Rounded Pyramid

10Apr07

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 x2+y2+z2+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!

Rounded Pyramid
What’s pictured is the surface x2+y2+z2+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…

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