Wikipedia:Reference desk/Archives/Mathematics/2019 February 22

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February 22

Dihedral angle of antiprisms

How does one compute the dihedral angles of the uniform antiprisms? I understand there are two different angles, those at the edges where a triangle meets the n-gon and those at the edges where two triangles meet. I need both angles for several specific uniform antiprisms because I want to know whether several convex uniform polyhedra can be arranged around an edge. Toshio Yamaguchi (talk) 15:13, 22 February 2019 (UTC)

Without going through all the calculations, one approach is to write down coordinates of all the vertices. Say, the top polygon (with n sides) has vertices $(\cos 2k\pi /n,\sin 2k\pi /n,c)$ for vertices $k=0,\ldots ,n-1,$ and the bottom has $(\cos(2k+1)\pi /n,\sin(2k+1)\pi /n,-c).$ Then, knowing that the lateral faces are equilateral triangles (so the distance between consecutive vertices on top is the same as the distances between a vertex on top and the neighboring vertices on the bottom), you can solve for c. At that point, you have the vectors that make up the edges, so you can use the cross product to find vectors normal to the lateral faces. And then finally, the angle between the normal vectors (which can be found with the dot product) is the same as the angle between the faces, which is what you're looking for. –Deacon Vorbis (carbon • videos) 19:30, 22 February 2019 (UTC)

The equation for c is given in the article (where it is called h) in the section Antiprism#Cartesian coordinates. --RDBury (talk) 22:58, 22 February 2019 (UTC)

Also, the vertices of each face lie on a sphere with center at the origin, and also the faces are regular polygons, so as a computational shortcut you can find a normal to each face by adding its vertices as vectors. --RDBury (talk) 10:49, 23 February 2019 (UTC)

Ok, I put everything in a spreadsheet and came up with the following for the angles. Let the angle between the top face and one of the side faces be a⋅2π, and the angle between adjacent side faces be b⋅2π. (In other words calculate the angles and divide by 2π to get a and b.) I get

n   a          b
3 0.304086724  0.304086724
4 0.2884337791 0.3543100081
5 0.2800342138 0.3838602364
6 0.2747206358 0.4033941426

and so on. It appears that no combinations of a and b sum to 1 for any integer value of n>2. It gets close for n=4 where a+2b=0.9970537953. --RDBury (talk) 21:52, 23 February 2019 (UTC)