Related papers: One Hundred and Twelve Point Three Degree Theorem
There has been recent work using Shape Theory to answer the longstanding and conceptually interesting problem of what is the probability that a triangle is obtuse. This is resolved by three kissing cap-circles of rightness being realized on…
The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed.
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle.…
Erd\H{o}s and Hajnal proposed a problem that: is it true that every $(2n+1)$-vertex graph with $n^2+n+1$ edges contains two vertices of equal degree connected by a path of length three? The edge bound is sharp by the complete bipartite…
We give a tree structure on the set of all periodic directions on the golden L, which gives an associated tree structure on the set of periodic directions for the pentagon billiard table and double pentagon surface. We use this to give the…
We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families…
An existence of an aperiodic point for outer billiard outside regular dodecagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic, and all possible periods are listed explicitly. The proof is…
In any periodic direction on the regular pentagon billiard table, there exists two combinatorially different billiard paths, with one longer than the other. For each periodic direction, McMullen asked if one could determine whether the…
Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins' theorem [Robbins, Am. Math. Monthly, 1939] states that such an orientation exists if and only if the graph is…
The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study the complex version of Ivrii's…
Given a right triangle ABC, with the ninety degree angle at A; consider the triangle O1OO2.Where the point O is the midpoint of the hypotenuseBC(and so the center of the triangle ABC's circumcircle), the point O1 being the triangle AOB's…
Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of…
The Road Coloring Theorem states that every aperiodic directed graph with constant out-degree has a synchronized coloring. This theorem had been conjectured during many years as the Road Coloring Problem before being settled by A. Trahtman.…
The periodic wind-tree model is a family T(a,b) of billiards in the plane in which identical rectangular scatterers of size axb are disposed at each integer point. It was proven by P. Hubert, S. Leli\`evre and S. Troubetzkoy…
In 1962, Tutte provided a formula for the number of combinatorial triangulations, that is, maximal planar graphs with a fixed triangular face and $n$ additional vertices. In this note, we study how many ways a combinatorial triangulation…
Iteration of the quadratic map produces sequences of polynomials whose degrees {\sl explode} as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree $4020$, while for the $52,377$ period-20…
Paul Erdos asked if, among sufficiently many points in general position, there are always $k$ points such that all the circles through $3$ of these $k$ points have different radii. He later proved that this is indeed the case. However, he…
We give lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. We apply a topological approach based on the…
In this paper, we derive the exact formula for the probability that three randomly and uniformly selected points from the interior of the unit cube form vertices of an obtuse triangle.
We show that the complexity of the billiard in a typical polygon grows cubically and the number of saddle connections grows quadratically along certain subsequences. It is known that the set of points whose first n-bounces hits the same…