Equilateral triangle | EPFL Graph Search

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac{\sqrt{3}}{4} a^2, *The perimeter is p=3a,! *The radius of the circumscribed circle is R = \frac{a}{\sqrt{3}} *The radius of the inscribed circle is r=\frac{\sqrt{3}}{6} a or r=\frac{R}{2} *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac{\sqrt{3}}{2} a Denoting the radius of the circum

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Michal Lason, Bartosz Maria Walczak

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set in that is not an axis-aligned rectangle and for any positive integer produces a family of sets, each obtained by an independent horizontal and vertical scaling and translation of , such that no three sets in pairwise intersect and . This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.

Springer US2013

Etude expérimentale de refuges à poissons aménagés dans les berges de rivières soumises aux éclusées hydroélectriques

Jean-Marc Ribi

Downstream of hydroelectric storage dams, the rivers suffer from rapid, frequent and intense flow variations linked to the power production constraints. Subjected to this hydropeaking phenomenon, the fish population suffers from a known decline. In the research of mitigation solutions, shelters in riverbanks are a protection measure against excessive flow velocities. This project gives the design basis for such shelters. The research process is based on experimentations with fish swimming in a canal built on purpose with a lateral embayment. The principle is to generate hydrodynamic conditions that are uncomfortable in the canal on one hand and attracts the fish to the shelter on the other hand. The significant parameter is the flow velocity distribution in the intersection between the main channel and the lateral fish shelter. The frequentation rate of fish in the shelter can be considered as the performance indicator. With a length of 12 m and a width of 1.2 m, the canal is supplied with freshwater from a river. At mid-length, the canal holds a one side enlargement of 1.2 m, shaped like a rectangular gap of 2 m long. For each test, after a steady flow of 10 l/s with a mean velocity of 0.1 m/s, 10 to 20 juvenile brown trout are suddenly exposed during 3 hours to a hydropeaking flow of 220 l/s. At that time, the mean velocity in the canal is of 1 m/s. The fish position is observed every 20 minutes and their movements are recorded continuously in the shelter sector. The first tests have been conducted with a basic rectangular shelter configuration. The frequentation rate was only of 35%. This rate has doubled by introducing a deviation groyne that enforced a water exchange between the canal and the shelter. Following this experiment, the position and the orientation angle of the groyne have been systematically varied with the purpose of maximising the frequentation rate. This way, 12 configurations have been tested 3 times each. The flows transiting in the shelter are calculated using a 2D numerical simulation. By analysing the fish positioning, the mean frequentation rate is determined for each configuration. The maximum rate is of 87%, the mean is of 75% and the standard deviation is of 8%. Using video pictures treatment, the preferential fish trajectories are updated and the fish entrance distribution is determined for each configuration at the interface between the shelter and the channel. On the same interface, ultrasonic velocity profiles (UVP) have been measured for the 12 configurations. By associating the fish entries distribution with the velocities distribution, the preferences are globally established. The entrances happen mostly from downstream of the derivation groyne, in a flow shear zone where the velocities are in a range of 0.0 to 0.2 m/s. Finally, the analysis of all the results confirmed the representativity of the configuration that gives the best frequentation rate, for the development of the experimental shelter's prototype to be realised in rivers. It is composed of a groyne shaped like an equilateral triangle. Finally, recommendations are given for the execution of an experimental prototype shelter to be realized in a river for validation tests.

EPFL2011

Two and a half billion years of distance research

Filip Moric, János Pach

In a seminal paper published in 1946, Erd ̋os initiated the investigation of the distribution of distances generated by point sets in metric spaces. In spite of some spectacular par- tial successes and persistent attacks by generations of mathe- maticians, most problems raised in Erd ̋os’ paper are still un- solved. GivenasetofnpointsinRd,letd1 >d2 >d3 >··· denote the sequence of all distances between pairs of points in P, listed in decreasing order. We raise some simple ques- tions related to a famous conjecture of Schur. For instance, is it true that any two regular (d − 1)-dimensional simplices of side length d1 induced by P share at least one vertex? We prove that if P is the vertex set of a convex polygon in R2, then the maximum number of equilateral triangles of side length dk induced by P is Θ(k).

2013

Etude expérimentale de refuges à poissons aménagés dans les berges de rivières soumises aux éclusées hydroélectriques

Jean-Marc Ribi

EPFL2011