7
We seek to classify. Classification is based on the definition of a species.
Two objects belonging to the same species have a common property.
-
Take a sphere, a particular sphere.
-
Look at the subgroup of the large group (Euclid's group) which leaves this sphere invariant. Souriau calls this subgroup the
regularity
of a sphere.
- Search for all the objects which are invariant under the action of this subgroup. You find all the spheres centered on a given point, including the sphere of zero radius: the point.
Then the point belongs to the species of "spheres centered at the origin".
Conversely:
-
Take a point of three-dimensional space.
-
Look at the subgroup of Euclid's group which leaves this point invariant. You find the orthogonal group O(3).
-
Then search for all the objects which are invariant under the action of the elements of this subgroup, under rotation around this point. You find all the spheres centered on this point and conclude that this point and all these spheres belong to a same species.
Objects such as a straight line, a plane, a cylinder, etc., can be "constructed" as a species linked to a certain particular subgroup.
In physics, we want to classify elementary particles. But you cannot take a particle between your thumb and index and look at it through a magnifying glass. You can only observe its behavior, its
movement
.
Tell me how you move, and I will tell you what you are.
I have an old good friend, Jean-Louis Philoche, who is a fine chess player. He can play blind (in French "jouer à l'aveugle", without seeing the chessboard). You just have to indicate to him the movement of a piece:
b1-c3
For non-players:
(90)
Horse's move
Jean-Louis is able to memorize all that in his head. I don't know how he does, but it works. It proves that chess pieces are not necessary to play (a computer does not use them).
Imagine you are in a room and you hear two neighbors who play "a game". You don't see them, but you hear when they announce their moves.
b2-b3
b7-b5
and so on...
You think: they are moving something. What is this game? You take a board, place small stones on it, and note their successive moves on a sheet of paper. Let C be the column index and L the line index. A move corresponds to:
( D C , D L )
If |DC| = 1 and |DL| = 1: this corresponds to a king's move.
If |DC| = |DL|: this corresponds to a bishop's move (along a diagonal).
If |DC| × |DL| = 0: this corresponds to a rook's move.
If |DC × DL| = 3: this corresponds to a knight's move.
If DL is strictly positive, this corresponds to a white pawn. If DL is strictly negative, this corresponds to a black pawn.
And so on. We build a classification of "objects"
based on their behavior
.
Another image. You have a box with mixed bolts. You want to classify these bolts. What do you need? Different nuts.
(91)
-
Take a bolt.
-
Find the nut that fits it.
-
Select all the bolts that fit this nut. You get a species of bolts.
Orthogonal group
O(3).
We can extend what was said above in the 2D context to the 3D context. We know how to perform a rotation in a 3D space, with respect to a fixed point, the origin of coordinates. It depends on three angles a, b, g, called Euler angles. We will not write such a matrix, but we will simply denote it:
(92)
det ( a ) = + 1
It is an orthogonal matrix:
(92b)
The orthogonal group O(3) is composed of all orthogonal matrices, including those whose determinant is -1. We call these matrices (93)
As in the previous section, we can obtain all the orthogonal matrices from SO(3) by:
(94)
L being the diagonal matrix:
(95)
(96)
All this is redundant. But it immediately shows the fundamental symmetries.
(97)
(98)
(98b)
(99)
There are "mirror matrices" which reverse the orientation of the objects, transforming these objects into their image in a mirror:
(100)
Give an example of oriented objects, whose orientation is reversed by this mirror symmetry:
(101)
This is the surface invented by Werner Boy, a student of Hilbert. Special attention will be paid to this interesting object in the section of the site devoted to mathematics. We have removed a portion of the surface in order to show the triple point T.
You can call any of these objects "right" or "left". Nobody has ever indicated what was the right rotation movement of Boy's surface. Anyway: why make a Boy's surface turn? Some claim it can fly, but I am skeptical.
Next:
(102)
(103)
(104)
As in 2D geometry (symmetry with respect to the origin), the symmetry with respect to the x-axis is equivalent to a rotation of
π
.
Finally:
(105)
which change the orientation of objects.
Index Dynamic Groups Theory