Catalan solid

Of the 13 Catalan solids, 7 have triangular faces. These are of the form Vp.q.r, where p, q and r take their values among 3, 4, 5, 6, 8 and 10. The angles ${\displaystyle \alpha _{p}}$, ${\displaystyle \alpha _{q}}$ and ${\displaystyle \alpha _{r}}$ can be computed in the following way. Put ${\displaystyle a=4\cos ^{2}(\pi /p)}$, ${\displaystyle b=4\cos ^{2}(\pi /q)}$, ${\displaystyle c=4\cos ^{2}(\pi /r)}$ and put

${\displaystyle S=-a^{2}-b^{2}-c^{2}+2ab+2bc+2ca}$.

Then

${\displaystyle \cos(\alpha _{p})={\frac {S}{2bc}}-1}$.

For ${\displaystyle \alpha _{q}}$ and ${\displaystyle \alpha _{r}}$ the expressions are similar of course. The dihedral angle ${\displaystyle \theta }$ can be computed from

${\displaystyle \cos(\theta )=1-2abc/S}$.

Applying this, for example, to the disdyakis triacontahedron (${\displaystyle p=4}$, ${\displaystyle q=6}$ and ${\displaystyle r=10}$, hence ${\displaystyle a=2}$, ${\displaystyle b=3}$ and ${\displaystyle c=\phi +2}$, where ${\displaystyle \phi }$ is the golden ratio) gives ${\displaystyle \cos(\alpha _{4})={\frac {2-\phi }{6(2+\phi )}}={\frac {7-4\phi }{30}}}$ and ${\displaystyle \cos(\theta )={\frac {-10-7\phi }{14+5\phi }}={\frac {-48\phi -155}{241}}}$.

Of the 13 Catalan solids, 4 have quadrilateral faces. These are of the form Vp.q.p.r, where p, q and r take their values among 3, 4, and 5. The angle ${\displaystyle \alpha _{p}}$can be computed by the following formula:

${\displaystyle \cos(\alpha _{p})={\frac {2\cos ^{2}(\pi /p)-\cos ^{2}(\pi /q)-\cos ^{2}(\pi /r)}{2\cos ^{2}(\pi /p)+2\cos(\pi /q)\cos(\pi /r)}}}$.

From this, ${\displaystyle \alpha _{q}}$, ${\displaystyle \alpha _{r}}$ and the dihedral angle can be easily computed. The faces are kites, or, if ${\displaystyle q=r}$, rhombi. Applying this, for example, to the deltoidal icositetrahedron (${\displaystyle p=4}$, ${\displaystyle q=3}$ and ${\displaystyle r=4}$), we get ${\displaystyle \cos(\alpha _{4})={\frac {1}{2}}-{\frac {1}{4}}{\sqrt {2}}}$.

Of the 13 Catalan solids, 2 have pentagonal faces. These are of the form Vp.p.p.p.q, where p=3, and q=4 or 5. The angle ${\displaystyle \alpha _{p}}$can be computed by solving a degree three equation:

${\displaystyle 8\cos ^{2}(\pi /p)\cos ^{3}(\alpha _{p})-8\cos ^{2}(\pi /p)\cos ^{2}(\alpha _{p})+\cos ^{2}(\pi /q)=0}$.

All of the formulae of this section apply to the Platonic solids, and bipyramids and trapezohedra with equal dihedral angles as well, because they can be derived from the constant dihedral angle property only. For the pentagonal trapezohedron, for example, with faces V3.3.5.3, we get ${\displaystyle \cos(\alpha _{3})={\frac {1}{4}}-{\frac {1}{4}}{\sqrt {5}}}$, or ${\displaystyle \alpha _{3}=108^{\circ }}$. This is not surprising: it is possible to cut off both apexes in such a way as to obtain a regular dodecahedron.