symetry
symetry as it is rigorously defined through mathematics is Invariance with respect to transformation. Reflectional symetry is only one of many kinds of symetry. A circle remains invariant on the transformation of reflecting itself across one of its diameters. Thus the circle has reflectional symetry or it is invariant under the transformation of reflection. This statement is not without limits however because all axes of symetry run through the diameter of the circle. So not all lines are axes of symetry to the circle. So it does not have perfect reflectional symetry. Because although it has an infinite number of axis's of symetry it does not have all possible axes. Under this strict criteria all finite shapes break down. because if they are not infinite we can pick a line that does not intersect with the shape which will never be an axis of symetry for it. furthermore any shape we pick that is regular can be reflected off the center of the period and also lacks perfect symetry. Is there any shape that has perfect reflective symetry? yes! The only shape with perfect reflectional symetry is in fact the plane. The plane exhibits invariance under every 2 dimensional transformation short of altering it's topology. Or in otherwords anything that we do to a plane will make it come out a plane. If we square the coordinates of every point on the plane we get a plane back. if we reflect it across any line what so ever we also get a plane rotation around any point magnification over any scale... etc. But I included the disclaimer that the transformation cannot change the topology of the shape. This is simply saying that you cannot do anything that changes the essential form of the plane. This is meant in the topological sense rather than any more conventional one. So for instance changing a circle into a square is perfectly allowable but cutting the circle in half and making two smaller circles is not. The other disclaimer is more simple No transforms that involve higher dimensional spaces are allowed. So for instance bending the plane into a pseudoshpere (a kind of infinite horse sadle shape) leaves us with something other than a plane but it only is capable of existing in a higher dimensional space. So more in general terms the most symetric shape in a D dimensional supporting space is the supporting space itself.
The plane is the most symetric object known. it has perfect reflectional symetry with every possible line constituting an axis of symetry. but the plane is not just reflectionally symetric, it also exhibits self similarity, translational symetry, rotational symetry, in fact it shows nearly every form of transform invariance. well this kind of invariance with operations is very important in physics and physical theoreys are chosen for their symetry. it is interesting to note that the world we live in seems to be governed by very symetric laws. however the world seems to be quite asymetric.
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