Symmetry and Counting Determine the number of ways the faces of a regular tetrah
ID: 3110903 • Letter: S
Question
Symmetry and Counting
Determine the number of ways the faces of a regular tetrahedron can be colored with three colors. Show all of your work and adequately explain your reasoning for your solution.
A.Type of problem: Identify the type of problem. Does this problem involve computations, problem solving, proving, or a mix? Explain.
B.Solution to problem where relevant:
1.Correctly solve the problem using a valid method of proof or verification using mathematical reasoning.
2.Write your solution using a logical sequence of steps. The proof or verification should include a comprehensive solution with a logical and sequential flow for the reader.
C.Description of method: In 1–2 paragraphs, describe the method you used. Explain why you chose that method.
D.Approach to problem: In 1–2 paragraphs, discuss the motivation behind your solution, including what research you conducted and how you approached the problem.
Explanation / Answer
This problem can be solved by using group theory and Burnside's Lemma.(problem soving)
Firstly we have to count rotationally distinct edge colorings (both proper and improper) in a regular tetrahedron with three colors.
First the improper colorings. There are 3^6 colorings in all. There are twelve rotations of a tetrahedron:
The identity rotation is okay for every coloring. There are 3^6 colorings invariant under it.
A 120-degree rotation requires all edges incident with the endpoint of the altitude around which we're rotating to be the same color, and all the rest of the edges to be another color. There are 3^2 colorings invariant under such a rotation.
The colorings invariant under an 180-degree rotation are such that in the pair of non-intersecting edges defining the rotation, the edges have any colors whatever, and in the other two pairs, both edges have the same color. For any such rotation, there are therefore 3^4 rotations invariant under it (because we need to color: the first edge in the defining pair, the second edge in it, the edges in the first of the remaining pairs and the edges in the second one).
Therefore, by Burnside's lemma, we have
1/12(1×3^6+8×3^2+3×3^4)=87
rotationally distinct, improper edge colorings of a regular tetrahedron with three colors.
For the proper colorings, there are six of them in all. In each of the three pairs of non-intersecting edges, both edges have to be the same color, and each of the pairs has to have a different color. Every such coloring is proper, so there are 3!=6 proper colorings.
Each of them is invariant under the identity rotation. None of them is invariant under a 120 degree-rotation, and all of them are invariant under a 180 degree-rotation. Therefore, there are
1/12(1×6+8×0+3×6)=2
rotationally invariant, proper edge colorings of a regular tetrahedron with three colors.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.