Symmetry Planes

The symmetry planes of an object are imaginary mirrors in which it can be reflected while appearing unchanged. A chiral polyhedron such as the snub cube or snub dodecahedron has all the axes of symmetry of its symmetry group, but no planes of symmetry. A reflexible polyhedron has at least one plane of symmetry. If there are more than one, they meet at its center.

You can interactively explore with symmetry planes by using the cylinder intersecter.

The Nine Planes of Symmetry of the Cube and/or Octahedron

This paper model illustrates the nine planes of symmetry of the cube and/or octahedron. It contains two different types of symmetry planes:

• three of the planes are orthogonal to the three 4-fold symmetry axes; each such plane is parallel to, and halfway between, two opposite faces of the cube; these three planes are mutually orthogonal.
• six of the planes are orthogonal to the 2-fold axes; each such plane contains two opposite edges of the cube (and so is an orthogonal bisector to two opposite edges of the octahedron). Two of these planes meet each other at either 90 degrees (if they share a 4-fold axis) or 60 degrees (if they share a 3-fold axis).

A plane from the first set and a plane from the second set will meet each other at either 45 degrees (if they share a 4-fold axis) or 90 degrees (if they share a 2-fold axis).

Together the planes divide the surface of a sphere into 48 triangular regions called Mobius triangles. Each is a 45-60-90 spherical triangle with a 4-fold, a 3-fold, and a 2-fold axis at the respective corners. Eight such triangles cover one square face of the cube; six such triangles cover one triangular face of the octahedron; four cover one rhombus of the rhombic dodecahedron.

The Fifteen Planes of Symmetry of the Icosahedron and/or Dodecahedron

This paper model illustrates the fifteen planes of symmetry of the icosahedron and/or dodecahedron (which being mutually dual, share their symmetry group and symmetry planes). There is only one type of plane in this case. Each plane contains two opposite edges of the icosahedron and two opposite edges of the dodecahedron. They can be colored as five sets of three mutually orthogonal planes. Incidentally, the paper model illustrated was made from brightly colored, inexpensive construction paper (three planes each of red, blue, green, yellow, and white) but has faded to dull shades of grey over 15 years.

The fifteen planes divide the sphere into 120 Mobius triangles. Each is a 36-60-90 spherical triangle with a 5-fold, a 3-fold, and a 2-fold axis at the respective corners. Ten such triangles cover one pentagon of the dodecahedron, six cover one triangle of the icosahedron, and four cover one rhombus of the rhombic triacontahedron.

If we place a cube inside a dodecahedron or an octahedron inside a dodecahedron, we can see how the symmetry planes of the cube/octahedron relate to the symmetry axes of the dodecahedron. The three mutually orthogonal planes of the cube are one set of the five sets making up the dodecahedron's planes of symmetry.

The Planes of Symmetry of the Tetrahedron

The tetrahedral symmetry group comes in three flavors, according to which planes of symmetry are present. The three cases are best explained with a typical example for each. in each of the following three cases, observe that all seven axes of tetrahedral symmetry are present.

So far, these two cases are exactly analogous to the octahedral and icosahedral situations. There is also a third case however:

The Planes of Symmetry of a Prism or Antiprism

An n-gonal prism has two kinds of planes of symmetry:

• one plane orthogonal to the n-fold axis, halfway between the two n-gons, and
• n planes each containing a 2-fold axis and the n-fold axis. If n is odd each plane contains one edge (between two adjacent squares) and bisects the opposite square. If n is even, half the planes contain two opposite edges and the other half bisect two opposite squares.

The antiprism leaves out the first plane and has its planes of symmetry half-way between its 2-fold axes.

Exercise: Which of the four symmetry types listed above, if any, does the pentagonal pyramid fall under ?

Answer: None of the above; pyramids have the same n planes of symmetry as the antiprism, but no 2-fold axes of symmetry.