What characteristics help you determine whether a polyhedron is regular or irregular? How can you check for symmetry and rotational symmetry of polyhedra?

This article is a 10 dollar wording, so you may have to read some of it twice, but it is very accurate.

http://en.wikipedia.org/wiki/Regular_polyhedron

To determine whether a polyhedron is regular or irregular, you need to examine certain characteristics. Here are some key factors to consider:

1. Faces: In a regular polyhedron, all the faces are congruent (identical) regular polygons. If you notice that the faces have different shapes or sizes, the polyhedron is irregular.

2. Edges: In a regular polyhedron, all the edges are congruent (identical) in length. If there are varying edge lengths, it indicates irregularity.

3. Vertices: In a regular polyhedron, all the vertices have the same number of faces meeting at each vertex. If you observe vertices where the number of faces meeting differs, the polyhedron is irregular.

To check for symmetry in a polyhedron, you can employ the concept of planes of symmetry. A plane of symmetry divides the polyhedron into two congruent halves. If you find multiple planes of symmetry that divide the polyhedron equally, it exhibits symmetry.

Rotational symmetry refers to the ability of a polyhedron to rotate around a central axis and still appear the same. To determine rotational symmetry, carry out the following steps:

1. Identify an axis around which the polyhedron can rotate without changing its appearance.

2. Count the number of times the polyhedron appears the same during one complete rotation (360 degrees).

- If it appears the same once during the full rotation, it has rotational symmetry of order 1.
- If it appears the same twice during one complete rotation, it has rotational symmetry of order 2.
- If it appears the same three times during one complete rotation, it has rotational symmetry of order 3.
- And so on...

By analyzing the faces, edges, and vertices, as well as checking for planes of symmetry and rotational symmetry, you can determine the regularity or irregularity of a polyhedron.