### Rounded Pyramid

**Updates**: Found an interactive Java applet with this shape (up to an affine transformation). My picture ignores four infinite “pointy dishes” that come out the corners. A nicer equation is x^{2}+y^{2}+z^{2}+2xyz=1: also see a 1933 paper by A.S. Merrill. Every conic section can be obtained as a cross-section!

A nice picture happened to come up in some problem I was working on, and since this blog has a sad lack of such, here it is!

What’s pictured is the surface x^{2}+y^{2}+z^{2}+4xyz=2(xy+yz+zx).

The surface closely resembles a triangle-based pyramid (a tetrahedron) and in fact the six edges of such a tetrahedron all belong to the surface (vertices labeled). It truly is slightly bigger than a pyramid, for example it contains the point (^{3}/_{4},^{3}/_{4},^{3}/_{4}) whereas the closest point to this in the tetrahedron is (^{2}/_{3},^{2}/_{3},^{2}/_{3}).

Unfortunately the problem I am *really* interested in has about 16 dimensions, so I can’t post a picture of it…

Filed under: geometry | Leave a Comment

## No Responses Yet to “Rounded Pyramid”