June 23, 2006
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Binary Geometry
There is currently no area of mathematics named “binary
geometry.” This is, therefore, a possible name for the geometry
of sets with 2n elements (i.e., a sub-topic of Galois geometry and of algebraic geometry over finite fields– part of Weil’s “Rosetta stone” (pdf)).Examples:
- Charles Sanders Peirce, “The Simplest Mathematics.”
- Donald E. Knuth’s discussion of binary hypercubes in “Boolean Basics,” a draft of section 7.1.1 in The Art of Computer Programming, Volume 4: Combinatorial Algorithms
- My own discussion of a binary hypercube in Geometry of the 4x4x4 Cube
- A more sophisticated example: the geometry of elliptic curves
over a binary Galois field. For an excellent introduction,
see the Certicom online elliptic curve tutorial. This has an applet
illustrating elliptic curves in a space of 256 points (256=16×16, with
the x and y variables of a curve each having 16 possible values). - In summary, apart from the fact that the native language of computers has
characteristic 2, “binary” mathematics, i.e. mathematics in
characteristic 2, is of special interest both in the study of finite geometry (Finite Geometry of the Square and Cube) and in algebraic geometry (see, for instance, the work of Brian Conrad).
