Spherical Geometry

The aim of this section of the website is to establish the relationships between different polyhedra and the sphere. This is an area in which an entire website may be devoted to later. For the purpose of an introduction to the relationships between the sphere and regular and semi-regular polyhedra, it was necessary to look at one polyhedron indepthly. The polyhedron choosen was the familiar cube (hexahedron).

The Sphere

Definition of a Sphere: A locus of points in three dimensional space all of which are at a given distance from a fixed point called the centre.

Characteristics of a Sphere

Every section of a spherical surface made by a plane is a circle. In particular, this means that each face of a polyhedron inscribed in a sphere lies on a plane that cuts the sphere in a circle. Its centre is the centre of the polyhedral face related to it and its radius is less than that of the sphere itself.

If a plane passes through the centre of the sphere, the section is called a great circle. Its centre and radius are the same as those of the sphere itself. In particular it should be noted that only in the case of some nonconvex uniform polyhedra will the faces lie on planes throughout the centre of the polyhedron. This centre coincides with the centre of the circumscribing sphere.

A unique great circle is determined by the centre of the sphere and any two points on its surface provided these points are not the extremities of a diameter. In particular this means that each edge of a polyhedron determines and arc of a great circle, since the two end points of an edge lie on the surface of the sphere.

The shortest path from one point to another on a spherical surface is along the arc of a great circle. This shortest path is called a geodesic. In particular the edges of a polyhedron can be replaced by arcs of great circles to obtain a spherical polyhedron. Each plane polygon that is a face of the polyhedron is thus transformed into a spherical polygon that is a face of the spherical polyhedron.

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Prerequisite Knowledge for Projection of a Polyhedron Face onto the Surface of a Sphere