Symmetry (geometry)
[5] The most common group of transforms applied to objects are termed the Euclidean group of "isometries", which are distance-preserving transformations in space commonly referred to as two-dimensional or three-dimensional (i.e., in plane geometry or solid geometry Euclidean spaces).
These isometries consist of reflections, rotations, translations, and combinations of these basic operations.
[7] A geometric object is typically symmetric only under a subset or "subgroup" of all isometries.
By the Cartan–Dieudonné theorem, an orthogonal transformation in n-dimensional space can be represented by the composition of at most n reflections.
Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical as mirror images of each other.
[12] Reflection symmetry can be generalized to other isometries of m-dimensional space which are involutions, such as in a certain system of Cartesian coordinates.
[15] This implies that for m = 3 (as well as for other odd m), a point reflection changes the orientation of the space, like a mirror-image symmetry.
That explains why in physics, the term P-symmetry (P stands for parity) is used for both point reflection and mirror symmetry.
Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law.
The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at a constant angular speed, while simultaneously translating at a constant linear speed along its axis of rotation.
At any point in time, these two motions combine to give a coiling angle that helps define the properties of the traced helix.
[26] When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°.
Conversely, if the object rotates slowly and translates quickly, the coiling angle will approach 90°.
[33] This self-similarity is seen in many natural structures such as cumulus clouds, lightning, ferns and coastlines, over a wide range of scales.
It is generally not found in gravitationally bound structures, for example the shape of the legs of an elephant and a mouse (so-called allometric scaling).
Similarly, if a soft wax candle were enlarged to the size of a tall tree, it would immediately collapse under its own weight.
A coast is an example of a naturally occurring fractal, since it retains similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand.
If you add required symmetries, you have a more powerful theory but fewer concepts and theorems (which will be deeper and more general).
A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers.
A geometric structure on a manifold M is a diffeomorphism from M to X/Γ for some model geometry X, where Γ is a discrete subgroup of G acting freely on X.
