ROLLING OF COXETER POLYHEDRA ALONG MIRRORS 1
... Also “mirrors” are hyperplanes of reflections. They divide the space Mn into chambers. The group G acts on the set of chambers simply transitively. We denote the reflection with respect to a mirror Y by sY . Each facet is contained in a unique mirror. 1.3. General Coxeter groups. Take a symmetric p ...
... Also “mirrors” are hyperplanes of reflections. They divide the space Mn into chambers. The group G acts on the set of chambers simply transitively. We denote the reflection with respect to a mirror Y by sY . Each facet is contained in a unique mirror. 1.3. General Coxeter groups. Take a symmetric p ...
A Variety of Proofs of the Steiner-Lehmus Theorem
... on the fact that the angles of any triangle sum to 180◦ . Many of those constructions and the justifications of the summation of a triangle’s angles are dependent on several theorems that are themselves limited to proofs by contradiction. To follow the most pure line of thought for direct proofs, a ...
... on the fact that the angles of any triangle sum to 180◦ . Many of those constructions and the justifications of the summation of a triangle’s angles are dependent on several theorems that are themselves limited to proofs by contradiction. To follow the most pure line of thought for direct proofs, a ...
Quadrilaterals
... b) Draw segment AD for an adjacent side to side AB (figure 1). c) Press F3 and select Parallel. To draw a line parallel to AB through point D outside the line, move the cursor to point D and press ENTER. Then move the cursor to AB and press ENTER. The line parallel to AB is shown in figure 1. d) Pre ...
... b) Draw segment AD for an adjacent side to side AB (figure 1). c) Press F3 and select Parallel. To draw a line parallel to AB through point D outside the line, move the cursor to point D and press ENTER. Then move the cursor to AB and press ENTER. The line parallel to AB is shown in figure 1. d) Pre ...
Smooth Maps
... and a “coordinate chart” is to be understood in the topological sense, as a homeomorphism from an open subset of the manifold to an open subset of Rn or Hn . If we wish to restrict attention to smooth manifolds or smooth coordinate charts, we will say so. Similarly, our default assumptions for many ...
... and a “coordinate chart” is to be understood in the topological sense, as a homeomorphism from an open subset of the manifold to an open subset of Rn or Hn . If we wish to restrict attention to smooth manifolds or smooth coordinate charts, we will say so. Similarly, our default assumptions for many ...
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f mapping a compact convex set into itself there is a point x0 such that f(x0) = x0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself.Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard.Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods.This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard and by Luitzen Egbertus Jan Brouwer.