Rational Points on the Unit Sphere

Points at Rational Distance from the Vertices of a Unit Polygon

Free groups of rotations acting without fixed points on the rational unit sphere

The purpose of this paper is to generalize an example of Sato of a free group of rotations of the Euclidean 3-space whose action on the rational unit sphere is fixed point free. Such groups are of interest, as they can be employed to construct paradoxical decompositions of spheres without assuming the Axiom of Choice.

Spatial statistics for lattice points on the sphere I‎: ‎Individual results

‎We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares‎. ‎We examine several statistics of these point sets‎, ‎such as the electrostatic potential‎, ‎Ripley's function‎, ‎the variance of the number of points in random spherical caps‎, ‎and the covering radius‎. ‎Some of the results are conditional on t...

A free group acting without fixed points on the rational unit sphere

We prove the existence of a free group of rotations of rank 2 which acts on the rational unit sphere without non-trivial fixed points. Introduction. The purpose of this paper is to prove that the group SO3(Q) of all proper orthogonal 3 × 3 matrices with rational entries has a free subgroup F2 of rank 2 such that for all w ∈ F2 different from the identity and for all ~r ∈ S2 ∩Q3 we have w(~r ) 6...

The appearance of multitude from unit in Farabi’s

  Farabi has described the problem of emanation with negative determiners and referred to the first intellect as the first case of emanation. For him, the appearance of multitude by intellect is described on a dual order; though the way of description is not accepted by the philosophers before him, specially by Avicenna. Farabi links the old physics to the problem of emanation and, therefore, ...