A few short combinatorial proofs in Geometry

We will treat the unit-distance graph over the real (or rational) points in Ed, discuss their connectivity and chromatic numbers, and use it in connection with the Beckman-Quarels Theorem ( Every mapping from Ed to itself, d >= 2, that preserves distance one is an isometry ). We will show that every closed curve on the unit sphere in E^d that meets all the d major hyperplanes has length of at least pi.