coordinates of icosahedron vertices with variable radius

Question

I was looking on the wikipedia page about icosahedrons and it says that for edge length $a$ the radius of the circumscribed sphere around the icosahedron is given by $r = a \times \sin(\frac{2\pi}{5})$. It then says that the vertices of an icosahedron of edge length 2 are given by:

$(0, ±1, ±\phi)$

$(±1, ±\phi, 0)$

$(±\phi, 0, ±1)$ where $\phi$ is the golden ratio.

How can I modify these coordinates to find the vertices of an icosahedron where the circumscribed sphere has variable radius?

Thanks

Answer 1

Converting comment to answer, as requested.

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Fact. Multiplying the coordinates by a number "scales" the figure's edge lengths by the absolute value of that number.

So, let's take this in stages:

• Multiply each of the given coordinates by $1/2$ to get the vertices of an icosahedron of edge length $1$.

• Then, multiply each resulting coordinate by $a$ to get the vertices of an icosahedron of edge length $a$.

• From there, you can replace each $a$ with $r/\sin(2\pi/5)$ if you must have the coordinates explicitly in terms of $r$.

In one step, that's just:

• Multiply each original coordinate by $\frac{r}{2\sin(2\pi/5)}$.

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You may or may not care to use the fact that $\sin\frac{2\pi}{5} = \frac{1}{2}\sqrt{\phi\;\sqrt{5}}$ to try and symbolically "simplify" the resulting coordinates. There's a bit (but not really much) that could be done; the whole reason Wikipedia gives the coordinates for edge-length $2$ is that that's the size with the prettiest coordinates. Since this is apparently for a computer program, the symbolic prettiness of the coordinates probably doesn't matter much.