Euclidean d-simplices with \(k\)-faces of the same volume
type: Item
instance of: scholarly article
title: Euclidean d-simplices with \(k\)-faces of the same volume
zbMATH Open document ID: 0965.51009
DOI: 10.1007/BF01237488
published in: Journal of Geometry
publication date: 2001-08-05T00:00:00Z
review text: Inspired by the well-known fact that a simplex in \(\mathbb{R}^3\) with equal 2-face areas has congruent 2-faces and by a corresponding question of H. Lenz referring to higher dimensions, the author proves the following theorem: For \(d\geq 4\) and each integer \(k\) with \(3\leq k\leq d-1\), there is a Euclidean \(d\)-dimensional simplex all whose \(k\)-faces have equal \(k\)-volumes but are not congruent to each other. The elegant proof is mainly based on linear algebra, using e.g. eigenvectors and eigenvalues of matrices corresponding to the simplices.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 1565906
zbMATH Keywords: simplices
zbMATH Keywords: \(k\)-faces
zbMATH Keywords: \(k\)-face volume
zbMATH Keywords: hyperbolic simplex
zbMATH Keywords: isosceles tetrahedron
zbMATH Keywords: equifacial tetrahedron
author: Bernulf Weissbach
MaRDI profile type: Publication
cites work: Determinantensätze und Simplexeigenschaften (Verallgemeinerung trigonometrischer Lehrsätze auf n‐dimensionale Simplexe)
cites work: Simplices with given 2-face areas
cites work: Regular Simplices in Spaces of Constant Curvature
cites work: Geometry II: spaces of constant curvature. Transl. from the Russian by V. Minachin
full work available at URL: https://doi.org/10.1007/bf01237488
OpenAlex ID: W2572101034
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description: scientific article; zbMATH DE number 1565906