This “paradox” was explain to me by a colleague. Take a square whose side have length $1$ (on blue on the figure below) and subdivide it in four squares of side $1/2$. In the middle of each smaller square place a disc which is tangent to the four sides (on white on the picture below). In particular the radius of these disc is $1/4$. Now place a fifth disc (red on the picture below), centered at the center of the original (blue) square which is tangent to the four other discs.

The Uniformization Theorem of complex analysis tells us that there are three important two-dimensional geometries: hyperbolic, euclidean and spherical. In contrast, the correct three-dimensional generalization, Thurston’s Geometrization Conjecture, singles out eight geometries fundamental to three-dimensional topology, namely $$\mathbf S^3,\ \mathbf E^3,\ \mathbf H^3,\ \mathbf S^2 \times \mathbf E,\ \mathbf H^2 \times \mathbf E,\ {\rm Nil},\ {\rm Sol},\ \widetilde{{\rm SL}(2,\mathbf R)}$$ The aim of this project is to develop accurate, real time, intrinsic, and mathematically useful illustrations of these geometries and more generally of homogeneous (pseudo)-riemannian spaces.

A (single) pendulum is a standard object studied in every mechanics textbook. Its tidy behavior is advantageously used to punctuate grandfather’s clocks. A grandfather’s clock. Image credit: © Arnaud Clerget A double Pendulum is a pendulum, with another pendulum attached to its end (see the main picture of this post). Its behavior is much more intriguing. Physically this object can be modeled as follow. We write $m_i$ and $l_i$ for the mass and the length of the $i$-th limb.