Finite
Geometry
Notes |

**Solomon's Cube**

**by Steven H. Cullinane
on May 28, 2003
**

(Last modified March 3, 2010)

In 1998, the Mathematical Sciences Research Institute
at Berkeley published a book, *The Eightfold Way*, inspired
by
a
new
sculpture
at
the
Institute.
This
note
describes
another
sculpture embodying some of the same concepts in a different guise.

*The Eightfold Way* deals with
Klein's quartic,
which,
like
all
non-singular
quartic
curves,
has
28 bitangents.
The
relationship
of
the
28
bitangents
to
the
27
lines
of a "
Solomon's
seal" in a cubic surface is sketched at the
Mathworld
encyclopedia.
For more details, see the excerpt below, from Jeremy Gray's
paper in *The Eightfold Way*.

Both the 28 bitangents and the 27 lines may be
represented within the 63-point space *Finite
Projective Spaces of Three Dimensions* (Clarendon Press,
Oxford, 1985).

The space PG(5,2) also contains a representation of the
*Klein quadric* (as opposed to the *Klein quartic* discussed
in *The Eightfold Way*). This representation, obtained via
the *Klein correspondence*, may be used to construct the Mathieu
group M_{24}. See The Klein
Correspondence, Penrose Space-Time, and a Finite Model.

Group actions on the *63* points of the finite *projective*
space *64*
points of the finite *affine* space

The theorem may be verified by manipulating a JavaScript version of the cube.

Those who like to associate mathematical with
religious entities may contemplate the above in the light of the 1931
Charles Williams
novel
*Many
Dimensions*. Instead of Solomon's *seal*, this
book describes Solomon's *cube.*

From
a
review: "Imagine 'Raiders of the Lost Ark' set in 20th-century
London, and then imagine it written by a man steeped not in Hollywood
movies but in Dante and the things of the spirit, and you might begin
to get a picture of Charles Williams's novel *Many Dimensions*."

From *
The Eightfold Way*,
a
publication
of

the
Mathematical Sciences Research
Institute

(MSRI Publications Vol. 35, 1998):

**From
the
History
of
a
Simple
Group**

**by Jeremy Gray**

Excerpt:** **

"Art isn't easy." -- Stephen Sondheim

For more on this theme, see

**ART WARS: Geometry as Conceptual Art****.**

For more about the group "*G*_{168}
in its

alternative guise as SL(3; Z/2Z)," see

The Eightfold Cube,

A Simple
Reflection Group of Order 168,

and the following--

Update of March 3, 2010 Plato's GhostJeremy Gray, "Here, modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated— indeed, anxious— rather than a naïve relationship with the day-to-day world, which is the de facto view of a coherent group of people, such as a professional or discipline-based group that has a high sense of the seriousness and value of what it is trying to achieve. This brisk definition…." Brisk? Consider Caesar's "The die is cast," Gray in "Solomon's Cube," and Symmetry, Automorphisms, and Visual Group Theory– This is the group of "8 rigid
motions
"… the action of – Jeremy Gray, "From
the History of a Simple Group," in Here MSRI, an acronym for Mathematical Sciences Research Institute, is pronounced "Misery." See Stephen King, K.C. Cole, and Heinrich Weber. *H. Weber, |