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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 orsnubdodecahedron 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/oroctahedron. 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/ordodecahedron (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

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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, andfour 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.

the tetrahedron itself is an example which has six planes of symmetry. Each plane contains

the center of the tetrahedron and one edge (and so bisects the opposite edge). These six

planes of symmetry are the same as the second group of planes listed above for the cube

and octahedron.

a chiral object with tetrahedral symmetry has no planes of symmetry. This is analogous to

the snub cube and snub dodecahedron cases described above. Interestingly, a snub

tetrahedron can be thought of as a three-colored icosahedron---red represents thetetrahedral faces; green the tetrahedral vertices, and blue the pairs of faces which replace

the tetrahedron's edges. Other examples are the tetrahedrally stellated icosahedron and

the tetrahedrally truncated dodecahedron, which are described on their own page, and

this canonical22-sided solid. The one with the fewest faces is the 12-sided tetartoid.

So far, these two cases are exactly analogous to the octahedral and icosahedral situations.

There is also a third case however:

it is possible to have the axes of symmetry of the tetrahedron with just three planes of

symmetry. These three planes of symmetry are the same as the first group of planes listed

above for the cube and octahedron---the three mutually orthogonal planes. This issometimes called pyrite symmetry as it is the symmetry of one form of the mineral pyrite, a

not-quite-regular dodecahedron called a pyritohedron. As other examples, consider

this cube with a stripe on each face, or this perforated trapezoid-faced variation on the

dodecahedron, or these compounds offour cubes and four octahedra. Be sure to observe

their three planes of symmetry.

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

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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.

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