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Symmetry elements |
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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. |
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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 |
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|
 |
 |
Required
parts |
6 large
triangles |
9 small
squares |
4 rectangles |
14 right
triangles |
33 pinges |
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For example, the vertex labeled 1,1,1 indicates that it is
one unit in length, height, |
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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. |
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Note in
Fig. B that this [111] axis passes through the center of the purple colored |
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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. |
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That is,
rotating the cube one complete revolution will give you three identical views |
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of it as you
look down these axes towards the opposite corners. Likewise the (111) |
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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). |
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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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Geometry rules!
- Introduction - Symmetry elements |