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I have been thinking my brain to find three different examples for a shape S in the plain, such that its group of symmetries is infinite.Also I was asked to draw each shape clearly why its group of symmetries is infinite.

my answer: I think circle is one shape that has its symmetries is infinite.

Can anyone help me to with hints to discover some more shapes such that its group of symmeties is infinite

Thank you in advance for help

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    $\begingroup$ What is the group of symmetries of an infinitely long string. (ie. $\mathbb{R} \subset \mathbb{R}^2$)? Or a single hexagon repeating itself indefinitely on all sides. $\endgroup$ – Prahlad Vaidyanathan Aug 28 '13 at 9:00
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If you are not limited to bounded shapes, you can think of shapes that have some translation symmetry; this immediately makes their symmtry group infinite. There are plenty examples of such shapes: a line, a horizontal infinite strip of finite width, a discrete infinite lattice of points.

Basically it is best to think first of which infinite symmetry group you want to have, and then adapt your shape to that. One general method is to take one chose subset of the plane (a single point will do) and add all the transforms of it by the group. For the rotation group this results in such stuff as one or more concentric circles or regions bounded by such circles. These shapes in fact get additional reflection symmetry for free, but that doesn't hurt.

You can have fun with other infinite groups. Think of the group generated by a single rotation by an angle that is irrational to the full rotation by $2\pi$.

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