Marking a Physical Sphere with a Projected Platonic Solid

Carolyn Yackel
Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture (2009)
Pages 123–130 Regular Papers


Although mathematicians often use coordinates for the vertices of the Platonic solids, a physical sphere is not a priori situated within this same coordinate system. In this paper, we describe how to locate points relative to one another on the surface of a sphere, in order to mark vertices and edges of each of the spherically projected Platonic solids without first coordinatizing. While certain methods for the cube and octahedron are standard and the tetrahedron method is known in the temari community, and is clear to any mathematician, in this paper, one procedure that leads to all three is given as a necessary first step to finding the dodecahedron. The author believes the dodecahedron and icosahedron procedures to be original to Western language scholarship. While some steps are theoretically exact, others only approximate; however at one step the approximation can be made as exact as one desires. Furthermore, the procedures allow for making adjustments to account for the fact that physical balls always fail to be perfect spheres.