#### Title

#### Date of Award

4-2022

#### Degree Type

Thesis

#### Degree Name

Master of Science

#### Department

Math and Computer Science

#### Program

Mathematics (MS)

#### First Advisor/Chairperson

Daniel Rowe

#### Abstract

In this paper the three dimensional kissing problem will be related to the Platonic and Archimedean solids. On each polyhedra presented their vertices will have spheres expanding such that the center of each of these outer spheres are the vertices of the polyhedron, and these outer spheres will continue to expand until they become tangent to each other. The ratio will be found between the radius of each outer sphere, and the radius of an inner sphere such that each inner sphere's center is the circumcenter of the polyhedron, and the inner sphere is tangent to each outer sphere. Every Platonic and Archimedean solid has a unique outer sphere to inner sphere ratio. The circumradius of the Platonic and Archimedean solids will be found by solving for the circumradius of the polyhedra's vertex figure. After the circumradius is found, the relation between the edge length of the solids, and the circumradius is converted to the radius of the outer spheres, r, and the radius of the inner sphere, R.

#### Recommended Citation

Webb, Anthony, "Kissing the Archimedeans" (2022). *All NMU Master's Theses*. 713.

https://commons.nmu.edu/theses/713

#### Access Type

Open Access