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- Thread starter khotsofalang
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one definition of symmetry is an isometry. I believe there are manifolds with trivial isometry groups i.e. the only isometry is the identity. Since any manifold can be embedded isometrically in space, you conjecture is false.

In any dimension, it should be easy to construct measurable sets in space that have no symmetries under reflection - but I am not sure

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Your hand is a 3D object that has no symmetries.

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f(x) = f(-x). For a 3-D object, a similar expression of symmetry might be

f(x,y,z) = f(-x,-y,z)

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http://en.wikipedia.org/wiki/Symmetry

I think you can glean from these scribblings how mathematics treats the definition of symmetry.

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