On Sun, Sep 25, 2016 at 12:01 PM, Hans Havermann <gladhobo@bell.net> wrote:
"The volume of a regular tetrahedron (triangular pyramid with unit edges) is exactly half the volume of a square pyramid with unit edges."
This is (I think) equivalent to stating that the volume of an octahedron is four times the volume of a tetrahedron (unit edges assumed).
Yes because you can slice an octahedron into two square pyramids. The geometrical "proof" of this is that you can assemble 4 unit tetrahedrons and a unit octahedron to produce a tetrahedron with edge length two, and therefore area equal to 8 tetrahedrons. I don't have the ability to draw a picture of this here, but start with an equilateral triangle of side 2, and divide it into 4 equilateral triangles of side 1. Place tetrahedrons on the 3 of these that are in the same orientation, and then add as a second layer a fourth tetrahedron whose bottom three vertices are the points of the first 3 tetrahedrons. These four tetrahedrons outline the shape of a side-2 tetrahedron, and the space in the large tetrahedron that isn't in any of the small ones is exactly a unit octahedron. Thanks to Kate Fractal for the idea of this construction. Andy