Nex7 (2,1) - Diagram
Uploading this worksheet gives following error message:
Error Opening file failed
error in <expression>: label=Bsolve, exp= (((-z(NSx(1))))/pi*180°)
I used a slightly different calculation than Chris Kitrick.
Nsolve provides a quick answer to this relatively simple formula.
Nsolve doesn't find solutions for more complicated formulas in larger nexorades.
ϕ = (1+sqrt(5))/2
c = atand(1/ϕ)
a_{Bc}(x) = atand(cos(x) tan(c))
b_{Bc}(x) = asind(sin(x) sin(c))
B36(x) = atand(tan(b_{Bc}(x)/2) / sin(a_{Bc}(x)))
NSx = NSolve(B36(x) = 36°, x = 0)
Bsolve = (-z(NSx(1))) /pi * 180°
B = 56.93190841766905°
a = a_{Bc}(B) / pi * 180°
b = b_{Bc}(B) / pi * 180°
a = 18.635196378034680°
b = 26.140549500762827° (two times Kitrick's b.)
Chris Kitrick's solution
New Nexorade/Rotegrity project : 15 jan 2019 02:53:13
Chris Kitrick,
Look at the simplest example in the diagram.
There are only two unique spherical triangles (0 and 1).
Spherical triangle 0 is a right spherical triangle.
Given the symmetry spherical triangle 1 is equilateral.
The arcs are greater circle segments composed of three (3) equal sub-arcs.
The three arcs form a great circle. Considering the equal division of the arcs
the following edge angle relationship is true:
0b = x / 2
Since the right spherical triangle 0 is at the pentagon,
the B face angle is always 36 degrees.
The sides (a,b,c) of the equilateral spherical triangle 1 is 2x.
1a = 1b = 1c = 2x
Because the three contiguous arcs form a great circle the following must be true:
0A + 0A + 1B = 180
Now the only question is what is the value of the edge angle x.
By simply iterating the edge angle x and solving the two spherical triangles
to solve the 180 degree requirement at the blue vertex you arrive at the value:
x 26.14054951 0.45623866
Here are the final spherical angle values for the two triangles:
Tri a b c A B C
00 18.63519637 13.07027475 22.62775606 56.15355653 36.00000000 90.00000000
01 52.28109899 52.28109899 52.28109899 67.69288693 67.69288693 67.69288693
You'll notice the derived number (0.45623866) differs slightly from Taff's (0.456158).
Again this is the simplest case but it gives you the idea. No twisting required.
NOTE: all numbers are angles in degrees except for 0.45623866 which is the length
of the arc segment for unit radius sphere.
0.45623866 = ( 26.14054951 / 360.0 ) * 2.0 * 3.141592654
Cheers, Chris