The Action of Symmetry Groups of Platonic Solids on their Respective Vertices
Waweru, R. Brian
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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.