Question

Realize all permutations of the four vertices of a regular tetrahedron by reflections in symmetry planes and rotations.

Short answer

All 24 permutations of the tetrahedron's vertices can be realized using its rotational and reflection symmetries.

Step-by-step solution

Understanding the Problem

We have a regular tetrahedron with four vertices labeled A, B, C, and D. A permutation of these vertices is a rearrangement of these labels. We need to identify all possible reconfigurations using symmetry operations like reflections and rotations.

Determining Permutations

Since a tetrahedron has four vertices, there are a total of 4! (factorial of 4) permutations, which accounts for 24 ways to rearrange the vertices.

Understanding Tetrahedron Symmetries

A regular tetrahedron has several symmetry operations including: identity rotation, rotations about axes through vertices and the center of the opposite face, and reflections in planes of symmetry. These symmetries can achieve all possible vertex permutations.

Symmetries in Detail

1. **Identity Rotation**: Keeps all vertices in their position, representing one permutation. 2. **Rotations**: There are rotations by 120° and 240° around an axis passing through a vertex and the centroid of the opposite face; each provides 3 permutations per vertex. 3. **Reflections**: There are 3 planes of symmetry, each offering 2 permutations by swapping two vertices separated by the plane.

Calculating All Permutations

Rotational symmetries contribute: 3 (axes) x 2 (angles) x 4 (vertices on axes) = 24 permutations. Reflection symmetries add more options but collectively with rotational symmetries, all 24 vertex arrangements are covered.

Key concepts

Permutations

A permutation is simply a way to rearrange a set of objects. For a tetrahedron with four vertices, each label can be rearranged in different sequences. Imagine you have the four vertices labeled as A, B, C, and D. Permutations involve changing the order of these labels.
For example, one permutation could be A-B-C-D and another could be B-A-D-C. Every different arrangement represents a unique permutation. Given four vertices, the number of permutations is calculated using the factorial, symbolized as 4!. This means calculating 4 factorial: 4 x 3 x 2 x 1, which equals 24. Therefore, there are 24 unique ways to arrange these four points.
In the context of a tetrahedron, these permutations can be achieved through symmetry operations, like rotations and reflections.

Symmetry Operations

Symmetry operations are actions that move an object in such a way that it appears unchanged. For a regular tetrahedron, symmetry operations include both reflections and rotations.
These symmetries are the tools enabling the reorganization of the tetrahedron's vertices. By applying these operations, you can find every possible permutation of its vertices. The key operations are:

• Identity Rotation
• Rotational Symmetries
• Reflection Symmetries

By using these operations, all permutations of the tetrahedron's vertices can be achieved without altering its geometric structure. This is because the operations maintain the fundamental shape of the tetrahedron while changing the vertex labels.

Rotational Symmetries

Rotational symmetries involve twisting the tetrahedron around an imaginary axis. Consider the different ways you can rotate the tetrahedron without changing its overall shape.
Rotations can be performed around axes that pass through a vertex and the center of the opposite face.

• 120° Rotation
• 240° Rotation

For each vertex, you can find three different rotations (120° and 240° in both clockwise and counterclockwise directions). Since there are four vertices offering rotational axes, you have multiple rotations that create various permutations of the vertices. The rotations alone can generate 12 of the 24 permutations.

Reflection Symmetries

Reflection symmetries flip the tetrahedron over a plane, as if mirrored. Three main planes of symmetry exist in a regular tetrahedron. Each plane involves swapping two vertices that are on opposite sides of the plane.
Imagine a mirror placed halfway through the tetrahedron, creating a reflection on the other side.

• Plane reflecting through vertex pairs (e.g., swapping vertices across the mirror plane)
• Each reflection swaps two vertices while keeping the others unchanged

This technique creates additional permutations beyond those available from rotations. By using symmetry planes, each reflection can achieve two permutations. The reflections therefore complete the set of permutations, working in tandem with rotational symmetries to cover all 24 possibilities.