Question

Show that each of the 12 rotations of a regular tetrahedron permutes the four vertices, and that to different rotations there correspond different permutations of the set of vertices.

Short answer

Each of the 12 rotations corresponds uniquely to a permutation of the vertices, confirming the one-to-one correspondence.

Step-by-step solution

Understand the Geometric Symmetry

A regular tetrahedron is a three-dimensional shape with four faces, all of which are equilateral triangles, and four vertices. Each rotation of the tetrahedron will result in a different permutation of its four vertices. We want to show that every rotation corresponds to a unique permutation of the vertices.

Count the Possible Rotations

A regular tetrahedron has 12 distinct rotational symmetries. This includes rotations around axes passing through any vertex and the opposite face, and rotations by 0°, 120°, and 240° around those axes.

Calculate Permutations

For any object with 4 distinct locations (like the vertices of a tetrahedron), there are 4! (4 factorial) possible permutations. Calculating 4! gives 24 permutations. However, due to the geometric constraints of the tetrahedron, only 12 rotational permutations are possible.

Match Rotations and Permutations

Each of the 12 rotations of the tetrahedron corresponds to a unique permutation of the vertices. These permutations are specific one-to-one mappings derived from the effect of the rotation on the tetrahedron's structure. To see this, consider rotating the tetrahedron through various axes and verifying that the new positions of the vertices do not repeat for any given rotation.

Verify One-to-One Correspondence

To ensure that each rotation leads to a unique permutation, check that no two different rotations result in the same arrangement of vertices. This can be done by visualizing or physically testing each rotation and mapping it to a permutation.

Key concepts

Tetrahedron

A tetrahedron is an intriguing 3D geometric shape consisting of four triangular faces, where each face is an equilateral triangle. It also has four vertices and six edges. A regular tetrahedron is special because of its perfect symmetry. All its edges are of equal length, making it one of the most symmetrical solids, which are known as Platonic solids. Understanding these properties is crucial for discussing symmetry and permutations related to its structure.

• Four faces, each with equal sides and angles.
• Four vertices, with equal angles meeting at each vertex.
• Six edges, all of the same length.

Because of this symmetry, the rotations of the tetrahedron can move the vertices into each other’s positions in a consistent way. This property is fundamental for the concept of rotational symmetry, such as the one displayed by the tetrahedron.

Permutations

Permutations refer to different arrangements of a set of objects. When exploring the rotational symmetry of a tetrahedron, permutations play a crucial role. Specifically, when the tetrahedron is rotated, the four vertices can be realigned in different sequences. Let's break down how permutations apply here:

• Consider four vertices labeled as 1, 2, 3, and 4.
• The total number of permutations for these vertices is 4!, which means 24 possible ways.
• However, not all these permutations result from the rotational symmetry of a tetrahedron, as very specific movements maintain the overall integrity and symmetry of the shape.

Rotating a tetrahedron results in only certain permutations, and among the 24 possible, there are just 12 that correspond to its symmetry. This indicates that not every permutation corresponds to a possible rotation, highlighting the unique nature of geometric constraints.

Geometric Symmetry

Geometric symmetry refers to the balanced and proportionally informed characteristic of shapes, and how they can maintain form through various transformations like rotations. For the tetrahedron, geometric symmetry involves understanding how each face and vertex can map onto another through rotations. This intrinsic symmetry allows the tetrahedron to have:

• 12 distinct rotational symmetries.
• Multiple axes of rotation, typically through a vertex opposite a face.
• Rotations of 0°, 120°, and 240° around these axes.

Geometric symmetry provides insight into why specific rotations yield unique vertex arrangements. Tetrahedron rotations are not arbitrary; each rotation must map vertices to vertices in ways that preserve the geometric structure. This principle demonstrates why every distinct rotation has a corresponding unique permutation of vertices that doesn’t repeat. It’s a smart accomplishment of nature’s balance in geometry, offering a fascinating glimpse into the beauty of mathematical structures.