Why quaternions?

Points [vectors] in the plane can be represented using complex numbers, and then can be “multiplied”, i.e. can be given the structure of an \mathbb{R}-algebra. Conversely, complex numbers can be viewed as geometric objects, and operations on them given geometric meaning, by the same construction.
Is there an analogue of this for points in 3-space? Hamilton tried to find such an analogue, and was led to the quaternions. There is a (possibly apocryphal) story of how his son asked him every morning: “Well, Papa, can you multiply triplets?” and always got the same answer: “No, I can only add and subtract them”, with a sad shake of the head. What Hamilton realized was that this is indeed not quite possible … unless we embed the triplets in a 4-dimensional algebra (the quaternions)—this is what Hamilton realized that morning and inscribed on Brougham Bridge.
Quaternions are a concise method of representing the automorphisms of three- and four-dimensional spaces. They have the technical advantage that unit quaternions form the simply connected cover of SO(3). For this reason, quaternions are used in computer graphics (Tomb Raider is often cited as the first mass-market computer game to have used quaternions to achieve smooth 3D rotation), control theory, signal processing, attitude control, physics, bioinformatics, and orbital mechanics.For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions.
Quaternion [algebra]s have received another boost from number theory because of their relation to quadratic forms.
Nevertheless, quaternions remain relatively obscure outside of, and even within large parts of, mathematics. Today, vector analysis—which was in fact born out of the subsequent development of quaternions—is used to do the mathematics and physics that could be done with quaternions.  Or, to steal Simon L. Altmann‘s words,
“… quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist.”  (1986.)


  • Wikipedia’s “History of quaternions” and “Quaternion
  • (see also) John Conway and Derek Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry

Free groups and topology

Any subgroup of a free group is free.

This result is quite interesting, because the statement is purely algebraic yet the simplest proof is topological. Namely, any free group G may be realized as the fundamental group of a graph X. 

The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X; but every such Y is again a graph. Therefore its fundamental group H is free.

Underlined technical details here