8
The general matrix:
(105b)
corresponds to a rotation around the origin, eventually combined with a symmetry with respect to a plane passing through the origin O.
Inhabitants of a 3D world are 3D structures. Remember that time does not exist in 3D Euclidean world.
Orientation exists in 2D worlds. Above, we presented a R meeting a Russian "ia" letter. In 3D, it gives:
(106)
(107)
(108)
(109)
Do a right and a left hand belong to distinct species? It depends on the group you choose. In 3D, you can use two groups. The first is what we called the
Special Euclidean group.
(110)
You cannot find any element of this group whose action can transform a right hand into a left hand, and vice versa.
But it becomes possible with the complete Euclidean group:
(111)
You will say: what is the group that rules my life today?
In today's world we see different objects. In particular our image in a mirror, when shaving, every morning. Then we have two choices:
1 - You decide that your image in the mirror is not real (although you use it daily), so that your geometric world corresponds to the first group.
2 - You decide the objects you see on the other side of your mirrors are real.
-
Then your world corresponds to the second group.
-
It drastically increases the volume of your flat.
-
Go to the psychiatric hospital as soon as possible.
Index Dynamic Groups Theory