Nice idea! Much more optimal (in terms of the size of the integers compared with accuracy of the angles) is to take all sign changes and even permutations of: (x,z,0) and (y,y,y) where x, y, z are three consecutive Fibonacci numbers. For instance, this dodecahedron is slightly more regular than yours, and much smaller: {{1597, 1597, 1597}, {-1597, -1597, -1597}, {1597,  1597, -1597}, {-1597, -1597, 1597}, {1597, -1597, 1597}, {-1597,  1597, -1597}, {-1597, 1597, 1597}, {1597, -1597, -1597}, {987, 2584,   0}, {-987, -2584, 0}, {2584, 0, 987}, {-2584, 0, -987}, {0, 987,  2584}, {0, -987, -2584}, {987, -2584, 0}, {-987, 2584, 0}, {-2584,  0, 987}, {2584, 0, -987}, {0, 987, -2584}, {0, -987, 2584}} I wonder whether this series gives the best rational approximations to the dodecahedron? Sincerely, Adam P. Goucher http://cp4space.wordpress.com

----- Original Message ----- From: James Buddenhagen Sent: 09/09/13 06:06 PM To: math-fun Subject: Re: [math-fun] near regular dodecahedron with integer vertices

On Mon, Sep 9, 2013 at 11:19 AM, Mike Stay <metaweta@gmail.com> wrote:

...

Is it a pyritohedron?

Yes. It is one of an infinite family given by 1 rational parameter. This one uses parameter value 1597/987 a continued fraction convergent to the golden ratio. Many curious forms including non-convex and self-intersecting arise for other parameter values, and not surprisingly, we can approximate regular ones as closely as desired.

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