On 2015-10-13 09:09, James Propp wrote:
Another 12-face solution is a throwing star made by joining four very acute triangular pyramids so that their equilateral sides form a tetrahedron.
That's a caltrop. Inferior aerodynamics.
But that'll be moot if Dan's 8-face solution checks out.
Dan wrote:
I think 8 faces is possible:
Connect two opposite edges of a regular tetrahedron ABCD with a segment I,
I assume that Dan has in mind that I joins the midpoints of those two edges (not that it matters).
and let the 1/3 and 2/3 points of I be 2 new vertices E and F.
Now throw away all faces and the 2 edges that I connects, and draw the 8 new edges between each of E and F with A, B, C, D. The total of 12 edges define 8 triangular faces.
I am having trouble picturing this. Can Dan or someone else suggest an alternative way to see this polyhedron (say by gluing pieces together)?
Jim
Instead of Jim's excavated tetrahedron, you could make nonstandard endododecahedra by, in the octahedral-tetrahedral tessellation, bulging the octahedra into deltoidal icositetrahedra. -rwg