Triangles are an important building block and distinguishing feature of real-world networks, but their structure is still poorly understood. Despite numerous reports on the abundance of triangles, there is very little information on what these triangles look like. We initiate the study of degreelabeled triangles — specifically, degree homogeneity versus heterogeneity in triangles. This yields n...

133[F].—F. L. Miksa, Table of primitive Pythagorean triangles with their perimeters arranged in ascending order from 119992 to 499998. 506 typewritten leaves on deposit in UMT File. This table is a continuation of UMT 111 \_MTAC, v. 5, p. 28], a table by A. S. Anema to 120000. The introduction gives a table showing the number of triangles whose perimeters do not exceed P for P = 120000 (10000)5...

We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues.

This paper presents an adaptive modification of the Edge spinning method for polygonization of implicit surfaces. The method insists on the shape of triangles and the accuracy of resulting approximation as well. The main advantages of the triangulation presented are simplicity and the stable features that can be used for next expanding. The implementation is not complicated and only the standar...

A perfect dissection of a polygon P into a polygon P ′ is a decomposition of P into internally disjoint pairwise incongruent polygons all similar to P ′. It is known that there is no perfect dissection of an equilateral triangle into smaller equilateral triangles. On the other hand, an equilateral triangle has trivial perfect dissections into any number n of right triangles where n ≥ 3. We give...

The final-state phases in B → Dπ, Dπ, and Dρ decays appear to follow a pattern similar to those in D → Kπ, K∗π, and Kρ decays. Each set of processes is characterized by three charge states but only two independent amplitudes, so the amplitudes form triangles in the complex plane. For the first two sets the triangles appear to have non-zero area, while for the Dρ or Kρ decays the areas of the tr...

We prove tight bounds on the complexity of bisectors and Voronoi diagrams on so-called realistic terrains, under the geodesic distance. In particular, if n denotes the number of triangles in the terrain, we show the following two results. (i) If the triangles of the terrain have bounded slope and the projection of the set of triangles onto the xy-plane has low density, then the worst-case compl...

In this article we present an algorithm to determine the intersection between rays and triangles based on the idea of the study of signs with respect to triangles. One of the advantages of this approach is its robustness due to its lack of trigonometric operations or complex divisions which might alter the result of the calculations. The algorithm is similar (or even better) in time to other ex...