List of definitions of a circle

Back in December of 2022 I visited a middle school in Tacoma, Washington. The kids and I talked about how, in some ways, a triangle is the same thing as a circle. Inspired by the famous list of Thurston found in On proof and the progress of mathematics describing various definitions of the derivative, I wanted to list out a few definitions of "the circle" accepted in various contexts. Please let me know if you think of any additions to this list or want to offer corrections.

1. The circle (centered at a point $P$) is the set of all points in the plane which are of a fixed distance from $P$. This is the purportedly the definition Groethendieck first heard at age 12 while living in an internment camp. The definition impressed him with its "simplicity and clarity"

2. The circle is the set of all pairs $(x,y)$ in the plane which satisfy the equation $x^2 + y^2 = 1$.

3. The circle is the set of all complex numbers which can be written $e^{i\theta}$ there $\theta$ is some real number.

4. A circle is a set which can be written as the image of a continuous function $\gamma:[0,1]\to \mathbb{R}^2$, where $\gamma(0) = \gamma(1)$ and $\gamma$ is injective on the open interval $(0,1)$.

5. A circle is the topological space $SO(2)$.

6. The circle is the unique compact one dimensional real manifold without boundary.

7. A circle is a one dimensional cell complex with one $0$-cell and one $1$-cell.

8. A circle is the affine scheme $\operatorname{Spec} R[x,y]/(x^2 + y^2 - 1)$.