Mathematics 192
Homework (due Oct. 1)
A. Hulpke
5) Find the converse (i.e. the – not equivalent – implication in the other direction) and the contrapositive (i.e. the equivalent statement with negated parts) of each of the following statements. (You
do not need to prove them!)
a) If two integers are odd, their sum is not odd.
b) if a quadrilateral is not a parallelogram, then its diagonals do not bisect.
6) Calculate coordinates for the corners of a regular tetrahedron (you may choose the edge length
and orient it in space to your liking). Explain your calculations!
7) A truncated octahedron is obtained from an octahedron by “cutting off ” a square pyramid at
every corner.
a) Draw the planar graph of edges and corners of a truncated octahedron.
b) Verify Euler’s theorem for the truncated octahedron.
8) Consider the structure (a trefoil-knot with square diameter) shown below Count its faces, edges
and corners. (You may, if you want, assume that each L-shaped face is cut into to convex faces by an
extra edge at its inner corner. Doing so will not change the result.) Does Euler’s theorem hold for
this object?