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Equilateral triangle

About: Equilateral triangle is a research topic. Over the lifetime, 2414 publications have been published within this topic receiving 25593 citations. The topic is also known as: regular triangle & equiangular triangle.


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Journal ArticleDOI
TL;DR: A graphical method, likelihood-mapping, is introduced, to visualize the phylogenetic content of a set of aligned sequences, based on an analysis of the maximum likelihoods for the three fully resolved tree topologies that can be computed for four sequences.
Abstract: We introduce a graphical method, likelihood-mapping, to visualize the phylogenetic content of a set of aligned sequences. The method is based on an analysis of the maximum likelihoods for the three fully resolved tree topologies that can be computed for four sequences. The three likelihoods are represented as one point inside an equilateral triangle. The triangle is partitioned in different regions. One region represents star-like evolution, three regions represent a well-resolved phylogeny, and three regions reflect the situation where it is difficult to distinguish between two of the three trees. The location of the likelihoods in the triangle defines the mode of sequence evolution. If n sequences are analyzed, then the likelihoods for each subset of four sequences are mapped onto the triangle. The resulting distribution of points shows whether the data are suitable for a phylogenetic reconstruction or not.

854 citations

Journal ArticleDOI
TL;DR: In this article, a simple random walk method for obtaining ab initio solutions of the Schrodinger equation is examined in its application to the case of the molecular ion H+3 in the equilateral triangle configuration with side length R=1.66 bohr.
Abstract: A simple random‐walk method for obtaining ab initio solutions of the Schrodinger equation is examined in its application to the case of the molecular ion H+3 in the equilateral triangle configuration with side length R=1.66 bohr. The method, which is based on the similarity of the Schrodinger equation and the diffusion equation, involves the random movement of imaginary particles (psips) in electron configuration space subject to a variable chance of multiplication or disappearance. The computation requirements for high accuracy in determining energies of H+3 are greater than those of existing LCAO–MO–SCF–CI methods. For more complex molecular systems the method may be competitive.

810 citations

Proceedings ArticleDOI
31 Jul 2005
TL;DR: A clean algorithm for determining whether a ray intersects a triangle which is comparable in speed to previous methods and is believed to be the fastest ray/triangle intersection routine for triangles which do not have precomputed plane equations.
Abstract: We present a clean algorithm for determining whether a ray intersects a triangle. The algorithm translates the origin of the ray and then changes the base of that vector which yields a vector (t u v)T, where t is the distance to the plane in which the triangle lies and (u, v) represents the coordinates inside the triangle.One advantage of this method is that the plane equation need not be computed on the fly nor be stored, which can amount to significant memory savings for triangle meshes. As we found our method to be comparable in speed to previous methods, we believe it is the fastest ray/triangle intersection routine for triangles which do not have precomputed plane equations.

705 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the dependence on the configuration in momentum space of the primordial three-point function of density perturbations in several different scenarios: standard slow-roll inflation, curvaton and variable decay models, ghost inflation, models with higher derivative operators and the DBI model of inflation.
Abstract: We study the dependence on the configuration in momentum space of the primordial three-point function of density perturbations in several different scenarios: standard slow-roll inflation, curvaton and variable decay models, ghost inflation, models with higher derivative operators and the DBI model of inflation. We define a cosine between the distributions using a measure based on the ability of experiments to distinguish between them. We find that models fall into two broad categories with fairly orthogonal distributions: models where non-Gaussianity is created on crossing the horizon during inflation and models in which the evolution beyond the horizon dominates. In the first case the three-point function is largest for equilateral triangles, while in the second the dominant contribution to the signal comes from the influence of long wavelength modes on small wavelength ones. We show that, because the distributions in these two cases are so different, translating constraints on parameters of one model to those of the other on the basis of the normalization of the three-point function for equilateral triangles can be very misleading.

618 citations

Journal ArticleDOI
TL;DR: In this paper, it is shown that the best way to cover a given area with circles of a given radius E is to place the centers of the circles on an equilateral triangle network, i.e., to circumscribe (inscribe) the circles about the hexagons of a regular hexagon network or honeycomb.
Abstract: It is well known that the " best " way to cover a given area with circles of a given radius E, or to pack such circles within a given region is to place the centers of the circles on an equilateral triangle network, i. e., to circumscribe (inscribe) the circles about the hexagons of a regular hexagon network or honeycomb. This, of course, is not a precise statement, and, in fact, it is difficult to make a precise statement in this direction that is true. Roughly, the statement becomes more true as E is taken smaller in relation to the area of the given region. The most usual 1 precise statement of this fact is that the densest plane Punktgitter is that of the equilateral triangle. This statement avoids the difficulties caused by the boundedness of the bounded region but is less general than might be desired in that permissible packings or coverings are limited to those in which the centers of the circles form a Punktgitter. The object of the present paper is to give, a new aiid elementary proof of a precise statement in this direction; a statement involving no restriction on the nature of permissible coverings or on the nature of the given region. Specifically the statement to be proved is the following:

547 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202371
2022208
202176
202087
201984