Category Archives: Euclidean Geometry

Liouville illiouminated

A couple of weeks ago, my student Yan Mary He presented a nice proof of Liouville’s theorem to me during our weekly meeting. The proof was the one from Benedetti-Petronio’s Lectures on Hyperbolic Geometry, which in my book gets lots … Continue reading

Posted in Complex analysis, Euclidean Geometry, Rigidity | Tagged , , , | 7 Comments


Last week while in Tel Aviv I had an interesting conversation over lunch with Leonid Polterovich and Yaron Ostrover. I happened to mention the following gem from the remarkable book A=B by Wilf-Zeilberger. The book contains the following Theorem and … Continue reading

Posted in Euclidean Geometry | Tagged , , | 11 Comments

Kenyon’s squarespirals

The other day by chance I happened to look at Richard Kenyon’s web page, and was struck by a very beautiful animated image there. The image is of a region tiled by colored squares, which are slowly rotating. As the … Continue reading

Posted in Complex analysis, Euclidean Geometry | Tagged , , , , | 19 Comments

Laying train tracks

This morning I was playing trains with my son Felix. At the moment he is much more interested in laying the tracks than putting the trains on and moving them around, but he doesn’t tend to get concerned about whether … Continue reading

Posted in Ergodic Theory, Euclidean Geometry | Tagged , , , , | 19 Comments

Hyperbolic Geometry Notes #2 – Triangles and Gauss Bonnet

In this post, I will cover triangles and area in spaces of constant (nonzero) curvature. We are focused on hyperbolic space, but we will talk about spheres and the Gauss-Bonnet theorem. 1. Triangles in Hyperbolic Space Suppose we are given … Continue reading

Posted in Euclidean Geometry, Geometric structures, Hyperbolic geometry, Surfaces, Uncategorized | 2 Comments

Hyperbolic Geometry (157b) Notes #1

I am Alden, one of Danny’s students. Error/naivete that may (will) be found here is mine. In these posts, I will attempt to give notes from Danny’s class on hyperbolic geometry (157b). This first post covers some models for hyperbolic … Continue reading

Posted in Commentary, Euclidean Geometry, Groups, Hyperbolic geometry, Lie groups, Overview, Visualization | 5 Comments

Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface

Hermann Amandus Schwarz (1843-1921) was a student of Kummer and Weierstrass, and made many significant contributions to geometry, especially to the fields of minimal surfaces and complex analysis. His mathematical creations are both highly abstract and flexible, and at the … Continue reading

Posted in Complex analysis, Euclidean Geometry, Surfaces | Tagged , , , , , , , | 4 Comments