Simplex - The Standard Simplex - Increasing Coordinates

Increasing Coordinates

An alternative coordinate system is given by taking the indefinite sum:

\begin{align} s_0 &= 0\\ s_1 &= s_0 + t_0 = t_0\\ s_2 &= s_1 + t_1 = t_0 + t_1\\ s_3 &= s_2 + t_2 = t_0 + t_1 + t_2\\ &\dots\\ s_n &= s_{n-1} + t_{n-1} = t_0 + t_1 + \dots + t_{n-1}\\ s_{n+1} &= s_n + t_n = t_0 + t_1 + \dots + t_n = 1 \end{align}

This yields the alternative presentation by order, namely as nondecreasing n-tuples between 0 and 1:

Geometrically, this is an n-dimensional subset of (maximal dimension, codimension 0) rather than of (codimension 1). The hyperfaces, which on the standard simplex correspond to one coordinate vanishing, here correspond to successive coordinates being equal, while the interior corresponds to the inequalities becoming strict (increasing sequences).

A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamental domain for the action of the symmetric group on the n-cube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the n-cube into mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume Alternatively, the volume can be computed by an iterated integral, whose successive integrands are

A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.

Read more about this topic:  Simplex, The Standard Simplex

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