I would like to calculate the closed form of some values relating to $U46$ and $U46'$ (especially angles and volumes).
I found this site where the values are given in term of root of equations
Premise: all solutions must depend on
$\xi:=\sqrt{\rho+\phi\rho^{2}}$ and $\rho$
where $\phi$ is the golden constant and $\rho$ is the plastic constant.
1. Dihedral angle of U46
As regards the calculation of the dihedral angles of $U46$, it is required to find the two real solutions of:
$$91125 x^6−425250x^5+888975x^4−607500x^3+53955x^2−690x+3481$$
I factored this polynomial into $$p_1(x)=135\sqrt{5}x^{3}-45(7\sqrt{5}\color{red}{+}10)x^{2}+3(147\sqrt{5}\color{red}{+}340)x-101\sqrt{5}\color{red}{-}218$$ and $$p_2(x)=135\sqrt{5}x^{3}-45(7\sqrt{5}\color{red}{-}10)x^{2}+3(147\sqrt{5}\color{red}{-}340)x-101\sqrt{5}\color{red}{+}218$$
The real solutions of $p_1(x)$ and $p_2(x)$ can be expressed in terms of $\rho$ and $\xi$
The real solutions are $x_1\approx 0.2565$ and $x_2\approx 0.9836$
Update
I solved this part: $x_1$ and $x_2$ are the solutions of
$$x^2-\frac{2(\rho+3)}{3(\rho+1)}x+\frac{51-13\rho}{45(3\rho+1)}=0$$
$$x_{1,2}=\frac{1}{3}\left(\frac{\rho+3}{\rho+1}\mp\frac{2}{\sqrt{5}}\frac{7\rho-4}{\rho+3}\right)$$
Technically instead of the $\sqrt{5}$ there should be a $\xi$, but as a result I think it is acceptable
2. Volume of U46
To calculate the volume of $U46$, however, I solved the equations given for the vertices:
"SRP"="square-root of a root of the polynomial"
C0 = 0.105398765906697216676314189282
C1 = 0.139623637868037118589881535187
C2 = 0.184961940339626297836961737414
C3 = 0.245022403774734335266195724469
C4 = 0.410877732043017261285800591418
C5 = 0.438898343962682737883306417824
C6 = 0.525190497798036582742263736641
C7 = 0.544297109869379954559620607106
C8 = 0.581416517652346986253630835588
C9 = 0.695729283407366307710093980810
C10 = 0.770212901572770918008459461110
C11 = 0.835352921275403426299975515997
C12 = 0.955174841912397215845421198524
C13 = 1.02031486161502972413693725341
C14 = 1.10660701545038356899589457223
C0 = SRP: 4096 x^6 - 5120 x^5 + 1536 x^4 - 512 x^3 + 544 x^2 - 96x + 1
C1 = SRP: 4096 x^6 - 5120 x^5+ 3840 x^4 - 1792 x^3 + 480 x^2 - 60x + 1
C2 = SRP: 4096x^6 - 1024x^5- 1024x^4 + 64x^3 + 80x^2 - 32x + 1
C3 = SRP: 4096x^6 - 2048x^5- 1536x^4 - 704x^3 - 96x^2 - 8x + 1
C4 = sqrt(6 * (12 - cbrt(12*(9 + sqrt(69))) - cbrt(12*(9 - sqrt(69))))) / 12
C5 = SRP: 4096x^6 - 5120x^5 + 1536x^4 - 512x^3 + 544x^2 - 96x + 1
C6 = SRP: 4096x^6 - 7168x^5+ 5120x^4 - 1664x^3 + 96x^2 + 24x + 1
C7 = sqrt(6*(2+cbrt(4*(101 + 15sqrt(69)))-cbrt(4(15*sqrt(69) - 101)))) / 12
C8 = SRP: 4096x^6 - 5120x^5+ 3840x^4 - 1792x^3 + 480x^2 - 60x + 1
C9 = SRP: 4096x^6 - 10240x^5+ 8960x^4 - 3648x^3 + 640x^2 - 20x + 1
C10 = SRP: 4096x^6 - 1024x^5 - 1024x^4 + 64x^3 + 80x^2 - 32x + 1
C11 = SRP: 4096x^6 - 7168x^5+ 5120x^4 - 1664x^3 + 96x^2 + 24x + 1
C12 = sqrt(3 * (3 + cbrt(12 * (9 + sqrt(69))) + cbrt(12 * (9 - sqrt(69))))) / 6
C13 = SRP: 4096x^6 - 2048x^5 - 1536x^4 - 704x^3 - 96x^2 - 8x + 1
C14 = SRP: 4096x^6 - 10240x^5+ 8960x^4 - 3648x^3 + 640x^2 - 20x + 1\
The solutions are: $$c_{0}=\frac{1}{2\rho^{3}\xi}\qquad c_{1}=\frac{1}{2\rho^{2}\xi}\qquad c_{2}=\frac{1}{2\rho\xi}\qquad c_{3}=\frac{1}{2\xi}\qquad c_{4}=\frac{\sqrt{2-\rho}}{2}$$ $$c_{5}=\frac{\xi}{2\rho^{3}}\qquad c_{6}=\frac{\phi\rho}{2\xi}\qquad c_{7}=\frac{\rho\sqrt{2-\rho}}{2}\qquad c_{8}=\frac{\xi}{2\rho^{2}}\qquad c_{9}=\frac{\phi\rho^{2}}{2\xi}$$ $$c_{10}=\frac{\xi}{2\rho}\qquad c_{11}=\frac{\rho\xi}{2\phi}\qquad c_{12}=\frac{\sqrt{2\rho+1}}{2}\qquad c_{13}=\frac{\xi}{2}\qquad c_{14}=\frac{\rho^{2}\xi}{2\phi}$$
Now the long part will be using these coordinates to get the volume, but it's just long and tedious work, it's not complicated
3. I'm having difficulty regarding the calculation of volume of $U46'$
C0 = 0.03581639493949013983380209116851
C1 = 0.0474466215401124369879652440389
C2 = 0.105398765906697216676314189282
C3 = 0.112586641242744945279481298926
C4 = 0.134722390669839585134028153001
C5 = 0.37604495239790042473910409039502
C6 = 0.438898343962682737883306417824
C7 = 0.486344965502795174871271661863
C8 = 0.537942209374520181919397458719
C9 = 0.544297109869379954559620607106
C10 = 0.608453514277639159564031071848
C11 = 0.621067356172634760005299814864
C12 = 0.656883751112124899839101906033
C13 = 0.725451430403416911803282769524
C14 = 0.822739079380407825858706772981
C15 = 0.844874828807502465713253627056
C16 = 0.870408778751185966979274045269
C17 = 0.89232145034761490270121887109501
C18 = 0.928137845287105042535020962264
C19 = 0.957461470050247410992734925983
C20 = 0.984498466675539584303135162243
C21 = 1.173805071579957404913880606454
C0 = SRP: 102400x^6 - 66560x^5+ 19456x^4 + 8896x^3 - 2768x^2 - 776x + 1
C1 = SRP: 102400x^6 - 10240x^5 - 18944x^4 - 13184x^3 - 64x^2 - 444x + 1
C2 = SRP: 4096x^6 - 5120x^5+ 1536x^4 - 512x^3 + 544x^2 - 96x + 1
C3 = sqrt(30 * (cbrt(4 * (7157 + 885 * sqrt(69)))- cbrt(4 * (885 * sqrt(69) - 7157)) - 28)) / 60
C4 = SRP: 102400x^6 - 143360x^5 + 97536x^4 - 40128x^3 + 1408x^2 - 68x + 1
C5 = SRP: 102400x^6 - 168960x^5+ 84736x^4 - 18432x^3 + 1808x^2 - 72x + 1
C6 = SRP: 4096x^6 - 5120x^5+ 1536x^4 - 512x^3 + 544x^2 - 96x + 1
C7 = SRP: 102400x^6 - 240640x^5+ 226816x^4 - 114176x^3 + 33152x^2 - 5304x + 361
C8 = SRP: 1478656x^6- 2528256x^5 + 694016x^4 - 24064x^3 - 240x^2 - 8x + 1
C9 = sqrt(6 * (2 + cbrt(4 * (101 + 15 * sqrt(69)))- cbrt(4 * (15 * sqrt(69) - 101)))) / 12
C10 = sqrt(30 * (22 + cbrt(4 * (367 + 15 * sqrt(69)))+ cbrt(4 * (367 - 15 * sqrt(69))))) / 60
C11 = SRP: 102400x^6 - 66560x^5+ 19456x^4 + 8896x^3 - 2768x^2 - 776x + 1
C12 = sqrt(15 * (31 - cbrt(4 * (101 + 15 * sqrt(69))) + cbrt(4 * (15 * sqrt(69) - 101)))) / 30
C13 = SRP: 1478656x^6- 2192384x^5 + 980736x^4 - 134656x^3 + 5200x^2 + 88x + 1
C14 = SRP: 102400x^6 - 10240x^5- 18944x^4 - 13184x^3 - 64x^2 - 444x + 1
C15 = SRP: 102400x^6 - 143360x^5 + 27136x^4 + 17472x^3 - 352x^2 - 448x + 121
C16 = SRP: 1478656x^6- 2192384x^5 + 980736x^4 - 134656x^3 + 5200x^2 + 88x + 1
C17 = SRP: 102400x^6- 143360x^5 + 97536x^4 - 40128x^3 + 1408x^2 - 68x + 1
C18 = SRP: 102400x^6 - 240640x^5+ 226816x^4 - 114176x^3 + 33152x^2 - 5304x + 361
C19 = SRP: 102400x^6- 143360x^5 + 27136x^4 + 17472x^3 - 352x^2 - 448x + 121
C20 = SRP: 102400x^6- 168960x^5 + 84736x^4 - 18432x^3 + 1808x^2 - 72x + 1
C21 = SRP: 1478656x^6- 2528256x^5 + 694016x^4 - 24064x^3 - 240x^2 - 8x + 1
The only values I found are:
$$c_{2}=\frac{1}{2\rho^{3}\xi}\qquad c_3=\frac{1}{2}\sqrt{\frac{5\rho-6}{4\rho+7}}\qquad c_{6}=\frac{\xi}{2\rho^{3}}\qquad c_9=\frac{1}{2}\sqrt{\frac{\rho+3}{2\rho+1}}\qquad c_{10}=\frac{1}{2}\frac{1}{\sqrt{1-\rho}}$$ $$\qquad c_{12}=\frac{1}{2}\sqrt{\frac{4\rho+1}{2\rho+1}}$$
The others have these properties:
$$c_0 c_{11}=\frac{\rho-1}{4(2\rho+1)}\qquad c_1 c_{14}=\frac{2-\rho}{4(\rho+3)}\qquad c_4 c_{17}=\frac{\rho-1}{4(2-\rho)}\qquad c_5 c_{20}=\frac{1}{4(2-\rho)}$$ $$c_7 c_{18}=\frac{10\rho-3}{4(7-\rho)}\qquad c_8 c_{21}=\frac{4\rho+5}{4(16-9\rho)}\qquad c_{13} c_{16}=\frac{4\rho+5}{4(16-9\rho)}\qquad c_{15} c_{19}=\frac{5\rho+3}{4(3\rho-1)}$$ $$\frac{c_{21}}{c_{13}}=\phi\qquad \frac{c_{16}}{c_8}=\phi$$
I need some help finding the some of them.
Both solids have symmetry Chiral Icosahedral, if it could be useful information.
4. Generalization
I should calculate these values for some other pair of solids (which also have more complicated equations).
For each of these pairs I know the characteristic constant that allows the solutions of the equations to be represented in a compact way. I would need a method to quickly find solutions in terms of the characteristic constant.