Symmetry in Platonic Solids

Date: 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
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