— 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)
The architecture of the recent post
Geometry of 6 and 8 is in part
a reference to the Klein quadric.
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.
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 —
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 —
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."
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:
“… 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
Metod Saniga, 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 Keywords:
Combinatorial Grassmannian − |
See also this journal on the above date — 17 September 2014.
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:
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."
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 |
"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 .
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. –>
“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 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
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 M24. 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
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."
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—
|
From MIT Commencement Day, 2011— A symbol related to Apollo, to nine, and to "nothing"— A minimalist favicon—
This miniature 3×3 square— |
Happy April 1.
| Edge on Heptads
Part I: Dye on Edge “Introduction: — “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: |
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
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.
and a Finite Model
Notes by Steven H. Cullinane
May 28, 2007
Part I: A Model of Space-Time
 |
Click on picture to enlarge.
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:
 |
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.
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