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=16x16, 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).
Page created June 23, 2006.