
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. 
. 
Fig. A
 Cartesian coordinates 
Fig. B
 [100], [110], and [111] axes and 
of the cube's
vertices 
(100)
red, (110) blue, and (111) purple 

planes
of cubic symmetry 
. 



Required
parts 
6 large
triangles 
9 small
squares 
4 rectangles 
14 right
triangles 
33 pinges 

. 
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 3fold
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 3fold rotational symmetry. 
The cube's [100] axes of 4fold 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 2fold 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. 

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 Introduction  Symmetry elements 