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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

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    $\begingroup$ Multiplying the coordinates by a (non-negative) number "scales" the figure's edge lengths by 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 $r$. $\endgroup$ – Blue Jul 11 '13 at 14:59
  • $\begingroup$ @Blue, Thanks that makes perfect sense. I just wanted to confirm as I have seen code that is just multiplying the coordinates by the radius, which didn't make sense to me. Please make it an answer and I shall accept. $\endgroup$ – Aly Jul 11 '13 at 15:03
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Converting comment to answer, as requested.


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 $\frac{r}{2\sin(2\pi/5)}$.

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.

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