Manifold


"Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. 

"But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus —a manifold. 

"Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. 

"This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. 

"This is also how they study the movements of fluids, robots, quantum particles and more.

"Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. 

"And they analyze high-dimensional datasets —such as those recording the activity of thousands of neurons in the brain —by looking at how those data points might sit on a lower-dimensional manifold."





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