Intereting Posts

Proving the functional equation $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ from the Poisson summation formula
Jointly continuous of product in $B(H)$
Are there any random variables so that E and E exist but E doesn't?
Meaning of “a mapping preserves structures/properties”
Checking whether a graph is planar
Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$
Greibach normal form conversion
Are the any non-trivial functions where $f(x)=f'(x)$ not of the form $Ae^x$
Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
How to prove that the numeric series $S := \sum_{n=0}^{\infty} x^n=\frac{1}{1-x}\text{ for any } x<1$
writing $pq$ as a sum of squares for primes $p,q$
For given prime number $p \neq 2$, construct a non-Abelian group with exponent $p$
Recursive solutions to linear ODE.
Uniqueness of a configuration of $7$ points in $\Bbb R^2$ such that, given any $3$, $2$ of them are $1$ unit apart
How to prove that the center of a group is not a maximal subgroup?

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.

- Simulating uniformly on $S^1=\{x \in \mathbb{R}^n \mid \|x\|_1=1\}$
- Equation of a rectangle
- New size of a rotated-then-cropped rectangle
- Area of Shaded Region
- The smallest 8 cubes to cover a regular tetrahedron
- Determining similarity between paths (sets of ordered coordinates).

- A triangle with vertices on the sides of a square, with one at a midpoint, cannot be equilateral
- Equation of a rectangle
- How fat is a triangle?
- Minimum operations to find tangent to circle
- find arc between two tips of vectors in 3D
- Calculating the area of an irregular polygon
- The shape of Pringles potato chip
- Dividing open domains in $\mathbb R^2$ in parts of equal area
- Inequality of length of side of triangle
- what's the equation of helix surface?

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]

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).

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