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### 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

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/
```
Associated Topics:
High School Geometry
High School Polyhedra
High School Symmetry/Tessellations

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