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Thursday, November 19, 2020

Set Design and the Schoolgirl Problem

Filed under: General — Tags: Kummerhenge — m759 @ 9:36 am

Underlying Structure of the Design —

Schoolgirl Problem —

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Wednesday, January 15, 2020

Paradigm Shift

Filed under: General — Tags: Interality — m759 @ 1:33 pm

Illustration, from a search in this journal for “Symplectic” —

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven) .

Some background: Rift-design in this journal and …

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Monday, August 27, 2018

Children of the Six Sides

Filed under: General,Geometry — Tags: Six-Set — m759 @ 11:32 am

From the former date above —

Saturday, September 17, 2016

A Box of Nothing

Filed under: Uncategorized — m759 @ 12:13 AM

(Continued)

"And six sides to bounce it all off of."

From the latter date above —

Tuesday, October 18, 2016

Parametrization

Filed under: Uncategorized — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia, seems useful for describing labelings that are not, at least at first glance, of a vector-space nature.

Examples: The labelings of a 4×4 array by a blank space plus the 15 two-subsets of a six-set (Hudson, 1905) or by a blank plus the 5 elements and the 10 two-subsets of a five-set (derived in 2014 from a 1906 page by Whitehead), or by a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization" is not limited to vector-space coordinates. He describes it as simply a mapping to a set of reproducible symbols .

(But Weyl does imply that these symbols should, like vector-space coordinates, admit a group of transformations among themselves that can be used to describe transformations of the point-space being coordinatized.)

From March 2018 —

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Sunday, December 10, 2017

Geometry

Filed under: G-Notes,General,Geometry — Tags: Dirac and Geometry, Interlacing Derrida, Quantum Tesseract Theorem — m759 @ 8:45 pm

See also Symplectic in this journal.

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of linear (or line ) complex
in the finite projective space PG(3,2) —

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Wednesday, February 15, 2017

Warp and Woof

Filed under: General,Geometry — m759 @ 3:00 pm

Space —

Space structure —

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 59-79 (1992):

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Space shuttle —

Related ethnic remarks —

… As opposed to Michael Larsen —

Funny, you don't look Danish.

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Wednesday, November 23, 2016

Yogiism

Filed under: General,Geometry — Tags: Diamond Theorem Correlation, Inner-Outer — m759 @ 12:31 pm

From the American Mathematical Society (AMS) webpage today —

From the current AMS Notices —

Related material from a post of Aug. 6, 2014 —

(Here "five point sets" should be "five-point sets.")

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 59-79 (1992):

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic structure* now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

* The phrase as used here is a deliberate
abuse of language . For the real definition of
“symplectic structure,” see (for instance)
“Symplectic Geometry,” by Ana Cannas da Silva
(article written for Handbook of Differential
Geometry , Vol 2.) To establish that the above
figure is indeed symplectic , see the post
Zero System of July 31, 2014.

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Tuesday, October 18, 2016

Parametrization

Filed under: General,Geometry — Tags: Six-Set — m759 @ 6:00 am

The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space nature.

Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by
a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem: to fix objectively
a class of equivalent coordinatizations and to
ascertain the group of transformations S
mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space coordinates. He describes it
as simply a mapping to a set of reproducible symbols .

(But Weyl does imply that these symbols should, like vector-space
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)

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Tuesday, June 7, 2016

Art and Space…

Filed under: General,Geometry — Tags: Barth Art, St. Bridget's Day — m759 @ 6:00 am

Continues, in memory of chess grandmaster Viktor Korchnoi,
who reportedly died at 85 yesterday in Switzerland —

IMAGE- Spielfeld (1982-83), by Wolf Barth

The coloring of the 4×4 "base" in the above image
suggests St. Bridget's cross.

From this journal on St. Bridget's Day this year —

"Possible title:

A new graphic approach
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M₂₄"

The narrative leap from image to date may be regarded as
an example of "knight's move" thinking.

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Monday, February 1, 2016

Historical Note

Filed under: General,Geometry — Tags: Natural Diagram, Priority — m759 @ 6:29 am

Possible title:

A new graphic approach
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M₂₄

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Saturday, January 30, 2016

Pope’s Geometry

Filed under: General,Geometry — m759 @ 10:21 am

From page 56 of The Science Fiction of Mark Clifton ,
Southern Illinois University Press, 1980 —

See also the following image in this journal —

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven) .

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Thursday, October 15, 2015

Contrapuntal Interweaving

Filed under: General,Geometry — Tags: Diamond Theorem Correlation — m759 @ 2:01 am

The title is a phrase from R. D. Laing's book The Politics of Experience .
(Published in the psychedelic year 1967. The later "contrapuntal interweaving"
below is of a less psychedelic nature.)

An illustration of the "interweaving' part of the title —
The "deep structure" of the diamond theorem:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven) .

The word "symplectic" from the end of last Sunday's (Oct. 11) sermon
describes the "interwoven" nature of the above illustration.

An illustration of the "contrapuntal" part of the title (click to enlarge):

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Saturday, February 21, 2015

High and Low Concepts

Filed under: General,Geometry — Tags: Diamond Theorem Correlation — m759 @ 4:30 pm

Steven Pressfield on April 25, 2012:

What exactly is High Concept?

Let’s start with its opposite, low concept.
Low concept stories are personal,
idiosyncratic, ambiguous, often European.
“Well, it’s a sensitive fable about a Swedish
sardine fisherman whose wife and daughter
find themselves conflicted over … ”

ZZZZZZZZ.

Fans of Oslo artist Josefine Lyche know she has
valiantly struggled to find a high-concept approach
to the diamond theorem. Any such approach must,
unfortunately, reckon with the following low
(i.e., not easily summarized) concept —

The Diamond Theorem Correlation:

From left to right …

http://www.log24.com/log/pix14B/140824-Diamond-Theorem-Correlation-1202w.jpg

http://www.log24.com/log/pix14B/140731-Diamond-Theorem-Correlation-747w.jpg

http://www.log24.com/log/pix14B/140824-Picturing_the_Smallest-1986.gif

http://www.log24.com/log/pix14B/140806-ProjPoints.gif

For some backstory, see ProjPoints.gif and "Symplectic Polarity" in this journal.

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Monday, November 3, 2014

The Rhetoric of Abstract Concepts

Filed under: General,Geometry — Tags: Symplectic Rhetoric — m759 @ 12:48 pm

From a post of June 3, 2013:

New Yorker editor David Remnick at Princeton today
(from a copy of his prepared remarks):

“Finally, speaking of fabric design….”

I prefer Tom and Harold:

Tom Wolfe in The Painted Word —

“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”

Harold Rosenberg in The New Yorker (click to enlarge)—

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 59-79 (1992):

Symplectic :

— Steven H. Cullinane,
diamond theorem illustration

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Monday, August 11, 2014

Syntactic/Symplectic

Filed under: General,Geometry — Tags: Articulation, Diamond Theorem Correlation, Langer Key, Rosenhain and Göpel, Springer — m759 @ 4:00 pm

(Continued from August 9, 2014.)

Syntactic:

Symplectic:

"Visual forms— lines, colors, proportions, etc.— are just as capable of
articulation , i.e. of complex combination, as words. But the laws that govern
this sort of articulation are altogether different from the laws of syntax that
govern language. The most radical difference is that visual forms are not
discursive . They do not present their constituents successively, but
simultaneously, so the relations determining a visual structure are grasped
in one act of vision."

– Susanne K. Langer, Philosophy in a New Key

For examples, see The Diamond-Theorem Correlation
in Rosenhain and Göpel Tetrads in PG(3,2).

This is a symplectic correlation,* constructed using the following
visual structure:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven) .

* Defined in (for instance) Paul B. Yale, Geometry and Symmetry ,
Holden-Day, 1968, sections 6.9 and 6.10.

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Saturday, August 9, 2014

Syntactic/Symplectic

Filed under: General,Geometry — m759 @ 3:00 pm

Syntactic Structure —

See the Lightfoot of today’s previous post:

Symplectic Structure —

See the plaited, or woven, structure of August 6:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven) .

See also Deep Structure (Dec. 9, 2012).

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Wednesday, August 6, 2014

Symplectic Structure*

Filed under: General,Geometry — Tags: Diamond Theorem Correlation — m759 @ 1:00 pm

From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens 54, 59-79 (1992):

"… what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic structure** now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Some related passages from the literature:

* The title is a deliberate abuse of language .
For the real definition of "symplectic structure," see (for instance)
"Symplectic Geometry," by Ana Cannas da Silva (article written for
Handbook of Differential Geometry, vol 2.) To establish that the
above figure is indeed symplectic , see the post Zero System of
July 31, 2014.

** See Steven H. Cullinane, Inscapes III, 1986

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Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: Change Logos, Eightfold Cube, four-set, Map Systems, Octad, Octad Generator, Strange Change — m759 @ 10:13 pm

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:

For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).

The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry