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..)


John Stillwell's book Naive Lie Theory motivates the subject in the first chapter using an exploration of quaternions acting by conjugation on the unit sphere. I think his explanation is great for someone at any point in their undergraduate career, however the subsequent chapters may be extremely slow for anyone already acquainted with lie theory, group theory or general topology.


There is a much more in depth coverage of the details of this in another book which I am trying to recall the name of.

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.


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