Galois Spaceby 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)—
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 4×4 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.)