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Galois Space

by Steven H. Cullinane on July 12, 2012

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional Galois projective space PG(3,2)—

(Click to enlarge.)

There are 15 different individual linear diagrams in the figure above. These are the points of the Galois space PG(3,2).  Each 3-set of linear diagrams represents the structure of one of the 35  44 arrays and also represents a line of the projective space.

The symmetry of the linear diagrams accounts for the symmetry of the 840 possible images in the kaleidoscope puzzle.

* For further details on the phrase "Galois space," see Beniamino Segre's "On Galois Geometries," Proceedings of the International Congress of Mathematicians, 1958  [Edinburgh]. (Cambridge U. Press, 1960, 488-499.)