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We seek to classify. Classification is based on the definition of species.
Two objects belonging to the same species share a common property.
- Take a sphere, a particular sphere.
- Examine the subgroup of the large group (Euclid's group) that leaves this sphere invariant. Souriau calls this subgroup the regularity of a sphere.
- Find all objects invariant under the action of this subgroup. You obtain all spheres centered at a given point, including the sphere of zero radius: the point.
Thus, the point belongs to the species of "spheres centered at the origin."
Conversely:
- Take a point in three-dimensional space.
- Examine the subgroup of Euclid's group that leaves this point invariant. You find the orthogonal group O(3).
- Then find all objects invariant under the action of the elements of this subgroup, under rotation about this point. You obtain all spheres centered at this point, and conclude that this point and all these spheres belong to the same species.
Objects such as a line, a plane, a cylinder, etc., can be "constructed" as species linked to certain specific subgroups.
...In physics, we aim to classify elementary particles. But you cannot take a particle between your thumb and index and examine it under a magnifying glass. You can only observe its behavior, its motion.
Tell me how you move, and I will tell you what you are.
...I have an old friend, Jean-Louis Philoche, who is an excellent chess player. He can play blind (in French: "jouer à l'aveugle," without seeing the board). You simply need to indicate the move of a piece:
b1-c3
For non-players:
(90) Knight's move
...Jean-Louis is able to memorize all this in his head. I don't know how he does it, but it works. This proves that chess pieces are not necessary to play (a computer doesn't need them either).
...Imagine you are in a room and hear two neighbors playing "some game." You cannot see them, but you hear when they announce their moves.
b2-b3 b7-b5 and so on...
...You think: they are moving something. What game is this? You take a board, place small stones on it, and record their successive moves on a sheet of paper. Let C be the column index and L the row index. A move corresponds to:
(DC, DL)
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 them. What do you need? Different nuts.
(91)
- Take a bolt.
- Find the nut that fits it.
- Select all bolts that fit this nut. You obtain 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 3D space with respect to a fixed point, the origin of coordinates. It depends on three angles α, β, γ, called Euler angles. We won’t write out such a matrix; we’ll just denote it as:
(92)
det(a) = +1
It is an orthogonal matrix:
(92b)
...The orthogonal group O(3) consists of all orthogonal matrices, including those with determinant equal to -1. We call these matrices (93)
As in the previous section, we can obtain all orthogonal matrices from SO(3) via:
(94)
L being the diagonal matrix:
(95)
(96)
All this is redundant. But it immediately reveals the fundamental symmetries.
(97)
(98)
(98b)
(99)
There are "mirror matrices" that reverse the orientation of objects, transforming them into their mirror images:
(100)
Give an example of an oriented object 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 part of the surface to reveal the triple point T.
...You can call one of these objects "right" or "left." No one has ever specified what the "right" rotation of Boy's surface is. Anyway: why rotate Boy's surface? Some claim it can fly, but I remain skeptical.
Next:
(102)
(103)
(104)
...As in 2D geometry (symmetry with respect to the origin), symmetry with respect to the x-axis is equivalent to a rotation of π. Finally:
(105)
which changes the orientation of objects.