# Parallelepipeds

Parallelepipeds have six parallelogram faces in three equal opposite pairs. They are the simplest zonohedra. We can classify parallelepipeds into five categories according to how much structure they have:
• A "random parallelepiped" can be constructed based on any three arbitrary vectors starting at a point in space. Each vector defines one zone of four faces. Each choice of two vectors from the three defines the directions and lengths of the edges of one pair of opposite faces. (One way to think of this parallelepiped is that its edges make the picture one would draw to show that the sum of three vectors in 3-space is commutative and associative.)
• Given such a random parallelepiped with unequal edge lengths, one can adjust the lengths of the four edges in a given direction until they are of unit length, without changing any angles. Doing this for each of the three edge directions results in a random rhombic parallelepiped, or rhombohedron, which has three pairs of rhombic faces. In general, each pair has its own face angles.
• If we further require that all six rhombic faces be congruent, the angles of the lines in space must be chosen carefully. For a given rhombus, there are sometimes two different ways to make a parallelepiped from it. Here are two different rhombic parallelepipeds which can be constructed from six copies of the same rhombus: The "acute" or "pointy-shaped" rhombic parallelepiped has two opposite vertices at which three acute angles meet. The "obtuse" or "flat-shaped" rhombic parallelepiped has two opposite vertices at which three obtuse angles meet.
• Instead of constraining the edge lengths, we can constrain the angles. A parallelepiped with 90 degree angles, e.g., a "brick," is called a right parallelepiped.
• As the most structured case, we can ask both that the angles be 90 degrees and that the edge lengths be equal, and get the cube.

Exercise: Sometimes a given rhombus can be the face of both a "pointy" and a "flat" version of parallelepiped, and sometimes there is only one version. What property of the rhombus determines whether it can be used to make one or two versions of parallelepiped? Under what condition is there only one, and then which one is it ?

Answer: Work on it by imagining you were building models from paper rhombi, before looking at the answer.

Exercise: What could you build with 20 "golden parallelepipeds" ? (A golden parallelepiped has a "golden rhombus" for each face. A golden rhombus has its two diagonals in the golden ratio, so the acute face angle is about 63.435 degrees.)

Answers: (1) With 10 flat and 10 pointy golden parallelepipeds, you can do this. (2) With 20 pointy golden parallelepipeds, you can make one of these.

Exercise: Show that a pointy parallelepiped (made from six rhombi each with 60 and 120 degree face angles) can be dissected into an octahedron and two tetrahedra. From this, because parallelepipeds are space-filling, it follows that all space can be filled with octahedra and tetrahedra.

Answer: Just take the center and two opposite small tetrahedra from this well-known stellation.