About 1900, Klein discovered an interesting perspective on geometry. He found that you could view a Euclidean space as a real space plus a group of transformations that leave figures congruent to one another. This group is formed by combining rotations, translations and reflections. We call this the Euclidean group. A natural question to ask was whether other spaces could be characterized in terms of symmetry groups? Klein showed the answer is yes.
For an affine geometry, it is characterized by symmetry under the action of the following transformations:
f(x)=A(x)+bA∈SO(R,2),b∈R2
For an affine geometry, it is characterized by symmetry under the action of the following transformations:
f(x)=A(x)+bA∈GL(R,2),b∈R2
The general linear group is bigger than the special orthogonal group, so correspondingly, we have stronger constraints on what kinds of structures can be preserved. And indeed, in affine geometry, we don’t preserve lengths, but rather: parallel lines, ratios of lengths along a line and straight lines themselves.
For hyperbolic geometries, the group structure is a bit harder to define in terms of familiar functions. We tend to call is SO(n,m), to denote the fact that we’ve got a psuedo metric with n minus signs and m plus signs. In the simplest case, this would mean “distances” are like |x−y|2=(x0−y0)2−(x1−y1)2
and it is these distances which are preserved. In this simple case, we find that group can be represented by transformations like
f(x)=±(cosh(α)I2+sinh(α)σ1)x+bα∈R,b∈R2
where σ1 is the first pauli matrix. The term in brackets acts like the hyperbolic equivalent of a rotation matrix, and α is a hyperbolic angle. And in fact, if we replace α with iβ, then cosh and sinh turn into cos and sin. So we can get back the Euclidean group! (I don’t, actually, understand why this isn’t a group isomorphism. So something must be going wonky here. )
Projective geometry is a bit trickier, as to frame it in terms of familiar Euclidean spaces you’ve got to deal with equivalence classes of lines through an origin. I don’t have time to explain how the Kleinian perspective applies here, but it does. Also, projective transforms don’t preserve angles or lengths. They do, however, preserve co-linearity.
OK, but how exactly does the Euclidean group characterize Euclidean space? We can understand that by looking at the 2d case. From there, it’s easy to generalize. Recall that we want to show the Euclidean group contains any transformation that leaves figures, or shapes, in our 2d Euclidean plane unchanged.
First, I want to point out that this transformation must be an isometry. The Euclidean metric defines Euclidean geometry, after all.
OK, now let’s see how this transformation must affect shapes. Let’s start with the simplest non-trivial shape, the triangle. We’re going to show that for any isometry, and any triangle, there is some element of the Euclidean group that replicates how the isometry transforms the triangle. If you consider an isometry mapping a triangle to another, they must be congruent. We can construct an element which respects this congruence by rotating till one of the corners is in the same orientation as its image, translating the triangle so that the corresponding vertex overlays its image, and then doing a mirror flip if the orientations of the two triangles don’t match.
Once we can get group elements mimicking the action of our isometry on triangles, it is a simple matter to get rectangles, as they’re made of triangles. And then rectangular grids of lines as they’re formed of rectangles. And then rectangular lattices of points, as they’re parts of the grids. Then dense collections of lattices. But an isometry and an element of the Euclidean group are continuous. Continuous maps are defined uniquely by their action on a dense subset of their domain. So they must be the same maps. Done.
OK, so that’s how we show the Euclidean group is the group of symmetries for Euclidean figures. What about other geometries? We can do it in basically the same way. Define a non-trivial simple figure and use that to pin down the action of a symmetry transformation in terms of simpler components. E.g. for the affine group, it is invertible linear maps and translations. Then show the actions of any transformation is uniquely determined by its actions on some simple figure. For the affine group, this is again a triangle.
Kleinian view of geometry
About 1900, Klein discovered an interesting perspective on geometry. He found that you could view a Euclidean space as a real space plus a group of transformations that leave figures congruent to one another. This group is formed by combining rotations, translations and reflections. We call this the Euclidean group. A natural question to ask was whether other spaces could be characterized in terms of symmetry groups? Klein showed the answer is yes.
For an affine geometry, it is characterized by symmetry under the action of the following transformations:
f(x)=A(x)+bA∈SO(R,2) , b∈R2
For an affine geometry, it is characterized by symmetry under the action of the following transformations:
f(x)=A(x)+bA∈GL(R,2) , b∈R2
The general linear group is bigger than the special orthogonal group, so correspondingly, we have stronger constraints on what kinds of structures can be preserved. And indeed, in affine geometry, we don’t preserve lengths, but rather: parallel lines, ratios of lengths along a line and straight lines themselves.
For hyperbolic geometries, the group structure is a bit harder to define in terms of familiar functions. We tend to call is SO(n,m), to denote the fact that we’ve got a psuedo metric with n minus signs and m plus signs. In the simplest case, this would mean “distances” are like |x−y|2=(x0−y0)2−(x1−y1)2
and it is these distances which are preserved. In this simple case, we find that group can be represented by transformations like
f(x)=±(cosh(α)I2+sinh(α)σ1)x+bα∈R , b∈R2
where σ1 is the first pauli matrix. The term in brackets acts like the hyperbolic equivalent of a rotation matrix, and α is a hyperbolic angle. And in fact, if we replace α with iβ, then cosh and sinh turn into cos and sin. So we can get back the Euclidean group! (I don’t, actually, understand why this isn’t a group isomorphism. So something must be going wonky here. )
Projective geometry is a bit trickier, as to frame it in terms of familiar Euclidean spaces you’ve got to deal with equivalence classes of lines through an origin. I don’t have time to explain how the Kleinian perspective applies here, but it does. Also, projective transforms don’t preserve angles or lengths. They do, however, preserve co-linearity.
OK, but how exactly does the Euclidean group characterize Euclidean space? We can understand that by looking at the 2d case. From there, it’s easy to generalize. Recall that we want to show the Euclidean group contains any transformation that leaves figures, or shapes, in our 2d Euclidean plane unchanged.
First, I want to point out that this transformation must be an isometry. The Euclidean metric defines Euclidean geometry, after all.
OK, now let’s see how this transformation must affect shapes. Let’s start with the simplest non-trivial shape, the triangle. We’re going to show that for any isometry, and any triangle, there is some element of the Euclidean group that replicates how the isometry transforms the triangle. If you consider an isometry mapping a triangle to another, they must be congruent. We can construct an element which respects this congruence by rotating till one of the corners is in the same orientation as its image, translating the triangle so that the corresponding vertex overlays its image, and then doing a mirror flip if the orientations of the two triangles don’t match.
Once we can get group elements mimicking the action of our isometry on triangles, it is a simple matter to get rectangles, as they’re made of triangles. And then rectangular grids of lines as they’re formed of rectangles. And then rectangular lattices of points, as they’re parts of the grids. Then dense collections of lattices. But an isometry and an element of the Euclidean group are continuous. Continuous maps are defined uniquely by their action on a dense subset of their domain. So they must be the same maps. Done.
OK, so that’s how we show the Euclidean group is the group of symmetries for Euclidean figures. What about other geometries? We can do it in basically the same way. Define a non-trivial simple figure and use that to pin down the action of a symmetry transformation in terms of simpler components. E.g. for the affine group, it is invertible linear maps and translations. Then show the actions of any transformation is uniquely determined by its actions on some simple figure. For the affine group, this is again a triangle.