In this work, we introduce the notion of triangle sparsifiers, i.e., sparse graphs which are approximately the same to the original graph with respect to the triangle count. This results in a practical triangle counting method with strong theoretical guarantees. For instance, for unweighted graphs we show a randomized algorithm for approximately counting the number of triangles in a graph G, wh...

We consider the problem of triangle guarding (4-guarding) a simple, 2D polygon Q. A polygon Q is 4-guarded if every point q of Q is contained in the convex hull of some three guards that can all see q. This rather odd condition approximates a desire, for example, to see all sides of q or to locate (via triangulation) q from at least two angularly well-separated views. If we restrict the guards ...

We calculate the values of the trigonometric functions for angles: π3 and π 6 , by [16]. After defining some trigonometric identities, we demonstrate conventional trigonometric formulas in the triangle, and the geometric property, by [14], of the triangle inscribed in a semicircle, by the proposition 3.31 in [15]. Then we define the diameter of the circumscribed circle of a triangle using the d...

We prove the hyperbolicity, ergodicity and thus the Bernoulli property of two hard balls in one of the following four polygons: the square, the equilateral triangle, the 45 − 45 − 90◦ triangle or the 30− 60− 90◦ triangle.

Among all bodies of constant width in the Euclidean plane, the Reuleaux triangle of the same width has minimal area. But Reuleaux triangles are also minimal in another sense: if a convex body can be covered by a translate of a Reuleaux triangle, then it can be covered by a translate of any convex body of the same constant width. The first result is known as the Blaschke-Lebesgue theorem, and it...

It is conjectured that given positive integers l, m, n with l−1 + m−1 + n−1 < 1 and an integer g > 0, the triangle group ∆ = ∆(l,m, n) = 〈X,Y,Z|X l = Y m = Z = XY Z = 1〉 contains infinitely many subgroups of finite index and of genus g. A slightly stronger version of this conjecture is as follows: given positive integers l, m, n with l−1 + m−1 + n−1 < 1 and an integer g > 0, there are infinitel...

A mathematical proof of how fast the diameters of a triangle mesh tend to zero after repeated trisection Francisco Perdomo a,b, Ángel Plaza a,b,∗, Eduardo Quevedo c, José P. Suárez a,d a Division of Mathematics, Graphics and Computation (MAGiC). IUMA, Information and Communication Systems, University of Las Palmas de Gran Canaria (ULPGC), Spain b Department of Mathematics, (ULPGC), Spain c Depa...

We study a combinatorial game called Bichromatic Triangle Game, defined as follows. Two players R and B construct a triangulation on a given planar point set V . Starting from no edges, they take turns drawing one straight edge that connects two points in V and does not cross any of the previously drawn edges. PlayerR uses color red and player B uses color blue. The first player who completes o...

Using Kummer’s theorem, we give a necessary and sufficient condition for a Narayana number to be divisible by a given prime. We use this to derive certain properties of the Narayana triangle. 1 The main theorem Let N denote the nonnegative integers and let k, n ∈ N. The Narayana numbers [10, A001263] can be defined as N(n, k) = 1 n (

The paper is focused on how chaotic patterns, occurring in nature, might be used by biological 7 organisms to perform computations. This issue is investigated in the context of neural systems. As a simple model of chaotic patterns, the world of Sierpinski triangles is analysed. The paper 9 introduces the Sierpinski basis functions and the Sierpinski brain, which is able to perform classi)cation...