today's Harvard Crimson ,
"programs and public space," we have

Groundhog Day 2014.

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: , , , , , , — m759 @ 12:24 pm

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, of the Miracle Octad Generator

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by "Dr. Parker" — Apparently Richard A. Parker —
Lecture 4, "Discovering M24," in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on "Sporadic and Related Groups."
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis's 35  4×6  1976 MOG arrays would use
Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35  4×6 arrays of the 1976
Curtis MOG would then reveal*  the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35  MOG arrays.  For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not  by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.

* "Then a miracle occurs," as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Saturday, March 8, 2014

Conwell Heptads in Eastern Europe

Filed under: General,Geometry — Tags: , , — m759 @ 11:07 am

“Charting the Real Four-Qubit Pauli Group
via Ovoids of a Hyperbolic Quadric of PG(7,2),”
by Metod Saniga, Péter Lévay and Petr Pracna,
arXiv:1202.2973v2 [math-ph] 26 Jun 2012 —

P. 4— “It was found that +(5,2) (the Klein quadric)
has, up to isomorphism, a unique  one — also known,
after its discoverer, as a Conwell heptad  [18].
The set of 28 points lying off +(5,2) comprises
eight such heptads, any two having exactly one
point in common.”

P. 11— “This split reminds us of a similar split of
63 points of PG(5,2) into 35/28 points lying on/off
a Klein quadric +(5,2).”

[18] G. M. Conwell, Ann. Math. 11 (1910) 60–76

A similar split occurs in yesterday’s Kummer Varieties post.
See the 63 = 28 + 35 vectors of R8 discussed there.

For more about Conwell heptads, see The Klein Correspondence,
Penrose Space-Time, and a Finite Model
.

For my own remarks on the date of the above arXiv paper
by Saniga et. al., click on the image below —

Walter Gropius

Monday, July 25, 2011

The Game

Filed under: General,Geometry — Tags: — m759 @ 9:00 am

Virginia Heffernan in Sunday's online New York Times

"… In the past, information on paper was something to read. Bricks and mortar were a place to be. But, since the first appearance of the Web in 1990, we have come to accept that information in pixels is something to read— and also a place to be . That familiar and yet still jaw-dropping metaphor takes energy to maintain. The odd shared sense that there’s three-dimensionality and immersion and real-world consequences on the Web as in no book or board game— that’s the Web’s sine qua non. Hence, cyberspace . And 'being on' the Internet….

… The dominant social networks are fantasy games built around rigged avatars, outright fictions and a silent— and often unconscious— agreement among players that the game and its somewhat creaky conceits influence the real world…."

— "The Confidence Game at Google+"

"It's just another manic Monday
I wish it was Sunday
'Cause that's my funday"
— The Bangles

"Accentuate the Positive"
— Clint Eastwood, soundtrack album
 for "Midnight in the Garden of Good and Evil"

http://www.log24.com/log/pix11/110410-Sugimoto-AndoChurch.jpg

This journal on All Saints' Day, Sunday, November 1, 2009

Suggested by the New York State lottery numbers on All Hallows’ Eve [2009]—

430 (mid-day) and 168 (evening)…

From 430 as a date, 4/30

Beyond Grief and Nothing: A Reading of Don DeLillo , by Joseph Dewey, University of South Carolina Press, 2006, page 123:

“It is as if DeLillo himself had moved to an endgame….”

For such an endgame, see yesterday’s link to a Mira Sorvino drama.

The number 168 suggested by the Halloween lottery deals with the properties of space itself and requires a more detailed exegesis…

For the full picture, consider the Log24 entries of Feb. 16-28 this year, esp. the entries of Feb. 27 and the phrase they suggest—

Flores, flores para los muertos.

      See also Pearly Gates of Cyberspace in this journal.

      For flores para los muertos , see today's Times .

Thursday, May 26, 2011

Life’s Persistent Questions

Filed under: General — Tags: , , — m759 @ 4:01 pm

This afternoon's online New York Times  reviews "The Tree of Life," a film that opens tomorrow.

With disarming sincerity and daunting formal sophistication “The Tree of Life” ponders some of the hardest and most persistent questions, the kind that leave adults speechless when children ask them. In this case a boy, in whispered voice-over, speaks directly to God, whose responses are characteristically oblique, conveyed by the rustling of wind in trees or the play of shadows on a bedroom wall. Where are you? the boy wants to know, and lurking within this question is another: What am I doing here?

Persistent answers… Perhaps conveyed by wind, perhaps by shadows, perhaps by the New York Lottery.

For the nihilist alternative— the universe arose by chance out of nothing and all is meaningless— see Stephen Hawking and Jennifer Ouellette.

Update of 10:30 PM EDT May 26—

Today's NY Lottery results: Midday 407, Evening 756. The first is perhaps about the date April 7, the second about the phrase "three bricks shy"— in the context of the number 759 and the Miracle Octad Generator. (See also Robert Langdon and The Poetics of Space.)

Sunday, April 10, 2011

Bedeviled

Filed under: General — Tags: , — m759 @ 10:30 pm

From tonight's online New York Times

John McCracken, Sculptor of Geometric Forms, Dies at 76

McCracken died in Manhattan on Friday, April 8.

From Christopher Knight in tonight's online LA Times

… the works embody perceptual and philosophical conundrums. The colored planks stand on the floor like sculptures….

McCracken was bedeviled by Stanley Kubrick's famously obscure science-fiction epic, "2001: A Space Odyssey," with its iconic image of an ancient monolith floating in outer space. The 1968 blockbuster was released two years after the artist made his first plank.

"At the time, some people thought I had designed the monolith or that it had been derived from my work," he told art critic Frances Colpitt of the coincidence in a 1998 interview.

Two photos of McCracken's 1967 Black Plank  seem relevant—

November 28, 2010 (Click to enlarge)

http://www.log24.com/log/pix11/110410-McCrackenPlank1967400w.jpg

December 28, 2010 (Click to enlarge)

http://www.log24.com/log/pix11/110410-McCracken-NatGallery-NothingToSeeHere-400w.jpg

Material that an artist might view as related, if only synchronistically—

Two posts in this journal on the dates the photos were taken—
The Embedding on November 28 and Dry Bones on December 28.

The photos are of an exhibition titled "There is nothing to see here" at the
National Gallery of Art, October 30, 2010-April 24, 2011 —

Click to enlarge.

http://www.log24.com/log/pix11/110410-NothingToSee-400w.jpg

For related nihilism from the National Gallery, see "Pictures of Nothing" in this journal.

Some less nihilistic illustrations—

The Meno  Embedding

Plato's Diamond embedded in The Matrix

A photo by one of the artists whose work is displayed above beside McCracken's—

http://www.log24.com/log/pix11/110410-Sugimoto-AndoChurch.jpg

"Accentuate the Positive."
 — Clint Eastwood

Wednesday, June 16, 2010

Geometry of Language

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

(Continued from April 23, 2009, and February 13, 2010.)

Paul Valéry as quoted in yesterday’s post:

“The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (Cahiers, 15:170 [2: 315])

The geometric example discussed here yesterday as a Self symbol may seem too small to be really impressive. Here is a larger example from the Chinese, rather than European, tradition. It may be regarded as a way of representing the Galois field GF(64). (“Field” is a rather ambiguous term; here it does not, of course, mean what it did in the Valéry quotation.)

From Geometry of the I Ching

Image-- The 64 hexagrams of the I Ching

The above 64 hexagrams may also be regarded as
the finite affine space AG(6,2)— a larger version
of the finite affine space AG(4,2) in yesterday’s post.
That smaller space has a group of 322,560 symmetries.
The larger hexagram  space has a group of
1,290,157,424,640 affine symmetries.

From a paper on GL(6,2), the symmetry group
of the corresponding projective  space PG(5,2),*
which has 1/64 as many symmetries—

(Click to enlarge.)

Image-- Classes of the Group GL(6,)

For some narrative in the European  tradition
related to this geometry, see Solomon’s Cube.

* Update of July 29, 2011: The “PG(5,2)” above is a correction from an earlier error.

Friday, April 30, 2010

Bridge to Nowhere

Filed under: General — Tags: — m759 @ 12:00 pm

(continued from April 26, 28, and 29):

http://www.log24.com/images/IChing/hexagram29.gif

Hexagram 29:
Water

http://www.log24.com/log/pix10A/100430-Commentary.jpg

http://www.log24.com/images/IChing/hexagram30.gif

Hexagram 30:
Fire

"Hates California,
it's cold and it's damp.
"

Image--'The Fire,' by Katherine Neville

Excerpt from The Fire,
by Katherine Neville —

"'Alaska's Aleutian Trench,' Key told us…. 'It's called the Ring of Fire because it boasts the largest collection of active volcanoes in the world.'….

'But you said that my father's not in Alaska…. So what does this Ring of Fire have to do with the place where we're actually going?'

'It's the Yellow Brick Road,' she told me."

Sarah Palin and friends-- Doonesbury, April 30, 2010

Friday, March 5, 2010

Space Case

Filed under: General — Tags: , , — m759 @ 9:48 am

Large ad, front page top, for Tom Brokaw's 'Boomers' in NY Times of March 4th, 2010

"And there we were all in one place,
A generation lost in space…"
— Don McLean, "American Pie

    Cybill Shepherd (born 1950) and Jeff Bridges (born 1949) in 'The Last Picture Show'

Today's NY Times says Robert T. McCall, space artist, died at 90 on Feb. 26.

"His most famous image may be the gargantuan mural, showing events from the creation of the universe to men walking on the Moon, on the south lobby wall of the National Air and Space Museum on the National Mall in Washington. More than 10 million people a year pass it.

Or it might be his painting showing a space vehicle darting from the bay of a wheel-shaped space station, which was used in a poster for Stanley Kubrick’s landmark 1968 film, '2001: A Space Odyssey.'"

Space station image by Robert T. McCall for '2001'

Cover art by McCall, with autograph dated
8/19/05, from a personal web page

Hal in "2010"– "Will I dream?"

Log24 on the day that McCall died

"Which Dreamed It?"
– Title of final chapter,Through the Looking Glass

"Go ask Alice… I think she'll know."
– Grace Slick, 1967  

Related material: James Joyce in this journal–

"Space: what you damn well have to see."

Monday, May 26, 2008

Monday May 26, 2008

Filed under: General — Tags: — m759 @ 10:00 pm
Sydney Pollack dies-- NY Times online front page

From Bloomberg.com:

Great Directors

“After his return to acting in ‘Tootsie,’ Pollack took movie roles under directors Robert Altman in ‘The Player’ (1992), Woody Allen in ‘Husbands and Wives’ (1992) and Stanley Kubrick in ‘Eyes Wide Shut’ (1999). He said he chose roles in part to study other great directors.”
 

Monday, April 28, 2008

Monday April 28, 2008

Filed under: General,Geometry — Tags: , , , — m759 @ 7:00 am
Religious Art

The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.

Black monolith, proportions 4x9

One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.

The following
figure does
allow such
  an epiphany.

A 2x4 array of squares

One approach to
 the epiphany:

"Transformations play
  a major role in
  modern mathematics."
– A biography of
Felix Christian Klein

The above 2×4 array
(2 columns, 4 rows)
 furnishes an example of
a transformation acting
on the parts of
an organized whole:

The 35 partitions of an 8-set into two 4-sets

For other transformations
acting on the eight parts,
hence on the 35 partitions, see
"Geometry of the 4×4 Square,"
as well as Peter J. Cameron's
"The Klein Quadric
and Triality" (pdf),
and (for added context)
"The Klein Correspondence,
Penrose Space-Time, and
a Finite Model
."

For a related structure–
  not rectangle but cube– 
see Epiphany 2008.

Tuesday, February 27, 2007

Tuesday February 27, 2007

Filed under: General,Geometry — Tags: — m759 @ 7:59 am
Continued from 2/06:

The Poetics of Space

Log24 yesterday:

“Imprimatur.
+John Cardinal Farley,
Archbishop of New York”

Tom Hanks as Robert Langdon in The Da Vinci Code

Tom Hanks as Robert Langdon
in “The Da Vinci Code”

“… and by ‘+’ I mean
artistic vision.”

New York State Lottery
yesterday, Feb. 26, 2007:

Mid-day 206
Evening 888


For more on the artistic
significance of 206,
see 2/06.

For more on the artistic
significance of 888, see
St. Bonaventure on the
Trinity at math16.com.

A trinity:

Click on picture for further details.

Tuesday, February 6, 2007

Tuesday February 6, 2007

Filed under: General,Geometry — Tags: , — m759 @ 8:00 am
The Poetics of Space

The title is from Bachelard.
I prefer Stevens:

The rock is the habitation of the whole,
Its strength and measure, that which is near, point A
In a perspective that begins again

At B:  the origin of the mango's rind.
It is the rock where tranquil must adduce
Its tranquil self, the main of things, the mind,

The starting point of the human and the end,
That in which space itself is contained, the gate
To the enclosure, day, the things illumined

By day, night and that which night illumines,
Night and its midnight-minting fragrances,
Night's hymn of the rock, as in a vivid sleep.

— Wallace Stevens,
   "The Rock," 1954

Joan Ockman in Harvard Design Magazine (Fall 1998):

"'We are far removed from any reference to simple geometrical forms,' Bachelard wrote…."

No, we are not. See Log24, Christmas 2005: 

Compare and contrast:

The image “http://www.log24.com/theory/images/EightfoldCubeCover.jpg” cannot be displayed, because it contains errors.

 

The image “http://www.log24.com/theory/images/EightfoldWayCover.jpg” cannot be displayed, because it contains errors.

 

(Click on pictures for details.)

More on Bachelard from Harvard Design Magazine:

"The project of discerning a loi des quatre éléments would preoccupy him until his death…."

For such a loi, see Theme and Variations and…

The image “http://www.log24.com/log/pix07/070206-Elements.gif” cannot be displayed, because it contains errors.

(Click on design for details.)

Thought for Today:
"If you can talk brilliantly
about a problem, it can create
the consoling illusion that
it has been mastered."
— Stanley Kubrick, American
movie director (1928-1999).

(AP, "Today in History,"
February 6, 2007)

Monday, July 26, 2004

Monday July 26, 2004

Filed under: General — Tags: , , , , — m759 @ 11:07 pm

Happy Birthday

to Kate Beckinsale
(star of Cold Comfort Farm)

and Kevin Spacey
(star of The Usual Suspects).

From a novel,
The Footprints of God,
published August 12, 2003

A tour guide describes
stations of the cross in Jerusalem:

"Ibrahim pointed down the cobbled street to a half circle of bricks set in the street.  'There is where Jesus began to carry the cross.  Down the street is the Chapel of Flagellation, where the Roman soldiers whipped Jesus, set on him a crown of thorns, and said, "Hail, King of the Jews!" Then Pilate led him to the crowd and cried, "Ecce homo!  Behold the man!" '

Ibrahim delivered this information with the excitement of a man reading bingo numbers in a nursing home."

In keeping with this spirit of religious fervor and with the spirit of Carl Jung, expositor of the religious significance of the mandala,

Behold —

The Mandala of Abraham

For the religious significance of this mandala,
see an entry of May 25, 2003:

Matrix of the Death God.

Wednesday, March 31, 2004

Wednesday March 31, 2004

Filed under: General — Tags: , , , — m759 @ 12:25 am

To Be

A Jesuit cites Quine:

"To be is to be the value of a variable."

— Willard Van Orman Quine, cited by Joseph T. Clark, S. J., in Conventional Logic and Modern Logic: A Prelude to Transition,  Woodstock, MD: Woodstock College Press, 1952, to which Quine contributed a preface.

Quine died in 2000 on Xmas Day.

From a July 26, 2003, entry,
The Transcendent Signified,
on an essay by mathematician
Michael Harris:

Kubrick's
monolith

Harris's
slab

From a December 10, 2003, entry:

Putting Descartes Before Dehors

      

"Descartes déclare que c'est en moi, non hors de moi, en moi, non dans le monde, que je pourrais voir si quelque chose existe hors de moi."

ATRIUM, Philosophie

For further details, see ART WARS.

The above material may be regarded as commemorating the March 31 birth of René Descartes and death of H. S. M. Coxeter.

For further details, see

Plato, Pegasus, and the Evening Star.

Saturday, July 26, 2003

Saturday July 26, 2003

Filed under: General,Geometry — Tags: — m759 @ 11:11 pm

The Transcendent
Signified

“God is both the transcendent signifier
and transcendent signified.”

— Caryn Broitman,
Deconstruction and the Bible

“Central to deconstructive theory is the notion that there is no ‘transcendent signified,’ or ‘logos,’ that ultimately grounds ‘meaning’ in language….”

— Henry P. Mills,
The Significance of Language,
Footnote 2

“It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato’s (realist) reaction to the sophists (nominalists). What is often called ‘postmodernism’ is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth.”

Simon Blackburn, Think,
Oxford University Press, 1999, page 268

The question of universals is still being debated in Paris.  See my July 25 entry,

A Logocentric Meditation.

That entry discusses an essay on
mathematics and postmodern thought
by Michael Harris,
professor of mathematics
at l’Université Paris 7 – Denis Diderot.

A different essay by Harris has a discussion that gets to the heart of this matter: whether pi exists as a platonic idea apart from any human definitions.  Harris notes that “one might recall that the theorem that pi is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to pi is injective.  In other words, pi can be identified algebraically with X, the variable par excellence.”

Harris illustrates this with
an X in a rectangle:

For the complete passage, click here.

If we rotate the Harris X by 90 degrees, we get a representation of the Christian Logos that seems closely related to the God-symbol of Arthur C. Clarke and Stanley Kubrick in 2001: A Space Odyssey.  On the left below, we have a (1x)4×9 black monolith, representing God, and on the right below, we have the Harris slab, with X representing (as in “Xmas,” or the Chi-rho page of the Book of Kells) Christ… who is, in theological terms, also “the variable par excellence.”

Kubrick’s
monolith

Harris’s
slab

For a more serious discussion of deconstruction and Christian theology, see

Walker Percy’s Semiotic.

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