Associating the symmetry of the Platonic solids

Symmetry elements

Geometers use the standard X,Y,Z Cartesian coordinate system to describe the

symmetry of polyhedra. As shown in Fig. A the location of each vertex of a cube

with an edge length of one is described in terms of its length - X, height - Y, and

depth - Z from the origin - 0,0,0.

For example, the vertex labeled 1,1,1 indicates that it is one unit in length, height,

and depth from the origin. The line shown connecting this vertex to the origin is

called the [111] axis* to indicate that it extends out from the origin in that direction.

Note in Fig. B that this [111] axis passes through the center of the purple colored

plane on a line that is perpendicular to that plane. This plane is therefore referred to

as the (111) plane*. The [111] axes are the cube's axes of 3-fold rotational symmetry.

That is, rotating the cube one complete revolution will give you three identical views

of it as you look down these axes towards the opposite corners. Likewise the (111)

planes are referred to as the cube's planes of 3-fold rotational symmetry.

The cube's [100] axes of 4-fold symmetry pass through the centers of the (100)

planes on lines perpendicular to those planes (the red plane in Fig. B is one). They are

called [100] axes because the primary [100] axis is parallel to the X axis which has

no Y or Z value.

The [110] axes of 2-fold symmetry pass through the centers of the (110) planes on

lines perpendicular to those planes (the blue plane in Fig. B is one). They are called

[110] axes because the primary [110] axis is parallel to a line passing midway between

the X and Y axes which therefore has no Z value.

If these planes are positioned so that they slice the cube in half, such as the

(100) and (110) planes do in Fig. B, they are also referred to as mirror planes. This

is because both halves of the cube on opposite sides of the plane are mirror images

of each other.

In this way the symmetry of polyhedra can be described in terms of their axes and

planes of rotational symmetry, and their mirror planes (i.e. their symmetry elements).

*Note: Axes are denoted by enclosing them in [brackets] and planes in (parentheses).

Exercise: Locate the 0,1,1 vertex of the cube pictured in Fig. A. Locate the

1,0,1 vertex. To check your answers move your mouse over the vertices.

Geometry rules! - Introduction - Symmetry elements