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Sunday, December 11, 2011

Fearless Symmetry 2/23: Groups

I read the second chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 2: Groups

Definition: A group G is a set with a composition defined on pairs of elements, as long as three axioms hold true:
1. For any three elements x,y,z in G: x*(y*z) = x*(y*z)
2. G contains an element e such that for all x in G: x*e = e*x = x.
3. For any element x in G, there is an element y in G such that x*y=e.

Example 1:
The group of rotations of the sphere in R3: SO(3) or the Special Orthogonal Group in 3 dimensions. -The set G is the collection of all rotational symmetries of the sphere, i.e. if we rotate the sphere by any angle, the sphere doesn't noticeably change. The group property basically means that if we rotate the sphere over any angle A, after this over an angle B, it is the same if we would have rotated it in one go, but over some different angle. Also any rotation has an inverse: rotating it over the opposite angle. This makes the rotations a group. SO(3) is in fact a Lie group because these rotations can be done arbitrary small which is not the case when considering the symmetry group of for example a cube. Lie groups capture the concept of "continuous symmetries".

For me personally, this is the time to review chapters 1,2 and 3 of Naive Lie Theory by John Stillwell, Springer 2008. There you will find that the ( 4 dimensional ) quaternions are intimately related to the group SO(3) and that the quaternions can be expressed as 'complex 2-dimensional rotations' or complex 2 by 2 matrices. - This explains why quaternions are frequently used in 3D-(game)-programming.

To be continued with 3. Permutations

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