Now out from behind a paywall . . .

The diamond theorem at SASTRA —

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The following IEEE paper is behind a paywall,

but the first page is now available for free

at deepdyve.com —

For further details on the diamond theorem, see

finitegeometry.org/sc/ or the archived version at . . .

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*A sequel to the post CP is for Consolation Prize (Sept. 3, 2016)*

An image from Log24 on this date last year:

A recent comment on a discussion of CP symmetry —

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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):

“… 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….”

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

Zero System of July 31, 2014.

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"Protection of digital content from being tapped by intruders is a crucial task in the present generation of Internet world. In this paper, we proposed an implementation of new visual secret sharing scheme for gray level images using diamond theorem correlation. A secret image has broken into 4 × 4 non overlapped blocks and patterns of diamond theorem are applied sequentially to ensure the secure image transmission. Separate diamond patterns are utilized to share the blocks of both odd and even sectors. Finally, the numerical results show that a novel secret shares are generated by using diamond theorem correlations. Histogram representations demonstrate the novelty of the proposed visual secret sharing scheme." — "New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure," by V. Harish, N. Rajesh Kumar, and N. R. Raajan.
Published in: |

**Excerpts —**

Related material — Posts tagged Diamond Theorem Correlation.

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Note that the six anticommuting sets of Dirac matrices listed by Arfken

correspond exactly to the six spreads in the above complex of 15 projective

lines of PG(3,2) fixed under a symplectic polarity (the *diamond theorem
correlation *). As I noted in 1986, this correlation underlies the Miracle

Octad Generator of R. T. Curtis, hence also the large Mathieu group.

**References:**

Arfken, George B., *Mathematical Methods for Physicists* , Third Edition,

Academic Press, 1985, pages 213-214

Cullinane, Steven H., *Notes on Groups and Geometry, 1978-1986*

**Related material: **

The 6-set in my 1986 note above also appears in a 1996 paper on

the sixteen Dirac matrices by David M. Goodmanson —

**Background reading:**

Ron Shaw on finite geometry, Clifford algebras, and Dirac *groups*

(undated compilation of publications from roughly 1994-1995)—

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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:

.

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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Some context for yesterday's post on a symplectic polarity —

This 1986 note may or may not have inspired some remarks

of Wolf Barth in his foreword to the 1990 reissue of Hudson's

1905 *Kummer's Quartic Surface* .

See also the diamond-theorem correlation.

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**The words:** "symplectic polarity"—

**The images:**

The Natural Symplectic Polarity in PG(3,2)

Symmetry Invariance in a Diamond Ring

The Diamond-Theorem Correlation

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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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**In the Miracle Octad Generator (MOG):**

The above details from a one-page note of April 26, 1986, refer to the

Miracle Octad Generator of R. T. Curtis, as it was published in 1976:

From R. T. Curtis (1976). A new combinatorial approach to M_{24},

*Mathematical Proceedings of the Cambridge Philosophical Society *,

**79**, pp 25-42. doi:10.1017/S0305004100052075.

The 1986 note assumed that the reader would be able to supply, from the

MOG itself, the missing top row of each heavy brick.

Note that the interchange of the two squares in the top row of each

heavy brick induces the diamond-theorem correlation.

Note also that the 20 pictured 3-subsets of a 6-set in the 1986 note

occur as *paired complements* in two pictures, each showing 10 of the

3-subsets.

This pair of pictures corresponds to the 20 *Rosenhain tetrads* among

the 35 lines of PG(3,2), while the picture showing the 2-subsets

corresponds to the 15 *Göpel tetrads* among the 35 lines.

See Rosenhain and Göpel tetrads in PG(3,2). Some further background:

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Some background for the part of the 2002 paper by Dolgachev and Keum

quoted here on January 17, 2014 —

Related material in this journal (click image for posts) —

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(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:

.

* Defined in (for instance) Paul B. Yale, *Geometry and Symmetry *,

Holden-Day, 1968, sections 6.9 and 6.10.

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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…."

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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Click image for a larger, clearer version.

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The title phrase (not to be confused with the film 'The Zero Theorem')

means, according to the Encyclopedia of Mathematics,

a *null system *, and

"A null system is also called null polarity,

a symplectic polarity or a symplectic correlation….

it is a polarity such that every point lies in its own

polar hyperplane."

See Reinhold Baer, "Null Systems in Projective Space,"

*Bulletin of the American Mathematical Society*, Vol. 51

(1945), pp. 903-906.

An example in PG(3,2), the projective 3-space over the

two-element Galois field GF(2):

See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.

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“The relevance of a geometric theorem is determined by what the theorem

tells us about space, and not by the eventual difficulty of the proof.”

— Gian-Carlo Rota discussing the theorem of Desargues

**What space tells us about the theorem : **

In the simplest case of a projective *space* (as opposed to a *plane *),

there are 15 points and 35 lines: 15 *Göpel* lines and 20 *Rosenhain* lines.*

The theorem of Desargues in this simplest case is **essentially a symmetry**

within the set of 20 Rosenhain lines. The symmetry, a reflection

about the main diagonal in the square model of this space, interchanges

10 horizontally oriented (row-based) lines with 10 corresponding

vertically oriented (column-based) lines.

*Vide* **Classical Geometry in Light of Galois Geometry**.

* Update of June 9: For a more traditional nomenclature, see (for instance)

R. Shaw, 1995. The “simplest case” link above was added to point out that

the two types of lines named are derived from a natural *symplectic polarity *

in the space. The square model of the space, apparently first described in

notes written in October and December, 1978, makes this polarity clearly visible:

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