For each of Problems 1-3, answer all the questions, but do it in the form of an explanation in paragraphs not using the letters labeling the questions (or even necessarily adhering strictly to their order). Instead it should be an explanation in complete sentences, maybe with some drawings. The explanation should be comprehensible to a reader who knows the tools of Part 1 but who has not attended the course or read these problems.
a) Explain clearly how 4 of the vertices of a cube can be chosen to form a regular tetrahedron.
b) If an edge length of the cube is s, what is the length of an edge of the tetrahedron?
c) Assume s = 2 and write the coordinates of all 8 vertices of the cube if the cube is in (x, y, z) space so that the center is at O = (0,0,0) and each edge is parallel to a coordinate axis.
d) In this case, what are the coordinates of the vertices of the tetrahedron?
e) What are the equations of the planes of the 4 faces of the tetrahedron?
f) Use the dot product and the normal vectors to these planes to compute the dihedral angle between any two faces. (Can you explain this in a way so that it is enough to do it once?)
Continuing with the situation in Question 1, the tetrahedron can be decomposed into four smaller "corner" tetrahedra, with edges half as long as the original edge and regular octahedron in the middle, as in Lab 2.
a) Find the coordinates of all the vertices of the octahedron.
b) Find the equations of the planes of all the faces of the octahedron. What planes are parallel?
c) Use dot product and the normals (perpendicular directions) to the planes to find the dihedral angle between two faces.
d) Based on the way the octahedron is sitting next to the corner tetrahedra, how is the dihedral angle of a regular tetrahedron related to the dihedral angle of a regular octahedron? Does this check with your answer?
Reference: http://www.mathaware.org/mam/00/master/essays/B3D/2/volume.html
a) Build a net for a cardboard model of a cube with square base ABCD.
b) Also, build a net for a pyramid with vertices ABCDE, where E is one of the other 4 vertices of the cube.
c) Build 3 congruent models of this pyramid and show how they can be fit together to form a cube congruent to the original cube. Conclude that the volume of the cube is equal to (1/3) the volume of its base area times its height.
d) Conclude that for ANY pyramid or cone that the volume = (1/3) (base area)* height.
e) Apply this to find the volume of the cube, the tetrahedron and the octahedron in Problems 1 and 2.