||It has been suggested that this article be merged into simplex. (Discuss) Proposed since April 2013.|
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.
- A Euclidean 0-simplex is a point.
- A Euclidean 1-simplex is a line segment.
- A Euclidean 2-simplex is a triangle.
- A Euclidean 3-simplex is a tetrahedron.
Standard Euclidean simplex
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, i.e.
- 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.
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. The simplex with vertices yp+1, yp+2, …, yk − namely [yp+1, yp+2, …, yk] − is called the opposite face to [y0, y1, …, yp]. 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.
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).