Small Stellated Dodecahedron, generating triangle vertices I have been trying to draw a small stellated dodecahedron (would post an image if I had enough rep) using OpenGL, and would like to generate the vertices programmatically. I'm looking for a way to map each of the "peak" vertices of this polyhedron to the "valley" vertices of the regular dodecahedron.
After reading everything I could find on Google (where most is about origami), I realize that in geometry (unlike when using only triangles) you just need 5 vertices to produce a pentagram, thus the definition of this polyhedron uses only 12 vertices, the pentagram "peaks", and doesn't include the 20 vertices of the original, regular dodecahedron inside. If I want my mesh to be empty in the center it seems like I need to combine the vertices of the SSD and the regular dodecahedron.
I found a useful site that lists the vertex coordinates. I have plotted some from both tables and found that they match (dodecahedron fits inside SSD), and begun to look into how to map them automatically.
Using trial and error I found that
peak vertex C ($0$, $-\phi^2$, $-\phi$) has valley vertices


*

*F ($1/\phi$, $-\phi$, $0$)

*G ($-1/\phi$, $-\phi$, $0$)

*K ($0$, $-1/\phi$, $-\phi$)

*S ($1$, $-1$, $-1$)

*T ($-1$, $-1$, $-1$).


I'm not sure if there's any useful pattern that I should know about, or if I should just create my own tables "by hand", such as peaks[0].valleys = {5, 6, 10, 18, 19 } from the above, and so on.
Triangle vertices will need to be in the right order (counter clockwise), and I will need to calculate the triangle faces' normals as well.
Is there any obvious symmetry, or something else to understand, that would make this simpler and/or more elegant, or should I just do it the hard way like I have so far? 
 A: Turns out there are indeed enough patterns to generate the 12+20 vertices pretty concisely.
I identified the 5 valley vertices for each of the 12 peak vertices and wrote them down in tables, and found the following:


*

*All vertices in the triangle based small stellated dodecahedron, use 3 out of 5 values: 0, and 4 successive $\phi$-based values in a Fibonacci-like series. For example, $0$, $1/\phi$ ($p_0$), $1$ ($p_1$), $\phi$ ($p_2$), and $\phi^2$ ($p_3$). (See Golden ratio.)

*The 12 peak vertices can be generated by rotating the sequence [0, $p_3$, $p_2$]. E.g. if the $Y$ coordinate is 0, $Z$ will be $p_3$ and $X$ will be $p_2$. For each of the 3 possibly configurations, find the 4 permutations of positive and negative for $p_2$ and $p_3$.

*The valleys' coordinates are generated per dimension. Depending on which value in [$0$, $p_3$, $p_2$] the peak has in a dimension, its valleys' 5 coordinates in that dimension will follow 1 out of 3 different sequences. The signs also follow patterns depending on the peak's value, and sign if not 0, in that dimension.

*Out of the 20 valleys, 8 are on the form ($\pm p_1$, $\pm p_1$, $\pm p_1$) and form a cube. Each peak has exactly 2 valleys that belong to this cube.

*To generate the valleys in the right order (counter clockwise) for correct OpenGL face orientation, observe if the peak has an odd or even number of negative coordinates.

*The normals follow similar rules and be deduced column-wise if we look at the valley coordinates for each (correctly oriented) face.
