Take a solid three-dimensional object. Facetize its surface. Now, extrude each facet one unit in a direction perpedicular to the three-space in which the object sits.
The analogous thing down a dimension is to take a solid, planar shape. Polygonalize its boundary. Extrude the boundary one unit perpendicular to the plane of the object.
If you duplicate the solid polygon on both top and bottom, you end up enclosing a three-dimensional volume. By enclosing, I mean that the shape divides three-space into two regions, a bounded interior and an unbounded exterior.
If you did the same thing in four dimensions, you will have enclosed a four-dimensional volume.
My problem is that I don't want to have to deal with the solid three-dimensional object. I only want to deal with its facetized surface. If I start from a point, taper each 2-facet of my 3-object up to full size as I extrude in the w-direction, then taper back to a point as I extrude further, have I enclosed a four-volume? Effectively, this is turning each facet into a bipyramid and all bipyramids share the same "upper" vertex and the same "lower" vertex.
The dimensional analogy says yes. My intuition says yes. All of the vectors that I've tried to cast "through" the object hit it. Am I missing something? or am I right?