The architecture of the recent post

Geometry of 6 and 8 is in part

a reference to the Klein quadric.

## Wednesday, December 11, 2019

### Klein Quadric

## Sunday, December 29, 2019

### Articulation Raid

“… 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]
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
Combinatorial Grassmannian − |

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

## Thursday, June 28, 2018

## Monday, May 28, 2018

### Skewers

A piece co-written by Ivanov, the author noted in the previous post, was cited

in my "Geometry of the 4×4 Square."

Also cited there — A paper by Pasini and Van Maldeghem that mentions

the Klein quadric.

Those sources suggested a search —

The link is to some geometry recently described by Tabachnikov

that seems rather elegant:

For another, more direct, connection to the geometry of the 4×4 square,

see Richard Evan Schwartz in *this* journal.

This same Schwartz appears also in the above Tabachnikov paper:

## Thursday, July 6, 2017

### A Pleasing Situation

The 4x4x4 cube is the natural setting

for the finite version of the Klein quadric

and the eight "heptads" discussed by

Conwell in 1910.

As R. Shaw remarked in 1995,

"The situation is indeed quite pleasing."

## Wednesday, July 5, 2017

### Imaginarium of a Different Kind

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). — User Rgdboer |

## Friday, May 13, 2016

### Geometry and Kinematics

"Just as both tragedy and comedy can be written

by using the same letters of the alphabet, the vast

variety of events in this world can be realized by

the same atoms through their different arrangements

and movements. Geometry and kinematics, which

were made possible by the void, proved to be still

more important in some way than pure being."

— Werner Heisenberg in *Physics and Philosophy*

For more about geometry and kinematics, see (for instance)

**"An introduction to line geometry with applications,"**

by Helmut Pottmann, Martin Peternell, and Bahram Ravani,

*Computer-Aided Design* 31 (1999), 3-16.

The concepts of line geometry (null point, null plane, null polarity,

linear complex, Klein quadric, etc.) are also of interest in *finite* geometry.

Some small finite spaces have as their natural models arrays of *cubes* .

## Sunday, November 15, 2015

### The Diamond and the Cube

Anyone who clicked on the Dirac search at the end of

the previous post, "Dirac's Diamond," may wonder why the

"Solomon's Cube" post of 11 AM Sunday, March 1, 2009,

appeared in the Dirac search results, since there is no

apparent mention of Dirac in that Sunday post.

<!– See also "a linear transformation of V6… which preserves

the Klein quadric; in this way we arrive at the isomorphism of

Sym(8) withthe full orthogonal group O+(6; 2)." in "The

Classification of Flats in PG(9,2) which are External to the

Grassmannian G1,4,2 Authors: Shaw, Ron;

 Maks, Johannes; Gordon, Neil; Source: Designs,

Codes and Cryptography, Volume 34, Numbers 2-3, February

2005 , pp. 203-227; Publisher: Springer.  For more details,

see "Finite Geometry, **Dirac** Groups and the Table of Real

Clifford Algebras," by R. Shaw (U. of Hull), pp. 59-99 in

Clifford Algebras and Spinor Structures, by By Albert

Crumeyrolle, Rafał Abłamowicz, Pertti Lounesto,

published by Springer, 1995. –>

## Saturday, March 8, 2014

### Conwell Heptads in Eastern Europe

“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 *Q *^{+}(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 *Q *^{+}(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 *Q *^{+}(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 **R**^{8} 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

## Wednesday, February 13, 2013

### Form:

**Story, Structure, and the Galois Tesseract**

Recent Log24 posts have referred to the

"Penrose diamond" and Minkowski space.

The Penrose diamond has nothing whatever

to do with my 1976 monograph "Diamond Theory,"

except for the diamond shape and the connection

of the Penrose diamond to the Klein quadric—

The Klein quadric occurs in the five-dimensional projective space

over a field. If the field is the two-element Galois field GF(2), the

quadric helps explain certain remarkable symmetry properties

of the R. T. Curtis *Miracle Octad Generator* (MOG), hence of

the large Mathieu group M_{24}. These properties are also

relevant to the 1976 "Diamond Theory" monograph.

For some background on the quadric, see (for instance)…

See also The Klein Correspondence,

Penrose Space-Time, and a Finite Model.

Related material:

"… one might crudely distinguish between philosophical – J. M. E. Hyland. "Proof Theory in the Abstract." (pdf) |

Those who prefer story to structure may consult

- today's previous post on the Penrose diamond
- the remarks of Scott Aaronson on August 17, 2012
- the remarks in this journal on that same date
- the geometry of the 4×4 array in the context of M
_{24}.

## Tuesday, October 16, 2012

### Cube Review

Last Wednesday's 11 PM post mentioned the

adjacency-isomorphism relating the 4-dimensional

hypercube over the 2-element Galois field GF(2) to

the 4×4 array made up of 16 square cells, with

opposite edges of the 4×4 array identified.

A web page illustrates this property with diagrams that

enjoy the Karnaugh property— adjacent vertices, or cells,

differ in exactly one coordinate. A brief paper by two German

authors relates the Karnaugh property to the construction

of a magic square like that of Dürer (see last Wednesday).

In a similar way (search the Web for *Karnaugh + cube *),

vertex adjacency in the 6-dimensional hypercube over GF(2)

is isomorphic to cell adjacency in the 4x4x4 cube, with

opposite faces of the 4x4x4 cube identified.

The above cube may be used to illustrate some properties

of the 64-point Galois 6-space that are more advanced

than those studied by enthusiasts of "magic" squares

and cubes.

See

- the 4x4x4 cube and An Invariance of Symmetry
- the 4x4x4 cube and the nineteenth-century

geometers' "Solomon's seal" - the 4x4x4 cube and the Kummer surface
- the 4x4x4 cube and the Klein quadric.

Those who prefer narrative to mathematics may

consult posts in this journal containing the word "Cuber."

## Sunday, April 1, 2012

### The Palpatine Dimension

A physics quote relayed at Peter Woit's weblog today—

"The relation between 4D N=4 SYM and the 6D (2, 0) theory

is just like that between Darth Vader and the Emperor.

You see Darth Vader and you think 'Isn’t he just great?

How can anyone be greater than that? No way.'

Then you meet the Emperor."

**Some related material from this weblog—**

**(See Big Apple and Columbia Film Theory)**

**The Meno Embedding:
**

**Some related material from the Web—**

**See also uses of the word triality in mathematics. For instance…**

A discussion of triality by Edward Witten—

Triality is in some sense the last of the exceptional isomorphisms,

and the role of triality for *n* = 6 thus makes it plausible that *n* = 6

is the maximum dimension for superconformal symmetry,

though I will not give a proof here.

— "Conformal Field Theory in Four and Six Dimensions"

and a discussion by Peter J. Cameron—

There are exactly two non-isomorphic ways

to partition the 4-subsets of a 9-set

into nine copies of AG(3,2).

Both admit 2-transitive groups.

— "The Klein Quadric and Triality"

Exercise: Is Witten's triality related to Cameron's?

(For some historical background, see the *triality* link from above

and Cameron's Klein Correspondence and Triality.)

Cameron applies *his * triality to the pure geometry of a 9-set.

For a 9-set viewed in the context of physics, see A Beginning—

A symbol related to Apollo, to nine, and to "nothing"—
This miniature 3×3 square— — may, if one likes, |

Happy April 1.

## Friday, October 2, 2009

### Friday October 2, 2009

Edge on Heptads
“Summary: ….we obtain various orbits of partitions of quadrics over GF(2 ^{a}) by their maximal totally singular subspaces; the corresponding stabilizers in the relevant orthogonal groups are investigated. It is explained how some of these partitions naturally generalize Conwell’s heptagons for the Klein quadric in PG(5,2).”“Introduction: — “Partitions and Their Stabilizers for Line Complexes and Quadrics,” by R.H. Dye,
“The Geometry of the Linear Fractional Group Note added by Edge in proof: |

## Sunday, March 1, 2009

### Sunday March 1, 2009

**Solomon's Cube
continued**

"There is a book… called *A Fellow of Trinity*, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."

— G. H. Hardy, *A Mathematician's Apology*

"* The Solomon Key* is the working title of an unreleased novel in progress by American author Dan Brown.

*The Solomon Key*will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were

*Angels & Demons*(2000) and

*The Da Vinci Code*(2003)." —Wikipedia

"One has O^{+}(6) ≅ S_{8}, the symmetric group of order 8! …."

— "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.

"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."

— George M. Conwell, "The 3-space *PG*(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A_{8}," in *Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference* (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97

## Monday, April 28, 2008

### Monday April 28, 2008

**Religious Art**

The black monolith of

Kubrick's *2001* is, in

its way, an example

of religious art.

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.

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:

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.

## Monday, May 28, 2007

### Monday May 28, 2007

**and a Finite Model**

Notes by Steven H. Cullinane

May 28, 2007

Part I: A Model of Space-Time

Part I: A Model of Space-Time

*Gravitation*

*and Geometry*:

*A Volume in Honor of Ivor Robinson*, Bibliopolis, 1987.

Part II: A Corresponding Finite Model

** **

G. M. Conwell, The 3-space

PG(3,2) and its group,Ann. of Math.11, 60-76.

Conwell discusses the quadric, and the related Klein *correspondence*, in detail. This is noted in a more recent paper by Philippe Cara:**
**

*heptads*. These play an important role in another correspondence, illustrated in the

*Miracle Octad Generator*of R. T. Curtis, that may be used to picture actions of the large Mathieu group M

_{24}.

Related material:

*PG*(5,2), home of the Klein quadric

*in the finite model, may be viewed as the set of 64 points of the affine space*

*AG*(6,2), minus the origin.

The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the *Classic of Transformation*, China's *I Ching*.

There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube. This correspondence leads to a natural way to generate the affine group *AGL*(6,2). This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.

*I Ching*into the Glass Bead Game. Elder Brother laughed. 'Go ahead and try,' he exclaimed. 'You'll see how it turns out. Anyone can create a pretty little bamboo garden in the world. But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'"

*The Glass Bead Game*,