A question on the disnub mentions
- golden ($x^2-x-1=0$) gives the dodecahedron + much more.
- tribonacci ($x^3-x^2-x-1=0$) gives the snub cube.
- plastic ($x^3-x-1=0$) gives the snub icosidodecadodecahedron.
All of these polyhedra can be built by generating 3D points in root $R$'s number with values $a + b R^c$ for integers $a$, $b$, $c$. After that, pick points with the same norm and use those to make a polyhedron.
Is there a nice $R$ for generating the snub dodecahedron?
Is there an interesting polyhedron generated by the number field of the Narayana cow sequence constant (root of $x^3-x^2-1=0$)? I found many pretty ones with octahedral symmetry, but none particularly noteworthy.
Take $R$ as the real root of ($x^3-6 x^2 + 4 x-2=0$). This value deserves a name, since $(R^{24} -24) - e^{\pi \sqrt{163}} <10^{-14}$. That $R$ can generate this figure, which seems more interesting than polyhedra I made with Narayana's cow.
When generating from $a + b R^c$, the bounds on $a$, $b$, $c$ matter. If all values are between -3 and 3, then we might say a particular polyhedron is easily generated by $R$. The higher the bound, the greater the difficulty. With a bound of 17, $$(0, 5, 17), (1, 12, 13), (3, 4, 17), (3, 7, 16), (5, 8, 15), (7, 11, 12), (8, 9, 13)$$ all have the same norm ($\sqrt{314}$), and can generate this figure:
What $R$ values yield a particularly high number of vertices with a relatively low bound? Are there $R$ values that make highly interesting polyhedra?