## On space in the context of curvatures, cusps, billiard maps

Learn from “Mathematical Omnibus: Thirty Lectures on Classic Mathematics”, Dmitry Fuchs, Serge Tabachnikov

– “The evolute of a curve is the locus of centers of curvature. A vertex of the curve corresponds to a singularity of the evolute, generically, a cusp”

– “The osculating circles of an arc with monotonic positive curvature are nested.”

– “If a differentiable function in the annulus is constant on each circle then this is a constant function.”

– “For every oval C, the outer billiard map is area preserving”

– “If two closed smooth curves have the same winding numbers then one can be continuously deformed to the other.”

– “Two generic smooth spherical curves can be continuously deformed to each other if and only if their numbers of double points are either both even or both odd.”

– “In space, typical curves have no points of zero curvature.”

– “The osculating plane is the plane through three infinitesimally close points of the curve.”

– “There are no non-planar triply ruled surfaces.”

– “A surface of degree 3 contains 27 or infinite number of straight lines.”

– Crofton formula (used for solving Hilbert’s fourth problem)

– Fary-Milnor: “If a closed spacial curve is knotted then its total curvature is greater than 4pi.”

– “There exist no simple closed geodesics on closed convex polyhedron”

– Gauss-Bonnet theorem, Dehn’s theorem

– “If a rectangle is tiled by squares then the ratio of its side lengths is a rational number.”

– “A rectangle R is tiled by rectangles each of which has an integer side. Then R has an integer side.”

– “If the corresponding faces of two convex polyhedra are congruent and adjacent in the same way then the polyhedra are congruent as well.”

– Cauchy: “Any convex polyhedron is rigid (cannot be bent).”

– Connelly: “There exists a (non-convex) flexible polyhedron.”

– “simple connectedness: every closed curve can be continuously deformed to one point.”

– “The equivipotential surfaces of the free distribution of charge on an ellipsoid are the confocal ellipsoids.”