Imaginary Numbers Are Real [Part 13: Riemann Surfaces]

More information and resources: http://www.welchlabs.com Supporting Code: https://github.com/stephencwelch/Imaginary-Numbers-Are-Real Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space.

It took over a year, but the Imaginary Numbers Series is finally complete. By far the most labor intensive parts were part 1 and part 13. When I began the series I had no idea where it would end up. I originally planned on 6 parts, but the deeper I got into imaginary numbers, the cooler things got - and I just couldn't deal with telling an incomplete story. 

I couldn't be happier with where the series ended up - I'm so happy I was able to talk about Riemann Surfaces. I'm sure a mathematician or two will take issue with my presentation (there's a reason Riemann Surface are a graduate level mathematics topic!), but I hope I was able to give a broad audience a taste for these beautiful mathematical structures without oversimplifying the meaning out of things.

I wanted to share a few of the visualizations I used in Part 13. I used the wonderful visualization tool plotly for all 3D graphics. Here are a few visualization from Part 13:

Riemann Surface for \(w=\sqrt{z}\) 

Paths on Riemann Surface. This one's a bit slow, it takes lots of points to make that path!

3D surface plot from opening scene. 

Thanks for watching!