Euclidean simplex

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In mathematics and especially algebraic topology and homology theory, a Euclidean simplex is a special type of convex set in Euclidean space. It generalises the idea of a triangle, and is used for triangulations.

Definition[edit]

A Euclidean 3-simplex in E3.

Let y0, y1, …, yk be linearly independent points in Euclidean n-space, denoted En. Let S be a subset of En given by

 S := \left\{ \sum_{i=0}^k \lambda_i\bold{y}_i : \lambda_i \ge 0 \ \text{and} \ \sum_{j=0}^k \lambda_j = 1 \right\} .

The set S is called a Euclidean k-simplex with vertices y0, y1, …, yk, and is often denoted as [y0, y1, …, yk]. Given a point y in S, the λi give barycentric coordinates on S.[1]

Examples[edit]

Standard Euclidean simplex[edit]

The standard Euclidean k-simplex, denoted by Δk, is taken to be a subset of Ek+1 and is given by [x0, x1, …, xk] where xi has a 1 in the (i+1)st position and a zero everywhere else,[1] i.e.

 \bold{x}_i = (\underbrace{0,\ldots,0}_{i},1,\underbrace{0,\ldots,0}_{k-i}) \ \ \text{for all} \ 0 \le i \le k .

Examples[edit]

  • Δ0 is the point 1 in E1.
  • Δ1 is the line segment joining (1,0) and (0,1) in E2.
  • Δ2 is the triangle with vertices (1,0,0), (0,1,0) and (0,0,1) in E3.
  • Δ3 is the tetrahedron with vertices (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) in E4.

Faces[edit]

Given a Euclidean k-simplex [y0, y1, …, yk], the Euclidean p-simplex with vertices y0, y1, …, yp − namely [y0, y1, …, yp] − is called a p-dimensional face of the k-simplex S.[1] The simplex with vertices yp+1, yp+2, …, yk − namely [yp+1, yp+2, …, yk] − is called the opposite face to [y0, y1, …, yp].[1] A Euclidean k-simplex has faces of all dimensions, from 0 to k. The 0-dimensional faces are the vertices, whilst the k-dimensional face is the k-simplex itself.

Examples[edit]

Consider the standard Euclidean 3-simplex Δ3.

  • The 0-dimensional faces are the vertices of Δ3. Consider the 0-dimensional face (1,0,0,0). The opposite face is a 2-dimensional face; namely the (non-standard) Euclidean 2-simplex given by the triangle with vertices (0,1,0,0), (0,0,1,0) and (0,0,0,1).
  • The 1-dimensional faces are the six edges of the tetrahedron. Consider the 1-dimensional face given by the line segment joining (1,0,0,0) to (0,1,0,0). The opposite face is a 1-dimensional face; namely the (non-standard) Euclidean 1-simplex given by the line segment joining (0,0,1,0) to (0,0,0,1).
  • The 2-dimensional faces are the four traditional triangular faces of the tetrahedron. Consider the 2-dimensional face given by the triangle with vertices (1,0,0,0), (0,1,0,0) and (0,0,1,0). The opposite face is a 0-dimensional face; namely the (non-standard) Euclidean 0-simplex with vertex (0,0,0,1).

References[edit]

  1. ^ a b c d Wallace, Andrew H. (Jan 2009), Algebraic Topology: Homology & Cohomology, Dover Publications Inc., ISBN 0-486-46239-0