There are five *Platonic solids*, two are *duals* to another two, while the tetrahedron is self-dual. As such, this gives rise to 3 polyhedral groups: the *tetrahedral group* of order 12, the *octahedral group* of order 24, and the *icosahedral group* of order 60.

Amazingly, Ramanujan found 3 continued fractions that can be associated with each group. See this article for more details. It turns out there are also corresponding hypergeometric formulas, and the numbers 12, 24, and 60 naturally appears.

First though, define the *j-function* as,

where,

This can be conveniently calculated in Mathematica as,

*NOTE*: In the formulas below, it will be assumed that,

**I. Tetrahedral group**

Given,

and,

then we have the simple relationship,

Example. Let , hence , then,

and *c(q)* can then be easily solved for as a cubic equation.

**II. Octahedral group**

Let,

then,

Example. Still using , then,

**III. Icosahedral group**

(To be discussed in the next post.)

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Posted by Al Waring on January 24, 2017 at 1:16 am

Succulent delicacies for the mathematical connoisseur.