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It is well-known that for Platonic solids:

  • The interior of cube a.k.a. hexahedron can be described with inequality $\max\{|x|,|y|,|z|\}<a$.
  • The interior of octahedron is $|x|+|y|+|z|<a$.

But what are such equations for tetrahedron, icosahedron and dodecahedron?

I doubt that the equations are elegant.

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  • $\begingroup$ check out coxeters polyhedra. has tons of nice information on all sorts. $\endgroup$ – abel Mar 3 '15 at 3:52
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Your equations are using "max" and "absolute value" to effectively compress many inequalities into one.

For example, the octahedron inequality you give could be expanded into 8 inequalities such as $-x + y + -z < a$. Four of those eight have an even number of negations and four have an odd number of negations — each of these groups of four defines a tetrahedron. Yes, an octohedron is the intersection of two tetrahedra.

In general, we can view your inequalities as being of the form $$\vec{v}\cdot\vec{x} < 1$$ where $\vec{x}$ represents $(x,y,z)$, and $\vec{v}$ is a member of a specific set $V$ of vectors corresponding to the faces of the object.

For example, for the cube, $V$ is all cyclic permutations of $(1,0,0)$ and $(-1,0,0)$.

For the octahedron, $V$ is the eight vectors $(±1,±1,±1)$.

For the tetrahedron, $V$ is all cyclic permutations of $(1,-1,-1)$ and $(1,1,1)$.

For the dodecahedron, $V$ is all cyclic permutations of $(0,±1,±\phi)$, where $\phi=\frac{\sqrt{5}+1}{2}$ is the golden ratio.

For the icosohedron, $V$ is all cyclic permutations of $(±1,±1,±1)$ and $(0,±\phi^{-1},±\phi)$.

What these equations are really doing is expressing a relationship between a platonic solid and a standard 3-dimensional coordinate system of measurements along 3 orthogonal axes. This works well when the group of symmetries matches well and the object is aligned with the axes, and less well in other cases.

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