Ah yes! It makes sense.
So, keeping things orthogonal and regular as possible, we see that
- line intervals are point-prisms
(x) (only one possibility);
- a square
(x2) /rectangle
(xy) are line prisms
(two possibilities);
- a cube
(x3)/square prism
(x2y)/rectangular prism
(xyz) are square/rectangle extrusions
(three possiblities).
It would, therefore, follow that there would be
four varieties/subspecies of animals in our geometric zoo of the genus/class/family of n-prisms:
But I find
FIVE:
- totally regular tesseracts
(x4) - all edges equal;
- cubic hyperprisms
(x3y) - 3 edges equal, 1 different;
- duoprisms
(x2y2) - 2 edges equal, the other 2 equal but different from former set;
- a square-rectangle Cartesian product thingy
(x2yz) (at least that's how I interpret it for now) - 2 edges equal, other 2 different; and
- an object whose side lengths are all unequal
(xyzw) - a run-of-the-mill "line-tetraprism"/Cartesian product of two differently-sized rectangles.
My logic appears to be breaking down at this point. Given that if we had a 4D measure polytope, we could discount the non-commutativity of the Cartesian product since our 4D object is rigid in R
4. But why FIVE subspecies that are topologically identical but qualitatively distinct in their metrics and not four? What am I missing?
A point may be extruded any length - say x, y, z or w linear units as above. There is only one distinct polytope that arises from this operation, namely {} (in Schlaefli notation). When the line interval is extruded, there are two possibilities: extrude it as far as exactly the length of the line interval to produce a square ({4}) or fall short or overshoot the interval length to produce a rectangle. So here we have two metrically distinct types of entities in R
2. We can then take these two toplogically identical but metrically distinct 2D objects and extrude again. The square extruded to exactly the length of the square will of course render us a cube; undershoot or overshoot the extrusion and we end up with a square prism. If we extrude the rectangle, we can extrude it to three possible lengths, namely to one of the side lengths of the rectangle or to neither side length; in the cases of extruding exactly to either side length, the operation will produce square prisms that are metrically identical in a qualitative sense to the overshooting or undershooting of the extrusion of the square. If we undershoot or overshoot the extrusion and avoid either side length measurement, then the object will have 3 distinct orthogonal measurements. Nothing more can be done here and as a result we have produced 3 distinct subpecies in R
3: a cube, a square prism and a rectangular prism. I see a pattern and a correlation here: in R
1 - only 1 possibility; R
2 - 2 possibilities; and R
3 - 3 possibilities.
I do believe I've discovered a hitherto unknown hole in my knowledge. Or have I nesciently erred somewhere?