Mathematics

Too little mathematics has been written in prose. Thus we prove here, via a fantasy novellette, that a locally L-bilipschitz mapping f:X→Y between uniformly Ahlfors q -regular, complete and locally compact path-metric spaces X and Y is an L -bilipschitz map when Y is simply connected. The motivation for such a result arises from studying the asymptotic values of BLD-mappings with an empty branch set; see e.g. [L17]. As far as the author is aware, the result is new, even though it would not be hard for specialists in the field to prove. The proof is essentially a modest extension of the ideas in [L17] in a more general setting when the branch set is empty.

Nicolas-Auguste Tissot (1824--1897) was a French mathematician and cartographer. He introduced a tool which became known among geographers under the name ``Tissot indicatrix'', and which was widely used during the first half of the twentieth century in cartography. This is a graphical representation of a field of ellipses, indicating at each point of a geographical map the distorsion of this map, both in direction and in magnitude. Each ellipse represented at a given point is the image of an infinitesimal circle in the domain of the map (generally speaking, a sphere representing the surface of the earth) by the projection that realizes the geographical map. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane, and he also developed a theory for the distorsion of mappings between general surfaces. His ideas are close to those that are at the origin of the work on quasiconformal mappings that was developed several decades after him by Gr{ö}tzsch, Lavrentieff, Ahlfors and Teichm{ü}ller. Gr{ö}tzsch, in his papers, mentions the work of Tissot, and in some of the drawings he made for his articles, the Tissot indicatrix is represented. Teichm{ü}ller mentions the name Tissot in a historical section in one of his fundamental papers in which he points out that quasiconformal mappings were initially used by geographers. The name Tissot is missing from all the known historical reports on quasiconformal mappings. In the present article, we report on this work of Tissot, showing that the theory of quasiconformal mappings has a practical origin. The final version of this article will appear in Vol. VII of the Handbook of Teichm{ü}ller Theory (European Mathematical Society Publishing House, 2020).

Some relations among Pythagorean triples are established. The main tool is a fundamental characterization of the Pythagorean triples through a chatetus which allows to determine relationships with Pythagorean triples having the same chatetus raised to an integer power.

The problem of the longest head run was introduced and solved by Abraham de Moivre in the second edition of his book Doctrine of Chances (de Moivre, 1738). The closed-form solution as a finite sum involving binomial coefficients was provided in Uspensky (1937). Since then, the problem and its variations and extensions have found broad interest and diverse applications. Surprisingly, a very simple closed form can be obtained, which we present in this note.

In an analogous construction as by Euler for 4x4 matrices, a parametrization of 8x8 magic squares of squares with orthogonal rows is shown to be obtainable by extending the quaternionic method, as shown by Hurwitz, to octonions, but not possible to be carried even further in the Cayley-Dickson construction series.

We give a probabilistic proof of the orbit-counting lemma.

This note offers a probabilistic proof of Girard's angle excess formula for the area of a spherical triangle, based on the observation that an unbounded 3-dimensional convex cone, with single vertex at the origin, has only three kinds of 2-dimensional orthogonal projections: a 2-dimensional convex cone with one vertex, a 2-dimensional half-plane (this is an outcome of probability zero), and a 2-dimensional plane.

"No two rainbows are the same. Neither are two packs of Skittles. Enjoy an odd mix!". Using an interpretation via spatial random walks, we quantify the probability that two randomly selected packs of Skittles candy are identical and determine the expected number of packs one has to purchase until the first match. We believe this problem to be appealing for middle and high school students as well as undergraduate students at University.

About Conway's surreal numbers: A letter to a friend (written in French). In memoriam John Horton Conway.

We proposed a purely 3-D geometrical solution to Mathematics Magazine Problem 2065. Let Q be a cube centered at the origin of R 3 . Choose a unit vector (a,b,c) uniformly at random on the surface of the unit sphere a 2 + b 2 + c 2 =1 , and let Π be the plane ax+by+cz=0 through the origin and normal to (a,b,c) . What is the probability that the intersection of Π with Q is a hexagon?