Associated and perspective simplexes

Abstract:

A set of $n + 1$ lines in $n$-space such that any $({\text {n}} - 2)$-dimensional flat which meets $n$ of the lines also meets the remaining line is said to be an associated set of lines. Two Simplexes are associated if the joins of corresponding vertices are associated. A simple criterion is given for simplexes to be associated and an analogous one for Simplexes to be perspective. These are used to give a brief proof of the following generalization of the theorem of Pappus. Let $\mathcal {A}$ and $\mathcal {B}$ be $n$-simplexes and let $p$ be a permutation on the vertices of $\mathcal {B}$. If $\mathcal {A}$ and $\mathcal {B}$ are associated (respectively perspective) and $\mathcal {A}$ and $\mathcal {B}p$ are associated (perspective) then $\mathcal {A}$ and $\mathcal {B}{p^k}$ are associated (perspective) for any integer $k$. Very short proofs are given of extensions to $n$-dimensions of many theorems from Neuberg’s famous Memoir sur le Tétraèdre, such as: the altitudes of a simplex are associated.