# Quaternions and Rotations

Two of the interesting achievements in Mathematics are Classification of platonic solids, and also classification of finite groups acting on the unit sphere in $\mathbb{R}^3$, and they are very nicely connected to each other. These objects also enter in the classification of finite subgroups of $GL(2,\mathbb{R}), GL(3,\mathbb{R})$.

When studying these groups with geometry, I visited their classification by various ways: by solving some Diophantine equations, and also using geometry of complex numbers; in particular multiplication by complex numbers.

Some finte subgroups of $GL(3,\mathbb{R})$ can be obtained from a multiplication in quaternions $\mathbb{H}$ by unit quaternions, and these are connected with rotations in $\mathbb{R}^3$; for a pure quaternion $a$, and unit quaternion $q$, the map $a\mapsto qaq^{*}$ is a rotation of $\mathbb{R}^3$, where $\mathbb{R}^3$ is identified with the space of pure quaternions. Many books/notes show this connection, but have not explained ideas behind considering multiplication by only unit quaternion, pure quaternions and multiplication in this specific way ($a\mapsto qaq^{*}$). Can anybody explains ideas behind them, and suggest good reference for them (except Conway's book). (Thanks in advance..)

edit: the book Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli by Toth begins with a chapter on platonic solids and finite rotation groups all very explicitly, in terms of Möbius transformations. Once you understand how to interchange between Möbius transforms and quaternions acting on $R^3$ from stillwell, this may be helpful.