Finite
Geometry Notes |

Penrose 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."

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-spacePG(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:**
**

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 M_{24}.

Some background on heptads:

Some background on heptads:

Edge on HeptadsPart I: Dye on Edge "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: 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)=A_{8} and O_{6}(2)=S_{8} where O_{6}(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. 173-194Part 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...." |

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.

The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the

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

"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.'"

Transcript of relevant text for search engines:

Transcript of relevant text for search engines:

**Part I--**

**From "Twistor cosmology and quantum space-time," by Dorje
C. Brody
(Imperial College, London) and Lane P. Hughston (King's College London):**

FIGURE 7. *Complexified, compactified Minkowski space-time as the
Klein quadric in complex projective 5-space*. The aggregate of
complex projective lines in P^{3} constitutes a nondegenerate
quadric *Q*^{4} in P^{5}.
The quadric contains two distinct systems of projective 2-planes,
called alpha-planes and beta-planes. Any two distinct planes of the
same type in *Q*^{4} intersect at a point. Two planes of
the opposite type in *Q*^{4} will in general not
intersect, but if they do, they intersect in a line. The points of P^{3}
correspond to alpha-planes in *Q*^{4}, and the 2-planes
of P^{3} correspond to beta-planes in *Q*^{4}.

Published in *XIX Max Born Symposium*, Wroclaw
(Poland), 28 September

**Part II--**

**From "RWPRI Geometries for the alternating group A_{8},"
by Philippe Cara, Department of Mathematics, Vrije Universiteit Brussel
**

It is well known that the lines of *PG*(3,2) correspond
to the 35 points of a hyperbolic quadric *Q*^{+}(5,2)
with equation *X*_{0}* X*_{1} + *X*_{2} *X*_{3}
+ *X*_{4} *X*_{5} = 0 *PG*(5,2). This
correspondence is known as the *Klein correspondence* and the
hyperbolic quadric is then called the *Klein quadric*. In what
follows, the Klein quadric will be denoted by *Q*. On *Q*
there are two families of 15 planes each, which correspond to the 15
points and the 15 planes of *PG*(3,2) respectively. Two different
planes of the same family have one point in common and two planes of
different families are either disjoint or share one line. A *p*P* of *PG*(3,2) appears on *Q*
as two planes from different families which intersect in a line,
representing the three lines in *PG*(3,2) which are included in *P*
and contain the point *p*. The Klein correspondence for *PG*(3,2)
is described in detail in [14].

_{
[14] G. M. Conwell, The 3-space PG(3,2) and its
group, Ann. of Math. 11 (1910), pp. 60-76
}

_{Published in Finite
Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 2000), Springer, 2001, ed. }_{Aart
Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A.
Thas, pp. 61-97
}