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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 conformal map, Liouville's theorem, Rigidity, umbilical surface
7 Comments
Orthocentricity
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
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
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 central limit theorem, local limit theorem, Markov chain, tiling, train tracks
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
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
Brianchon-Gram-Sommerville and ideal hyperbolic Dehn invariants
A beautiful identity in Euclidean geometry is the Brianchon-Gram relation (also called the Gram-Sommerville formula, or Gram’s equation), which says the following: let be a convex polytope, and for each face of , let denote the solid angle along the … Continue reading
Geometry and the imagination 
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 →