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It is well-known that for Platonic solids:
But what are such equations for tetrahedron, icosahedron and dodecahedron?
I doubt that the equations are elegant.
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.
