3
Group of translations:
...Consider the 2D space (x, y). In this space, a translation corresponds to the pair of scalars (Δx, Δy), and it is customary to write:
x' = x + Δx
y' = y + Δy
We then use addition. Could we manage to encode the translation using multiplication instead?
Consider the matrices:
and the group action:
Note that this is no longer simple matrix multiplication (g × r), but rather the group action:
We can also consider translations in three, four, or more dimensions:
The corresponding group action is then:
...Incidentally, the group of translations is commutative, and the identity element is "the zero translation." In 3D, the group's dimension is three; in 4D, it is four.
Interest of matrix groups. Example: the Euclidean group.
...The advantage of a matrix group is that we can simultaneously handle several different types of operations that previously seemed fundamentally distinct—such as rotation and translation. It suffices then to consider matrices:
and apply the group element matrix to the column vector, observing that this corresponds to the combination of a rotation by angle α and a translation along the vector (Δx, Δy).
...As we see, the matrix g does not act "directly" on the points (x, y) in this 2D space, but through what is called a "group action," which obeys certain axioms.
...Thus, a group "acts" and "transports" points. This is the Euclidean group. Associated with a 2D space (x, y), it is defined by three parameters. It is g(α, Δx, Δy): the dimension of this group is 3. Specifically:
g(0, Δx, Δy) represents the subgroup of translations.
g(α, 0, 0) represents the subgroup of rotations about the origin.
g(0, Δx, 0) represents the subgroup of translations parallel to a line (the OX axis).
...The Euclidean group transports points that, by themselves, have no intrinsic attributes (whereas dynamical groups assign "attributes" such as mass, energy, momentum, spin to a simple "material point").
...With the Euclidean group, we are obliged to consider sets of points. As if, in chemistry, atoms were indistinguishable from one another, and only the geometry of molecular assemblies carried meaningful information.
...A geometric figure—such as a triangle (considered as a set of three points or three segments), or a square (considered as a set of four points or four segments)—can be transported by the group. This is where the fundamental idea of "species" arises. Two "objects" are said to belong to the same species if there exists a group element that can map one onto the other.
With respect to the Euclidean group, squares with the same side length a form a species:
Squares of the same species.
...If the sides a and b are different, these objects do not belong to the same species. There is no group element that allows transformation from one to the other.
With respect to the Euclidean group, these squares do not form the same species.
Euclid does not allow homotheties. To handle them, we would need to move to another group—the group of Descartes:
a four-parameter group
g(λ, α, Δx, Δy),
where λ is a homothety coefficient. Thus, the dimension of this group is 4.
From here, we can easily imagine that there exists a 3D Euclidean group acting on objects in three dimensions.
...We are not aiming to launch into a full course on groups, but rather to grasp a few key ideas. What is zoology? A science devoted to studying animals and classifying them. If we limit ourselves to shape alone, the Euclidean group allows us to classify adult rabbits. To classify rabbits of different sizes within the same species, we would need to resort to the Descartes group, since no element of the 3D Euclidean group can transform a small rabbit into a large one.
...You're smiling? You're mistaken. Perhaps in your apartment or house, there is a baby learning to play in a corner. You've given them a classic toy: a shape-sorting box with holes for cylinders, cubes, or triangular prisms.
...What is the baby doing? He or she is becoming familiar with the 3D Euclidean group. They are classifying objects by species, which will later enable recognition—form recognition.
...Even though they are different colors, the baby verifies that there exist group actions (movements of these objects in 3D space) that can bring cylinder A and cylinder B into coincidence, using the hollow shape of the cylinder or prism as a template—the entry slot into the compartment of the sorting box. Thus, the baby learns that cylinders A and B, with respect to the shape criterion (Euclidean group), belong to the same species.
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