Let's call a subset of $\mathbb{R}^2$ that is bounded bad if the only symmetry it has is the identity, i.e it cannot be rotated or reflected in any way that gets back to the original other than doing nothing. Then two natural questions arise:
- What is the cardinality of all non-congruent bad sets?
- What is the cardinality of all non-similar bad sets?
I'm not sure how to find either of these - but just from intuition I except the answer to be greater than that of $\mathbb{R}$.
EDIT: By a symmetry, I mean a distance preserving bijection from the subset to itself (with the usual distance metric)
The answer for both questions is $2^{|\mathbb R|}$. Consider the set of sets of the form $$\{(0,0)\}\cup A\cup \{(2,0)\},$$ where $A$ is any nonempty subset of the open line segment from $(0,0)$ to $(1,0)$. These sets all have a diameter of 2, so they are similar iff they are congruent. With some thought, these sets are pairwise non-congruent. Since there are $2^{|\mathbb R|}$ choices for $A$, we are done.