Is a $3D$ volume possible with only $3$ faces?

Question

I was wondering if a $3D$ volume with only $3$ faces was possible. I know that in the Euclidean space, it is technically not possible (the minimum being $4$ faces), but maybe there was some other way.

Like the moebius strip that only has one side and one face.

Maybe by bending the space?

Thank you!

Answer 1

I prepared this image for the version of your question asking for a volume with three vertices, which emphasizes mathreadler's comment "sure if you allow bending of surface":

A version with three faces is obtained by intersecting three spheres:

If you do not want to have bent faces, but you allow infinite faces: imagine a trianglular pyramid that extends infinitely in one direction. It has three faces only.

If you want to avoid bent and infinite faces, you might find a solution in sphereical geometries similar to the one in my second picture. I think that in spherical geometries, the bent faces of this shape are actually intrisically "flat" because bounded by great circles. Not completely sure though.

Answer 2

Suppose we have a polyhedron with $3$ (planar) faces $F_i = \{x\in \mathbb{R}^3:\, x.\xi_i = a_i\}$ for some $\xi_i$ of norm $1$ and some $a_i$. The result is trivial if any distinct $\xi_i$ agree (i.e., if two of the faces are parallel); otherwise, there exists some $g\in O(3)$ mapping $(\xi_1, \xi_2, \xi_3)$ to the standard basis of $\mathbb{R}^3$. Such an action preserves volume (modulo sign), so we're left with the case where the faces are parallel to the coordinate axes. The same argument works in arbitrary $\mathbb{R}^n$.

Answer 3

If you want to do something really cool, make a torus with one face.

Start with a hexagon in which opposite sides are congruent and parallel, for example by cutting opposite corners from a parallelogram. For ease of manipulation one pair of opposite sides should be much longer than the other two pairs.

Glue the long sides together so they become a single joint, with the other pairs of sides forming the jagged ends of a tube. Now give one end of the tube a half-twist and bend the ends together. You need to do some stretching and squeezing to get the fits exact, for you are creating a noneuclidean surface, but you end up with a torus derived from one hexagonal face!

For a video deminstration see here.