Maths - Symmetry

The property of a geometry of being the same in all directions.

A symmetry operation will not change the object.

For example, if an object has mirror symmetry, then the operation of reflecting it on a mirror will not change it. But there are other types of symmetry than mirror symmetry which we will show on this page.

2D Symmetry

haxagon

As an example in 2 dimensions, we could do an operation on this pentagon which maps:

a —» b
b —» c
c —» d
d —» e
e —» a

In other words we can rotate it by a fifth of a turn. Since the object is completely symmetrical in this way we will not be able to distinguish the new position from the original, they are identical. There are other symmetrical operations that we can do in this object, for instance, rotate it by two fifths of a turn.

3D Symmetry - Platonic Solids

In 3 dimensions there are platonic solids or regular polyhedron, which are solid 3 dimensional objects whose outer surface is made up of flat polygons, if we restrict ourselves to shapes whose faces are all the same then there are only 5 types.

name   vertices edges faces
Tetrahedron tetrahedron 4 6 4
Cube
(Exahedron)
exahedron 8 12 6
Octahedron Octahedron 6 12 8
Icosahedron Icosahedron 12 30 20
Dodecahedron Dodecahedron 20 30 12

There is a duality between pairs of these solid objects.

For instance

Eulers Theorem on Polyhedron

v - e + f = 2

Where:

Analysis of Rotation Symmetry

On this page we take the example of a cube and show how we can analyse its rotations using a permutation group.


metadata block
see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 

cover Applied Geometry for Computer Graphics...

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2017 Martin John Baker - All rights reserved - privacy policy.