Boxes and Tangled Tetrahedra

H. Katsuura

Journal for geometry and graphics, Tome 28 (2024) no. 2, pp. 155-168

Résumé

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.

Détail

Classification : 51M04
Mots-clés : 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