2. This question involves has many equivalent cases, in the sense that notjust the answer is the same but even the calculation is nearly identical. To keep the problem manageable, you should combine repeated cases, and (for full credit) you should explain why a given calculation handles many cases at the same time. That is, I'm not asking you to do the same thing dozens of times; rather, I'm asking you to explain why you don't have to. The calculation is about a polyhedron known as a regular icosahedron, whose vertices are given in this gure: (-1:07) (0'' 1) In the gure, : V52\" , a number which is is also known as "the golden ratio". (It satises the equation2 = .1) + 1, which is useful in calculations.) The gure has 12 vertices, which in the spirit of abbreviation can be described as the four points (:l:>, :l:1, 0) together with the cyclic permutations of their coordinates. - (a) The icosahedron has 30 edges in two types (relative to the coordinates). There are 6 type A edges that are parallel to one of the three axes, and 24 type B edges that are diagonal. Prove that all 30 edges have the same length. - (b) The icosahedron has 20 triangular faces, again in two types. There are 12 types C faces that have one A edge and two B edges, and 8 type D faces that have three B edges. Calculate the interior angle when two of these faces meet and show that it is always the same. (Hint: There is such an angle at every edge, but there are really only two types of cases. Also, from looking at the picture, should the interior angle be acute or obtuse?) - (c) Find the volume of this icosahedron and compare it to the volume of the circumscribed sphere. (Hint: Each triangular face is the base of a tetrahedron whose apex is the origin. These 20 tetrahedra make the icosahedron, and I discussed the calculating the volume of a tetrahedron in class.)