Exact Dihedral angle for Disdyakis Triacontahedron I've tried calculating the exact dihedral angle of a Disdyakis Triacontahedron, with no success. I cannot seem to find it online either. What is the correct approach to trying to figure out this value?
Thank you.
 A: I am learning MathJax yet, i could not find any tool online to convert text into MathJax as desmos.com does but that format is not supported here..
Apologies, but i will convert it to required format in some time.

First an expression for the dihedral angle is must for this problem.

w.r.t the above figure 
Cos(Dihedral angle)  =  1 - 2*((sin(Dihedral Angle/2))^2)  = 1 - 2*((Cos pi/n)/( cos theta))^2 = 2*asin((Cos pi/n)/( cos theta))
or
sin(Dihedral Angle/2) = (cos pi/n)/(cos theta)
Its a long derivation to show this, but to cut short the steps, lets start from this
In the below figure showing a disdyakis tricontahedron, let the angle of the unitary scalen triangle which goes up in making this be as below

now there can be three dihedral angles, each calculated taking one of the vertices of this scalen triangle as point 'O' of figure 1.
cos (Dihedral angle from A) = 1 - 2((Cos pi/10)/( cos angle A/2))^2, as n = 10 around A --(i)
cos (Dihedral angle from B) = 1 - 2((Cos pi/6)/( cos angle B/2))^2, as n = 6 around B
cos (Dihedral angle from C) = 1 - 2((Cos pi/4)/( cos angle C/2))^2 as n = 4 around C
since all of the three dihehdral angles are equal, we get the folowing equation
sin(Dihedral Angle/2) = (cos pi/10)/(cos angle A/2) = (cos pi/6)/(cos angle B/2) = (cos pi/4)/(cos angle C/2)
But angle (A + B + C) = pi & we have cos pi/10 = ((5^0.5)/(2(5^0.5 - 1)) )^0.5 (=a say) & cos pi/6 = (3^0.5)/2 (=b say) & cos pi/4 = 1/(2^0.5) (=c say)
from the above equations and after following long process of simplification, we get
cos(angle A/2) = ((((b+c)^2 - a^2)(a^2 - (b-c)^2))^0.5)/(2bc) = ((33 + (125^0.5))/48)^0.5
So, from (i) --> Cos(Dihedral angle)  =  1 - 2((Cos pi/10)/( cos angle A/2))^2 = - acos((179 + (24*(5^0.5)))/241) on substitution of above values
or  Dihedral angle = pi - acos((179 + (24*(5^0.5)))/241)
Also all the angles of the scalen triangle i.e. angles "A, B & C" can be found
as angle A = 2*acos(((33 + (125^0.5))/48)^0.5) = 32.7702785 degrees
and substituting this value in above equations we get
angle B = 2*acos((241/(280 + (1280^0.5)))^0.5) = 58.2379196 degrees &
angle C = 2*acos((241/(420 + (2880^0.5)))^0.5) = 88.9918019 degrees
