2022-08-30 11:38:54 +0100 received badge ● Popular Question (source) 2020-07-26 16:15:12 +0100 received badge ● Supporter (source) 2020-07-13 22:15:33 +0100 received badge ● Student (source) 2020-07-12 01:48:31 +0100 asked a question Defining $S^1$ with four charts Hello, I am trying to implement the construction of the manifold $S^1$ using the four charts, as described at en.wikipedia.org/wiki/Manifold#Circle . It's not clear to me how to define the charts in terms of the projection operators within SageMath. I'm also not sure how to isolate open sets that I would like to define the charts over. The circle is parameterized using polar coordinates, and the four charts have the form: \begin{align} \varphi_1(r, \theta) &= \theta && \theta \in (-\frac{\pi}{3}, \frac{\pi}{3}) \ \tag{1} \end{align} \begin{align} \varphi_2(r, \theta) &= \theta && \theta \in (\frac{\pi}{6}, \frac{5\pi}{6}) \ \tag{2} \end{align} \begin{align} \varphi_3(r, \theta) &= \theta && \theta \in (\frac{2\pi}{3}, \frac{4\pi}{3}) \ \tag{3} \end{align} \begin{align} \varphi_4(r, \theta) &= \theta && \theta \in (\frac{7\pi}{6}, \frac{11\pi}{6}) \ \tag{4} \end{align} Once we have these charts, how would I define the four transition functions to complete the description? Thank you for your help!