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 - , , and  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
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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  axis* to indicate that it extends out from the origin in that direction.

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Note in Fig. B  that this  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  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  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  axes because the primary  axis is parallel to the X axis which has

no Y or Z value.

The  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

 axes because the primary  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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Geometry rules! - Introduction - Symmetry elements

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