Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 143
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
Expert verified
Each rotation corresponds to a unique permutation of vertices.
Step by step solution
01
Understand the Regular Tetrahedron
A regular tetrahedron is a 3-dimensional shape with 4 identical triangular faces, 4 vertices, and 6 edges. Each vertex is connected to the three other vertices by edges, creating a symmetrical structure.
02
Recognize Rotational Symmetries
The tetrahedron has 12 rotational symmetries. These symmetries include: retaining a vertex and rotating the opposite face in three ways, and rotations around the edges between faces.
03
Correspond Rotations to Permutations of Vertices
Each rotational symmetry can be viewed as a permutation of the set of vertices \( V = \{ A, B, C, D \} \). A permutation reorders elements in the set. List each rotation's effect on the vertices to see these permutations.
04
Apply Group Theory Basics
In group theory, these permutations form a group under composition (doing one permutation after another). The permutation group of 4 elements without fixing any is denoted \( S_4 \), the symmetric group of degree 4.
05
Identify Different Permutations
Verify that for each of the 12 rotations of the tetrahedron, there is a unique resulting permutation of the vertices. No two rotations produce the same permutation, as each is distinct within \( S_4 \).
06
Conclusion
Every rotation of the tetrahedron can be associated with a unique permutation of its vertices. The distinct rotations cover every possible permutation, confirming the connection.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Permutations
Permutations refer to the different ways in which a set of objects can be ordered or arranged. In our case, we are dealing with the four vertices of a regular tetrahedron. By applying a rotation, these vertices are shuffled into a new order, thus creating a permutation.
- Each permutation is a unique arrangement.
- Permutations are crucial in understanding how rotating the tetrahedron changes the order of the vertices.
Group Theory
In mathematics, group theory provides us with a framework to study the algebraic structures known as groups. A group consists of a set of elements equipped with a certain operation that combines any two of its elements to form another element within the same set.
Groups must satisfy four key properties: closure, associativity, having an identity element, and being commutative (though not every group has this property, it is common in simpler groups).
Groups must satisfy four key properties: closure, associativity, having an identity element, and being commutative (though not every group has this property, it is common in simpler groups).
- Closure: Combining any two elements in the group results in another element in the group.
- Associativity: Grouping of operations does not change the outcome.
- Identity element: An element that, when combined with any element of the group, does not change it.
- Inverse element: Every element has an inverse that combines to form the identity element.
Symmetric Group S4
The symmetric group, denoted as \( S_4 \), is a major player in group theory. It represents all the possible permutations of a set of four elements. For the tetrahedron's vertices, \( S_4 \) accounts for every conceivable way in which these vertices can be rearranged.
- There are 24 elements in \( S_4 \) because 4! (4 factorial) equals 24.
- Each element of \( S_4 \) represents a particular permutation of the four vertices.
3-Dimensional Geometry
3-dimensional geometry focuses on objects with depth, width, and height, such as our regular tetrahedron. This simple 3D shape consists of four triangular faces, six edges, and four vertices.
- A regular tetrahedron is symmetric and highly uniform.
- The rotational properties are a direct consequence of its geometric symmetry and balance.
