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 PG(5,2), as noted by J. W. P. Hirschfeld in Ch.20 of 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 M₂₄.  See The Klein Correspondence, Penrose Space-Time, and a Finite Model.

Group actions on the 63 points of the finite projective space PG(5,2) are derived from group actions on the 64 points of the finite affine space AG(6,2).... which may, as my Diamond Theory points out, be visualized as a 4x4x4 cube. 

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 "G₁₆₈ 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 Ghost Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics, Princeton, 2008– "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 generated by reflections in midplanes" of "Solomon's Cube." Related material: "… the action of G168 in its alternative guise as SL(3; Z/2Z) is also now apparent. This version of G168 was presented by Weber in [1896, p. 539],* where he attributed it to Kronecker." – Jeremy Gray, "From the History of a Simple Group," in The Eightfold Way, MSRI Publications, 1998 Here MSRI, an acronym for Mathematical Sciences Research Institute, is pronounced "Misery." See Stephen King, K.C. Cole, and Heinrich Weber. *H. Weber, Lehrbuch der Algebra, Vieweg, Braunschweig, 1896. Reprinted by Chelsea, New York, 1961.