quickfur wrote:Now look at something like the snub square antiprism. There are 4 layers of vertices, the top and bottom are o4x and x4o. To fix the points in the middle two layers, you need to simultaneously solve for the heights of the x4o (resp. o4x's) and the height of the middle layers, with unit edge lengths constraining the height between the middle layers and between the middle layer to the outer layer. There is no direct constraint that lets you solve for one height first, because the circumradius of the middle layers is dependent on the height. Almost all of the parts are interdependent, so I'm expecting that you need to solve at least a cubic equation in order to get everything fixed. So in a sense, the snub square antiprism is somehow "inherently" more complex than the bilunabirotunda or the triangular hebesphenorotunda.

I actually tried to compute its coordinates in Mathematica and you're right: the set of 3 quadratic equations needed to set constraints for it leads to a cubic equation.

I set coordinates of the bottom layer to (0,+-1,+-1), second layer to (x,+-y,0) and (x,0,+-y), third layer to (z,+-y*Sqrt(2)/2,+-y*Sqrt(2)/2) and fourth layer to (x+z,+-Sqrt2/2,0) and (x+z,0,+-Sqrt2/2) and tried to solve by setting all distances in triangle (0,1,1)-(x,y,0)-(z,y*Sqrt(2)/2,y*Sqrt(2)/2) equal to 2. I got extremely complicated cubic expression as solution, with numerical values (if all 3 are positive) of x = 0.759133, y = 1.65094 and z = 1.39426. With this values, the triangle is equilateral.

For snub disphenoid, I can take vertices (0,0,+-1), (x,+-y,0), (z,0,+-y) and (x+z,1,0) and search for equilateral triangle (0,0,1)-(x,y,0)-(z,0,y). Unlike snub square antiprism, this one is actually quadratic, though with very complicated expressions.

{x -> 1/2 (2 Sqrt[-1 + Sqrt[7 - 2 Sqrt[2]] + Sqrt[

2 (7 - 2 Sqrt[2])]] - Sqrt[

2 (-1 + Sqrt[7 - 2 Sqrt[2]] + Sqrt[2 (7 - 2 Sqrt[2])])]),

z -> Sqrt[-1/2 + Sqrt[1/2 (7 - 2 Sqrt[2])] +

1/2 Sqrt[7 - 2 Sqrt[2]]],

y -> 1/2 (1 - Sqrt[2] + Sqrt[7 - 2 Sqrt[2]])}

or numerically

{x -> 0.580704, z -> 1.40194, y -> 0.814115}