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
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.