الدوران والتماثل في الفضاء ثلاثي الأبعاد

a4108

The general matrix :
(105b)

corresponds to a rotation around the origin, eventually combined to a symmetry with respect to a plane which contains the origin O.

Inhabitants of a 3d's world are 3d structures. Remember times does not exist in 3d Euclid's 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 . Depends on the group you choose. In 3d you can use two groups. The first is what we called the Special Euclid's group. (110)

You can't find any element of this group whose action can transform a right and into a left and, and vice-versa.
But it becomes possible with the complete Euclid's group :
(111)

You will say : what is the group that runs my today's life ?

...In today's world we see different objects. In particular our image in a looking glass, when shaving, each morning. Then we have two choices:

1 - You decide that your image in the looking glass is a not real (although you use it daily) so that you geometrical world corresponds to the first group.

2 - You decide the object you see on the other side of you mirrors are real.

• Then your wold corresponds to the second group.

• It increases drastically the volume of your flat.

• Join a psychiatric hospital as fas as you can.

Index Dynamic Groups Theory