Publication Date

1-1-2024

Document Type

Article

Publication Title

Journal for Geometry and Graphics

Volume

28

Issue

2

First Page

155

Last Page

168

Abstract

The twin tetrahedron of a given tetrahedron is obtained by circumscribing it by a parallelepiped. However, in general, it is not easy to construct a box that circumscribes a tetrahedron. Actually, constructing a box is equivalent of finding two tangled tetrahedra. We first establish a theorem to construct tangled tetrahedra circumscribed in a box with concurrent diagonals. This generalizes the idea of twin tetrahedra circumscribed in a parallelepiped. And we show that two tetrahedra are twins if and only if they are tangled with concurrent diagonals at the centroid of one of the tetrahedra. We establish a theorem in order to give an alternate proof of this theorem, which we think is a new characterization of the centroid of a tetrahedron. Then we prove that there is a tetrahedron that tangles a reversible tetrahedron with concurrent diagonals such that these two tetrahedra are congruent after relabeling vertices. In addition, both of these tetrahedra can be circumscribed by the same sphere.

Keywords

Skew quadrilateral, quadrilateral, tetrahedron, hexahedron with eight vertices, box, tangled tetrahedra, tangled tetrahedra with concurrent diagonals, parallelepiped, twin tetrahedra, isosceles tetrahedron, reversible tetrahedron, trapezoidal box

Comments

© 2024 by the author.

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution-Share Alike 4.0 License.

Department

Mathematics and Statistics

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