5
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The formula for the j-function which employs polynomial invariants of the icosahedron,

$$j(\tau)=-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5}$$

where,

$$r^{-1}-r=\frac{\eta{(\tau/5)}}{\eta{(5\tau)}}+1$$

and $r = r(\tau)$ is the Rogers-Ramanujan continued fraction and the modular form Dedekind eta function $\eta{(\tau)}$ is well-known.

Question: Is there a similar relationship between any of the six regular 4D polytopes and some known modular form?

P.S. Somebody asked me this question.

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  • $\begingroup$ I believe the tag 'gt.geometric-topology' should be replaced with the tag 'mg.metric-geometry'. You may also want to add the tag 'convex-polytopes'. $\endgroup$ Dec 7, 2013 at 16:57
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    $\begingroup$ I don't know the answer but I would guess not. The connection between the level-5 j-function and the icosahedron is ultimately due to the fact that $PSL(2, 5)$ is isomorphic to the group of rotations of the icosahedron. I don't think any of the symmetry groups of the 4-polytopes are of the form $PSL(2, q)$. The 5-cell does have symmetry group $S_5$ and so contains $PSL(2, 5)$ but this is just the group of the icosahedron anyway. $\endgroup$ Dec 7, 2013 at 20:04

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