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Associated and perspective simplexes


Author: Leon Gerber
Journal: Trans. Amer. Math. Soc. 201 (1975), 43-55
MSC: Primary 50B10
DOI: https://doi.org/10.1090/S0002-9947-1975-0355788-5
MathSciNet review: 0355788
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0355788-5
Keywords: Associated lines, hyperbolic group of lines, perspective, simplex, orthological simplexes, polar simplex, Pappus theorem
Article copyright: © Copyright 1975 American Mathematical Society

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