I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with the standard chordal metric.

The simplest functions I have found posessing these symmetries are rational functions, with zeroes at the vertices of a certain inscribed polyhedron (tetrahedron, octahedron and icosahedron) of order equal to the vertex degrees, and poles of order 3 centered at their faces (i.e. at the vertices of the dual polyhedron). Up to rotation, these functions are:

$$F_{3,3}(z) = f_{3,3} \: z^3 \frac{(z^3-2\sqrt{2})^3}{(z^3+1/(2\sqrt{2}))^3}$$

$$F_{4,3}(z) = f_{4,3} \: z^4 \frac{(z^4-1)^4}{(z^8+14z^4+1)^3}$$

$$F_{5,3}(z) = f_{5,3} \: z^5 \frac{(z^{10}-11z^5-1)^5}{(z^{20}+228z^{15}+494z^{10}-228z^5+1)^3}$$

where the $f_{a,b}$ are in principle arbitrary normalization constants. As an example, here is a color wheel graph of $F_{5,3}$ stereographically projected into the complex plane where the icosahedral symmetry is apparent:

Icosahedrally symmetric rational function F_(5,3)

One can follow a similar procedure in the case of dihedral symmetry, obtaining

$$F_{p,2}(z) = f_{p,2} \frac{z^p}{(z^p-1)^2}$$

However, playing with the plots in this website I noticed that there is a certain special value for the constants $f_{a,b}$, for which the subset of the image of $F$ consisting of points with unit modulus becomes connected, tesellating the sphere. These values are $$f_{3,3} = \dfrac{1}{16\sqrt{2}}, \: f_{4,3}=108, \: f_{5,3}=1728$$ and $f_{p,2}=4$ for any $p$. They seem to be universal constants in some sense: they are unique (up to phase) with that property, and $f_{p,2}$ doesn't depend on $p$. But they don't seem to follow any obvious relationship with the corresponding polyhedra.

My question is, where do these constants come from? I found them by trial and error, but it is possible to calculate or characterize them in some way?

Edit: I noticed that the last constant $f_{5,3} = 1728$ also appears in the definition of the $j$-invariant function. But I know nothing about modular form theory aside from the very basics, so I can't tell whether there's some relationship here or it's just coincidence.


I finally found an answer online. It turns out that Felix Klein was also aware of these functions in 1878: he used them to construct his famous solution to the quintic equation!

I'll post here the parts of the derivation that are relevant to my question. It is clear from the form of the functions $F_{a,b}$ that each one is defined in terms of two monic polynomials, which we can call $G(z)$ and $H(z)$ (the references call it $f$ and $H$, but I will keep the notation consistent with my question to avoid any confusion). As I said above, in the Riemann sphere $G$ has zeros at the vertices of the corresponding regular polyhedron, and $H$ has zeros at the face centers. We have

$$F_{a,b}(z) \propto \frac{G(z)^a}{H(z)^b}$$

There is another monic polynomial that can be defined, called $T(z)$, which has zeros at the edge midpoints of the polyhedron. It can be constructed, for example, as the Jacobian of $G$ and $H$, expressed in symmetric form by substituting $z=x/y$ and multiplying by $y$ until there are no factors of $y^{-1}$ left:

$$T(x,y) \propto \begin{vmatrix} G_{,x} & G_{,y} \\ H_{,x} & H_{,y} \end{vmatrix}$$

It can be demostrated that there exists a relation or syzygy:

$$H^b + T^2 = f_{a,b} \: G^a$$

where the $f_{a,b}$ are the constants from my question. Thus knowing $G, H$ and $T$, using the relation above and comparing the two sides, one can directly calculate the value of the constants.

For references, see this MathOverflow post.


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