A remarkable result on the ArXiv tonight, due to Kedlaya, Kolpakov, Poonen, and Rubinstein, at arxiv.org/pdf/2011.14232…. One of the highlights is that they fully classify tetrahedra (= triangular pyramids) with rational dihedral angles. Here's what they prove... (1/n)
A tetrahedron is a 3-dimensional shape built out of 4 triangular faces. These faces meet at 4 vertices, and 6 edges. At each edge, two faces meet at an angle. These are called the dihedral angles of the tetrahedron. (2/n)
(Excuse the pic ripped off from The Math Forum in the last tweet.) Knowing all 6 dihedral angles is really enough to know the tetrahedron. Those 6 numbers determine the similarity class of the tetrahedron, just as the 3 angles of a triangle determine its similarity class. (3/n)
So what could those 6 angles be? There are some constraints. In their recent paper, K-K-P-R determine all possibilities where the angles are rational numbers of degrees, e.g. 60 degrees or 45 degrees or 120 degrees or 1/7 of a degree or whatever. (4/n)
They find two infinite families. One with angles: (90, 90, 180-2x, 60, x, x), for *an rational angle x between 30 and 90 degrees. The other with angles: (150-x, 30+x, 120-x, 120-x, x, x), for any rational x between 30 and 60 degrees. (5/n)
The first infinite family was found by Hilbert in 1895. The second infinite family seems new. Then, they find 59 isolated examples, of which 44 seem new. One example has angles (180/7, 540/7, 60, 60, 720/7, 720/7). Sevenths! (6/n)
All this and more is in the paper at arxiv.org/pdf/2011.14232…. Including, of course, the big theorem that these are all of the examples, with links to computer code used along the way. These tetrahedra could make a wonderful poster someday. (7/7)

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