## The Action of Symmetry Groups of Platonic Solids on their Respective Vertices

##### Abstract

Platonic solids are 3-dimensional regular, convex polyhedrons. Each of the faces
are equidistant and equiangular to each other in any of the solids. They derive
their name from the ancient Greek philosopher, P lato who wrote about them in
his dialogue, the Timaeus as reported by Cornford (2014). The solids features have
fascinated mathematicians for decades including the renown geometer, Euclid: In
his Book XIII of the Elements, as rewrote by Heath et al. (1956), he successfully
determined the exact number of solids that qualify to be Platonic Solids; tetrahedron,
cube, octahedron, dodecahedron and icosahedron. In group theory, the symmetry
group of an object is the group of all transformations under which the object
remains unchanged, endowed with the group operation of composition. Due to their
inherent symmetry of these solids many mathematicians have attempted to derive
their symmetry groups. For instant, Foster (1990) who successfully enumerated
the symmetry groups of the dodecahedron and recently Morandi (2004) attempted
to compute these symmetric groups of the solids using a computer program called
Maple. Although such contributions are noteworthy, a few attempts have been made
to explore other features such as the symmetry groups of the platonic solids. Thus,
this project investigates the properties of the group action of the symmetry groups of
these platonic solids acting on their respective vertices. We embark on constructing
the symmetry groups of each of the solids then employ the orbit-stabilizer and other
theorems to determine the ranks and sub-degrees of each solid. The action of G on V
shows that tetrahedron has a rank of 2, the octahedron has a rank of 3, dodecahedron
has a rank of 6 while the cube and icosahedron have a rank of 4.