hw6-solutions

# 2 b how many rotational symmetries does it have

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(b) How many rotational symmetries does it have? Justify your answer. (c) Your answer to (b) should be a number we’ve seen before in a similar context. Where have we seen this number before, and why are we seeing it again? Solution (a) There is one face for each face of the cube and one face for each vertex of the cube, so 6 + 8 = 14 faces. There is one edge for each edge of the cube, plus three edges for each vertex of the cube, so 12 + (3)(8) = 36 edges. There are three vertices for each vertex of the cube, so (3)(8) = 24 vertices. (b) There are 24 rotational symmetries. One way to see this is to notice that deciding which vertex a single vertex is taken to by a rotation uniquely specifies the rotation. There are 24 vertices, thus 24 rotational symmetries. (c) This is the same as the number of symmetries of the cube; in a sense a truncated cube is just a cube where the vertices have been “made larger”, i. e. expanded into small triangles. 5. Take a cube. Put a point in the middle of each face. Now draw straight lines to the middles of each of the sides of that fact, producing a plus sign + on each face. The kinked line that goes from the center of one face to the center of an adjacent face forms a bent edge on this cubical world. Thus we have created eight “bent” triangles whose vertices are the centers of the faces of the cube. (a) What is the sum of the angles for each of those triangles? (b) What is the sum of all the angles of all the triangles, and how does this compare to the sum of all the angles of eight ordinary triangles in the plane? (B+S 4.6.32) Solution.
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