Log24

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:

Those who prefer story to structure may consult 

1. today's previous post on the Penrose diamond
2. the remarks of Scott Aaronson on August 17, 2012
3. the remarks in this journal on that same date
4. the geometry of the 4×4 array in the context of M₂₄.