The ICOSAHEDRON and Dodecahedron's axis of symmetry and their circles of revolution. 

All the regular solids have associated circles that are defined when the solids are rotated about their Vertices (points where the faces join up), Faces or mid point of their Edges.

When the ICOSAHEDRON is rotated from each of its Vertices, a circle is defined like the equator around the earth. The ICOSA has 12 of these Apices (Vertices) in two pairs symmetrically opposite each other, therefore 6 circles in total are defined. As the DODECAHEDRON is the ICOSAHEDRONS duel, these circles are the same as when the DODECA is rotated on a face. [move over the image to see an animation] 

A further 10 Circles are defined when when an ICOSAHEDRON is rotated from the centre of each of its faces. The ICOSA has 20 faces in two pairs, symmetrically opposite each other, hence 10 circles are defined. As the DODECAHEDRON is the ICOSAHEDRONS duel, these circles are the same as when the DODECA is rotated on its vertices. [move over the image to see an animation] 

There are a further 15 circles of the ICOSAHEDRON which are defined when it's rotated from the mid point of it's edges. These form 120 identical ‘right angle' triangles, Illustrated by the animation on the right. [Right angle is used loosely here as in spherical geometry the angles of a triangle can add up to more than 180°] [move over the image to see an animation] 

That brings the total to 31 circles aligned to the axis of symmetry of the ICOSAHEDRON and DODECAHEDRON. 


