Symmetry in Platonic SolidsDate: 01/24/97 at 05:22:21 From: Ian Thompson Subject: Symmetry Dr. Math, How many planes of symmetry does each of the platonic solids have? Is there an 'algorithm' for finding planes of symmetry? Do you know a good reference book for 3-D symmetry? All the best, Ian Thompson Date: 01/24/97 at 13:01:10 From: Doctor Rob Subject: Re: Symmetry I cannot answer this question in full, but perhaps a partial answer would be helpful. Every plane of symmetry must be the perpendicular bisector of a line segment (not necessarily an edge!) connecting two vertices. It must be perpendicular to every face that it intersects at an interior point. It must be perpendicular to every edge that it intersects at an interior point. It must bisect every dihedral angle formed by faces which meet at an edge contained in the plane. For the tetrahedron, there are 6 pairs of vertices, and the line segments connecting them are all edges. The planes bisecting those edges are all planes of symmetry, and are all distinct. Thus there are exactly 6 planes of symmetry for the tetrahedron. For the octahedron, there are 15 pairs of vertices. They fall into two classes: those pairs of vertices joined by an edge (12), and those diametrically opposite (3). Each plane which is the perpendicular bisector of an edge is also the same for one other edge, so there are 6 planes of symmetry in this class. Each plane in the second class works, so there are exactly 9 planes of symmetry for the octahedron. For the cube, there are 28 pairs of vertices. They fall into three classes: those pairs of vertices joined by an edge (12), those joined by two adajent edges (12), and those diametrically opposite (4). Each plane which is a perpendicular bisector of an edge is also the same for three other edges, so there are just 3 planes of symmetry of this class. Each plane from the second class bisects the face that the two adjacent edges determine and is perpendicular to that face. It also bisects the opposite face in the same way, so there are 6 planes of symmetry in this class. Each plane from the third class is not a plane of symmetry. Thus the total number of planes of symmetry for the cube is 9. Similar analyses can be performed for the dodecahedron and the icosahedron. Probably they will have the same number of planes of symmetry, since they are duals of each other, just as the cube and octahedron did. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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