- published: 05 Dec 2015
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In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. 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 are affinely independent, which means are linearly independent. Then, the simplex determined by them is the set of points
For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a 5-cell. A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices.
A regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.
In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices.
Some people wake up on Monday mornings
Barring maelstroms and red flare warnings
With no explosions and no surprises
Perform a series of exercises
Hold your fire
Take your place around an open fire
Before your neurons declare a crisis
Before your trace Serotonin rises
Before you're reading your coffee grounds
And before a pundit can make a sound
And before you're reading your list of vices
Perform the simplest exercises
So here we are at the end
The war is over
There's nothing left to defend
No cliffs of Dover
So let us put down our pens
And this concludes the test
Our minds are scattered about
From hell to breakfast
Hold your fire
Take your place around an open fire