A collection of definitions for the 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.
- 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 purportedly the definition Grothendieck first heard at age 12 while living in an internment camp. The definition impressed him with its “simplicity and clarity”.
- The circle is the set of all pairs \((x,y)\) in the plane which satisfy the equation \(x^2 + y^2 = 1\).
- The circle is the set of all complex numbers which can be written \(e^{i\theta}\) where \(\theta\) is some real number.
- A circle is a set which can be written as the image of a continuous function \(\gamma:[0,1]\to \bR^2\), where \(\gamma(0) = \gamma(1)\) and \(\gamma\) is injective on the open interval \((0,1)\).
- A circle is the topological space \(SO(2)\).
- The circle is the unique compact one dimensional real manifold without boundary.
- A circle is a one dimensional cell complex with one \(0\)-cell and one \(1\)-cell.
- A circle is the affine scheme \(\Spec \bR[x,y]/(x^2 + y^2 - 1)\).