Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams

This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures.

A Beckman-Quarles Type Theorem for Coxeter's Inversive Distance

We prove that a bijective transformation on the set of circles in the real inversive plane which preserves pairs of circles a fixed inversive distance ρ > 0 apart must be induced by a Möbius transformation.

The Geometry of GF(q3)

1. Introduction. Inversive geometry involves as basic entities points and circles [2, p. 83; 4, p. 252]. The best known examples of inversive planes (the Miquelian planes) are constructed from a field K which is a quadratic extension of some other field F. Thus the complex numbers yield the Real Inversive Plane, while the Galois field GF(q2)(q = pe, p prime) yields the Miquelian inversive plane M(q) [2, chapter 9; 4, p. 257]. The purpose of this paper is to describe an analogous geometry of M(q) which derives from GF(q3), the cubic extension of GF(q).The resulting space, is three-dimensional, involving a class of surfaces which include planes, some quadric surfaces, and some cubic surfaces. We explore these surfaces, giving particular attention to the number of points they contain, and their intersections with lines and planes of the space .

Five Mutually Tangent Spheres and Various Associated Configurations

If five spheres σ0, σ1, …, σ4 touch each other externally and have radii in geometrical progression, there is a dilative rotation mapping σ0, σ1, σ2, σ3, to σ1, σ2, σ3, σ4; the dilatation factor is shown to be negative. The ten points of contact of the spheres lie by fours on 15 circles, forming a (154106) configuration in inversive space. In the corresponding configuration in the inversive plane, the 15 circles meet again in 60 points, which lie by fours on 45 circles touching by threes at each of the 60 points, and forming a configuration isomorphic to that of 60 Pascal lines (associated with six points on a conic) meeting by fours at 45 points. The 45 circles arise from ten Money-Coutts configurations of nine anti-tangent cycles. Conjectures are made about other circles through the 60 points.

On Singular Points of Normal Arcs of Cyclic Order Four

In [5] N. D. Lane and P. Scherk discuss arcs in the conformai (inversive) plane which are met by every circle at not more than three points; i.e., arcs of cyclic order three. This paper is concerned with the analysis of normal arcs of cyclic order four in the conformai plane.