How to prove there are exactly eight convex deltahedra? A deltahedron is a polyhedron whose faces are equilateral triangles. It is well-known that there are exactly eight convex deltahedra, and it is easy to find out that this was first proved by Freudenthal and van der Waerden in 1947.
Unfortunately, the paper is in a rather obscure journal , and also is written in Dutch. (Freudenthal, H; van der Waerden, B. L. (1947), "Over een bewering van Euclides ("On an Assertion of Euclid")", Simon Stevin 25: 115–128).  I was not able to obtain this article.  I have spent a lot of time searching elsewhere for proofs. Most books and papers that I looked at that discussed the matter just referred back to the Freudenthal-van der Waerden paper.  The only proof I found was quite ad-hoc and also unpersuasive: it depended on a lot of rather handwavy assertions about the geometric form of a deltahedron that I found not at all obvious.
If you have seen the Freudenthal-van der Waerden proof, how does it go?  If you have not, but you have an idea for how to prove this, I would be glad to see that too.
 A: Our library had a copy of the Freudenthal/van der Waerden paper. The proof is straightforward and uses only elementary geometric arguments. My native tongue is German, but I could easily make out what they are saying. I have put up a pdf-version of the paper here, so you can see for yourself.
[Addendum: The paper is no longer on Blatter's site, so I have placed a copy on my own server, where it is likely to remain for a long time. —MJD]
A: According to Math Reviews, the deltahedra are discussed in Chapter 8 of A R Rajwade, Convex polyhedra with regularity conditions and Hilbert's third problem, Texts and Readings in Mathematics, 21, Hindustan Book Agency, New Delhi, 2001. viii+120 pp. ISBN: 81-85931-28-3, MR1891668 (2003b:52007). 
A: You even could embed your quest into the search for convex regular faced polyhedra (i.e. with any type of regular faces). Those encounter the regular ones, the archimedeans, the prisms and antiprisms, plus the 92 Johnson solids. - The latter list from 1966, in those days a mere conjecture, shortly thereafter had been confirmed by an exhaustive research on partial facial complexes, cf. e.g. Zalgaller.
Thus given these sets your quest becomes the mere enumeration of the subset of deltahedra. These then are


*

*Tetrahedron (4 triangles, regular)

*Triangular Bipyramid (6 triangles, J12)

*Octahedron (8 triangles, regular)

*Pentagonal Bipyramid (10 triangles, J13)

*Snub Disphenoid (12 triangles, J84)

*Triaugmente Triangular Prism (14 triangles, J51)

*Gyroelongated Square Bipyramid (16 triangles, J17)

*Icosahedron (20 triangles, regular)


Thus a more direct proof could be done in the same sense, but restricting directly to regular triangles only.
--- rk
