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Tetrahedra passing through a triangular hole, and tetrahedra fixed by a planar frame

Barany, I; Maehara, H; Tokushige, N; (2012) Tetrahedra passing through a triangular hole, and tetrahedra fixed by a planar frame. COMP GEOM-THEOR APPL , 45 (1-2) 14 - 20. 10.1016/j.comgeo.2011.07.004.

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Abstract

We show that a convex body can pass through a triangular hole iff it can do so by a translation along a line perpendicular to the hole. As an application, we determine the minimum size of an equilateral triangular hole through which a regular tetrahedron with unit edge can pass. The minimum edge length of the hole is (1 + root 2)/root 6 approximate to 0.9856. One of the key facts for the proof is that no triangular frame can hold a convex body. On the other hand, we also show that every non-triangular frame can fix some tetrahedron. (C) 2011 Elsevier B.V. All rights reserved.

Type:Article
Title:Tetrahedra passing through a triangular hole, and tetrahedra fixed by a planar frame
DOI:10.1016/j.comgeo.2011.07.004
Keywords:Frame, Holding a convex body, Fixing a convex body, Regular tetrahedron, Minimal embedding, COVER
UCL classification:UCL > School of BEAMS > Faculty of Maths and Physical Sciences > Mathematics

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