Log24

Wednesday, December 11, 2019

Klein Quadric

Filed under: General — Tags: , — m759 @ 1:06 AM

The architecture of the recent post 
Geometry of 6 and 8 is in part
a reference to the Klein quadric.

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.

Thursday, June 28, 2018

Eine Kleine Nachtmusik

Filed under: General — m759 @ 1:00 AM

See  Klein Quartic  and  Klein Quadric.

Monday, May 28, 2018

Skewers

Filed under: General,Geometry — m759 @ 1:16 PM

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

Filed under: General,Geometry — m759 @ 9:20 PM

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

Filed under: General,Geometry — 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

Friday, May 13, 2016

Geometry and Kinematics

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

"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

Filed under: General,Geometry — m759 @ 8:00 PM

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.

Use the source

<!– 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;
&#160;Maks, Johannes;&#160;Gordon, Neil; Source: Designs,
Codes and Cryptography, Volume 34, Numbers 2-3, February
2005 , pp. 203-227; Publisher: Springer.&#160; 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, Rafa&#322; Ab&#322;amowicz, Pertti Lounesto,
published by Springer, 1995. –>

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

Wednesday, February 13, 2013

Form:

Filed under: General,Geometry — Tags: , — m759 @ 9:29 PM

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

IMAGE- The Penrose diamond and 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 M24. These properties are also 
relevant to the 1976 "Diamond Theory" monograph.

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

IMAGE- Stroppel on the Klein quadric, 2008

See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model
.

Related material:

"… one might crudely distinguish between philosophical
and mathematical motivation. In the first case one tries
to convince with a telling conceptual story; in the second
one relies more on the elegance of some emergent
mathematical structure. If there is a tradition in logic
it favours the former, but I have a sneaking affection for
the latter. Of course the distinction is not so clear cut.
Elegant mathematics will of itself tell a tale, and one with
the merit of simplicity. This may carry philosophical
weight. But that cannot be guaranteed: in the end one
cannot escape the need to form a judgement of significance."

– J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.

Those who prefer story to structure may consult 

  1. today's previous post on the Penrose diamond
  2. the remarks of Scott Aaronson on August 17, 2012
  3. the remarks in this journal on that same date
  4. the geometry of the 4×4 array in the context of M24.

Tuesday, October 16, 2012

Cube Review

Filed under: General,Geometry — Tags: — m759 @ 3:00 PM

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

Those who prefer narrative to mathematics may
consult posts in this journal containing the word "Cuber."

Sunday, April 1, 2012

The Palpatine Dimension

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

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."

— Arkani-Hamed

Some related material from this  weblog—

(See Big Apple and Columbia Film Theory)

http://www.log24.com/log/pix12/120108-Space_Time_Penrose_Hawking.jpg

The Meno Embedding:

http://www.log24.com/log/pix10B/101128-TheEmbedding.gif

Some related material from the Web—

IMAGE- The Penrose diamond and the Klein quadric

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

From MIT Commencement Day, 2011—

A symbol related to Apollo, to nine, and to "nothing"

A minimalist favicon—

IMAGE- Generic 3x3 square as favicon

This miniature 3×3 square— http://log24.com/log/pix11A/110518-3x3favicon.ico — may, if one likes,
be viewed as the "nothing" present at the Creation. 
See Feb. 19, 2011, and Jim Holt on physics.

Happy April 1.

Friday, October 2, 2009

Friday October 2, 2009

Filed under: General,Geometry — m759 @ 6:00 AM
Edge on Heptads

Part I: Dye on Edge

“Summary:
….we obtain various orbits of partitions of quadrics over GF(2a) 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:
In 1910 Conwell… produced his heptagons in PG(5,2) associated with the Klein quadric K whose points represent the lines of PG(3,2)…. Edge… constructed the 8 heptads of complexes in PG(3,2) directly. Both he and Conwell used their 8 objects to establish geometrically the isomorphisms SL(4,2)=A8 and O6(2)=S8 where O6(2) is the group of K….”

— “Partitions and Their Stabilizers for Line Complexes and Quadrics,” by R.H. Dye, Annali di Matematica Pura ed Applicata, Volume 114, Number 1, December 1977, pp. 173-194

Part II: Edge on Heptads

The Geometry of the Linear Fractional Group LF(4,2),” by W.L. Edge, Proc. London Math Soc., Volume s3-4, No. 1, 1954, pp. 317-342. See the historical remarks on the first page.

Note added by Edge in proof:
“Since this paper was finished I have found one by G. M. Conwell: Annals of Mathematics (2) 11 (1910), 60-76….”

Some context:

The Klein Correspondence,
Penrose Space-Time,
and a Finite Model

Sunday, March 1, 2009

Sunday March 1, 2009

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

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) ≅ S8, 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 A8," 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

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.

Monday, May 28, 2007

Monday May 28, 2007

Filed under: General,Geometry — Tags: — m759 @ 5:00 PM
Space-Time
and a Finite Model

Notes by Steven H. Cullinane
May 28, 2007

Part I: A Model of Space-Time

The following paper includes a figure illustrating Penrose's model of  "complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space."
 
The image “http://www.log24.com/log/pix07/070528-Twistor.jpg” cannot be displayed, because it contains errors.
Click on picture to enlarge.

For some background on the Klein quadric and space-time, see Roger Penrose, "On the Origins of Twistor Theory," from Gravitation and Geometry: A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.


Part II: A Corresponding Finite Model

The Klein quadric also occurs in a finite model of projective 5-space.  See a 1910 paper:

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:

The image “http://www.log24.com/log/pix07/070528-Quadric.jpg” cannot be displayed, because it contains errors.

As Cara goes on to explain, the Klein correspondence underlies Conwell's discussion of eight 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 M24.


Related material:

 

The projective space 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.

Geometry of the I Ching.
 
"Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the 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.'"
 
— Hermann Hesse, The Glass Bead Game,
  translated by Richard and Clara Winston
 

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