Conformal Geometric Algebra in Stochastic Optimization Problems ...

Chapter 2
Geometric Algebra
The beginning is a tale of a mathematical success story ...
Historical note
The quest for a mathematical language suitable to express geometrical relationships
in an algebraic way can be traced back to the ancient Greeks:
it was Euclid with his seminal work ’Elements’ in the 3rd century B.C.
who first wrote down geometrical laws for a world as he perceived it.
It took a long time, until the eighteenth century, that mathematicians
caused a furor by discovering geometries distinct from the Euclidian
one. The predominant task was then to find an algebraic framework
which would unify all different geometries...
Inspired by the algebra of complex numbers, which allows the division by
a vector, William Rowan Hamilton (1805-1865) was preoccupied with a
generalization to three dimensions: he had been trying to find a reasonable
way to multiply three-dimensional points for a couple of years, in
such a way as to allow division. Finally, on 16th of October 1843 Hamilton
discovered the numbers later called Quaternions. Considering an
anti-commutative multiplication and the idea of using four dimensions
instead of three were the crucial factors making his finding possible.
At that time, another pivotal question was how best to represent rotations
in 3D. Here Quaternions emerged as a very clear to handle and
very efficient way for carrying out rotations and are still in use today.
Despite the then positive impact the role of Quaternions diminished with
the introduction of the more straightforward vector algebra of Josiah
Willard Gibbs (1839-1903) in the 1880s. The framework promoted by
Gibbs is basically the classical vector algebra being taught high school
students nowadays; it distinguishes between the scalar product and the
vector cross product. The Quaternion product, in contrast, combines
both of them. Eventually, the Quaternions were, due to Gibbs’s established
reputation, displaced by his hybrid vector algebra.
The probably most important step towards geometric algebra was taken
by the German mathematician and schoolteacher Hermann Günther
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