Related papers: Average angles of triangle in regular polygons
We compute the generalized triangle inequalities explicitly for all rank 3 symmetric spaces. We find that for Sp(6) the corresponding polyhedral cone has 102 facets and 51 edges.
We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find…
The concept of the cyclic averages are introduced for a regular polygon $P_n$ and a Platonic solid $T_n$. It is shown that cyclic averages of equal powers are the same for various $P_n(T_n)$, but their number is characteristic of…
We introduce a bulging triangle like the generalization of the Reuleaux triangle. We may be able to propose various ways to bulge a triangle, but this paper presents the way so that its vertices are the same as them of the original…
We apply a new notion of angle between projections to deduce criteria for uniform convergence results of the alternating projections method under several different settings: averaged projections, cyclic products, quasi-periodic products and…
We improve the estimates of the subgraph probabilities in a random regular graph. Using the improved results, we further improve the limiting distribution of the number of triangles in random regular graphs.
Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…
The equality constraint a+b+c=1 for random triangle sides corresponds to breaking a stick in two places. An analog a^2+b^2+c^2=1 has a remarkable feature: the bivariate density for angles coincides with that for 3D Gaussian triangles.…
A triangle with rational sides and rational area is called a rational triangle. In this paper we consider three problems of finding pairs of rational triangles which have a common circumradius as well as either a common perimeter or a…
This paper presents formulae for calculation the solid angle of intersecting spherical caps, conical surfaces and polyhedral cones.
In this survey article, we are interested on minimal triangulations of closed pl manifolds. We present a brief survey on the works done in last 25 years on the following: (i) Finding the minimal number of vertices required to triangulate a…
In this paper we consider the problem of finding pairs of triangles whose sides are perfect squares of integers, and which have a common perimeter and common area. We find two such pairs of triangles, and prove that there exist infinitely…
We examine implications of angles having their own dimension, in the same sense as do lengths, masses, {\it etc.} The conventional practice in scientific applications involving trigonometric or exponential functions of angles is to assume…
We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of…
We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.…
Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of $a^3b$-quadrilaterals with some irrational angle: there are a sequence of…
Starting from any given rational-sided, right triangle, for example the $(3,4,5)$-triangle with area $6$, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show…
We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.
Let a polygon be composed of equal rectangles. We find all quadratic irrationals r for which the polygon can be tiled by similar rectangles with given side ratio r.
We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. A focus is given to dimension 3 where slicings are normal surfaces. In the case of 2-neighborly 3-manifolds and quadrangulated slicings, a…