## Response from Prof. B.

Let's see, I should probably ask you what you think the answer should be. If we take anything in the plane of a fixed area C square units and we shift it perpendicular to itself for a distance of one unit, then we obtain a slab with volume equal to C cubic units. Similarly shifting a body in three-space with volume D cubic units in a perpendicular direction for a distance of one unit should give a four-space object with hypervolume D "hypercubic" units, right? Thus coning a square of area A so that the cone point is one unit from the plane of the square gives volume A/3 cubic units and shifting this in four-space gives a hypervolume of the same number of hyperunits, namely A/3. However if we first shift the square of area A in a direction perpendicular to itself for a distance of one unit, we obtain a three-dimensional cube again with A cubic units. Coning this in four-space gives only one-fourth of a hypercube of the appropriate sidelength, so we get hypervolume A/4. Thus it looks as if the process is not "commutative".

We could have predicted this by a lower-dimensional experience. If we start with a line segment of length L, then coning it at a distance one unit away gives a triangle of area L./2 square units, and then shifting this gives a triangular prism with volume L/2 cubic units. On the other hand shifting it first gives a rectangle with L square units, and coning this gives a pyramid of volume L/3 cubic units. Thus the non-commutativity of the process is apparent even at a lower level.

I like that problem. It's a good exercise for the expanded version of B3D, whenever we get the go-ahead to work on that.