Question

Question 0.3. What is the size of the symmetry group of a regular tetrahedron? By "symmetry" here, let's mean "rigid (distance- and orientation-preserving) linear transformations of R3" --so, no flips, to make it easier. You may consider flips too if you like. Once you know the size, can you list all the distinct symmetries?

Answer 1

Let A denote the tetrahedron sitting in R3, centred at the origin, with vertices labelled 1,2,3,4. Let T denote the set of symmetries of A. There are two distinct types of symmetries in T- reflections and rotations. We will denote a symmetry s in 7T as a list (, 2, i3, i4), with i; € {1, 2,3, 4}, and where i is the vertex position that vertex j is sent to via s. For example, if rį is the symmetry that fixes vertex 1 and rotates the plane containing the remaining three vertices by 27/3 counterclockwise (viewed from vertex 1), then we write (1, 3,4, 2) uan This is the list of all 24 elements in T, with the above notation: (1,2, 3,4) (1,3,4, 2) rotation (fix vertex 1) (1,4,2,3) rotation (fix vertex 1) (4,2, 1,3 rotation (fix vertex 2) (3, 2,4,1) (2,4,3, 1) (4,1,3, 2) rotation (3, 1,2,4) rotation (fix vertex 4) (2, 3, 1,4) rotation (fix vertex 4) (2,1,4,3) identity rotation (fix vertex 2) rotation (fix vertex 3) vertex 3 rotation

(3,1,2,4) (2, 3, 1, 4) (2,1,4, 3) (3,4, 1,2) (4,3, 2, 1) (1,2,4,3) (1,3, 2, 4) (2,1,3,4) (1,4, 3, 2) (4, 1, 2, 3) (2,4, 1,3) (3, 1,4, 2) (3, 2, 1,4) (2, 3, 4,1) (4,3, 1,2) (4, 2,3, 1) (3, 4, 2,1) rotation (fix vertex 4) rotation (fix vertex 4) rotation rotation rotation reflection reflection reflection reflection reflection reflection reflection reflection reflection reflection reflection reflection It is a fact that you shall prove later in this course that each of these symmetries is a product of the reflections (2,1,3,4), (1,3, 2,4), (1,2,4,3). For example, the rotation (1,3, 4, 2) = (1,3,2,4) * (1,2,4,3) Let W4 be the set of wiring diagrams on four vertices. Define (h./2, i3, i4) > X izisis f: TW where X s the wiring diagram that has a line between j and i;: for example, X2143 is the wiring

diagram 4 XX 3 2 1 4 3 2 f is surjective: let X e W4 be a - so that h is the endpoint of line starting at 1 etc. Then, f(i, i2, is, ia) = X, so that f is surjective. Hence, because f is surjective and |T = |W4|, f is bijective. wiring diagram. Let (i, iz, ig, ia) be the list of endpoints of lines in x You can check that (2, 1, 3,4) (1, 3, 2,4) (2,3,1,4), (2,3,1,4) * (2,1,3, 4) = (3,2,1,4), (2,3, 1,4)* (2, 3, 1, 4) = (3,1,2,4), (2,1,3,4) * (1,2,4, 3) (2,1,4,3) (1,3, 2,4) (1,2,4, 3) (1,3,4, 2). It is straightforward to verify that f(s*t) = f(s)f(t), for s*t one of the five products appearing above. For example, f(2,1, 3,4) = X2134, f(1,3,2,4) = X1324, and X2134 X1324 = X2314 3 4 2 1 2 3 3 1 2 3 4 2 1 4 3 2 1