On 10/12/07, Bill Thurston <wpthurston@mac.com> wrote:
... You can make a similar origami type embedded construction for the square. Make concave-upward creases on the diagonal lines, and concave-downward creases on the altitudes. Add extra vertices on the altitudes at some fixed distance from the center, with new concave- downward creases from these vertices to the two adjacent corners, and reverse the direction of the fold on the distal portions of the altitudes. This should fold into a figure in upper half-space whose base looks like a 4-pointed star in the plane. (But I haven't actually done it)
I've been attempting to add more creases to the construction above, so as to increase the (maximum attainable) volume --- so far without success. Is it the case that an isometric embedding is always approximable by polyhedral isometric embeddings? WFL