In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space.
For example,
a 0-simplex is a point,a 1-simplex is a line segment,a 2-simplex is a triangle,a 3-simplex is a tetrahedron,a 4-simplex is a 5-cell.Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u 0 , … , u k ∈ R k {\displaystyle u_{0},\dots ,u_{k}\in \mathbb {R} ^{k}} are affinely independent, which means u 1 − u 0 , … , u k − u 0 {\displaystyle u_{1}-u_{0},\dots ,u_{k}-u_{0}} are linearly independent.Then, the simplex determined by them is the set of points
C = { θ 0 u 0 + ⋯ + θ k u k | ∑ i = 0 k θ i = 1 and θ i referenceEver curious about what that abbreviation stands for? fullforms has got them all listed out for you to explore. Simply,Choose a subject/topic and get started on a self-paced learning journey in a world of fullforms.
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