### Family I

Each tetrahedral vertex is opposite a mid-face. Joining these creates four axes of spin - four great circles of which any face exhibits parts of three. Equivalent symmetry is created by Family III of the Octahedron, for marked upon the sphere is the spherical equivalent of the cuboctahedron.

[Ref: ‘Geodesic Math And How To Use It’ p.53]

### Family II

In ‘Geodesic Math And How To Use It’ on p.53 it states that making axes through pairs of mid-edge points creates *four* great circles of which any single face exhibits parts of three. However, I can only see three great circles using this method (after all, the tetrahedron has six edges, so using pairs of edges will only create three great circles). I presume this is a misprint. These three great circles mark on the sphere the outline of a spherical octahedron, thus creating equivalent symmetry with Family I of the Octahedron.

[Ref: ‘Geodesic Math And How To Use It’ p.53]

### Family III

This is not an official great circle familiy of the Tetrahedron as far as I know, and I have yet to find documentation for it. I devised it when trying to generate Möbius Triangles for this Platonic solid. Presumably it is not considered an official family because the poles of spin do not pass through vertices, mid points of edges, or even the centroids of a face. It was generated in the following way: each edge forms a co-incident plane that cuts through the mid-point of the opposite edge. This plane bisects the tetrahedron. The axis of the great circle is a line perpendicular to this plane that passes through the polyhedron center. Each bisected half of the tetrahedron is a reflection in this great circle plane. This generates six great circles that divide each face in six by the medians and edges. Even though this is not an official family, the symmetry of the tetrahedron is manifest by it.

### Families I & II

### Families I & III

### Families II & III

### Families I, II & III