Log24

The natural geometric setting for the "bricks" in the Miracle Octad Generator
(MOG) of Robert T. Curtis is PG(5,2), the projective 5-space over GF(2).

The Klein correspondence mirrors the 35 lines of PG(3,2) — and hence, via the 
graphic approach below, the 35 "heavy bricks" of the MOG that match those
lines — in PG(5,2), where the bricks may be studied with geometric methods,
as an alternative to Curtis's original MOG combinatorial construction methods.

The construction below of a PG(5,2) brick space  is analogous to the
"line diagrams"  construction of a PG(3,2) in Cullinane's diamond theorem.

Exercise:  The eight-part diagrams in the graphic "brick space"
model of PG(5,2) below need to be suitably labeled with six-part
GF(2) coordinates to help illustrate the Klein correspondence that
underlies the large Mathieu group M₂₄.

A possible approach:  The lines  separating dark squares from light
(i.e., blue from white or yellow) in the figure above may be added
in XOR fashion (as if they were diamond theorem  line diagrams)
to form a six  dimensional vector space, which, after a suitable basis
is chosen, may be represented by six-tuples of 0's and 1's.

Related reading —

log24.com/log24/241221-'Brick Space « Log24' – m759.net.pdf .

This is a large (15.1 MB) file.  The Foxit PDF reader is recommended.

The PDF is from a search for Brick Space  in this journal.

Some context:  http://m759.net/wordpress/?s=Weyl+Coordinatization.

"Drawing the same face from different angles sounds fun,
but let me tell you – it’s not. It’s not fun at all. It’s HARD!!"

— Loisvb on Instagram, Dec. 18, 2024

Likewise for PG(5,2).

Exercise:  The eight-part diagrams in the graphic "brick space"
model of PG(5,2) below need to be suitably labeled with six-part
GF(2) coordinates to help illustrate the Klein correspondence that
underlies the large Mathieu group M₂₄.

— Cartoon by S. Harris

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:

Articulation Day.