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J. Geom. (2019) 110:49
c
2019 Springer Nature Switzerland AG
0047-2468/19/030001-11
published online September 17, 2019
https://doi.org/10.1007/s00022-019-0506-y Journal of Geometry
Duality of isosceles tetrahedra
Jan Brandts and Michal Kˇr´ıˇzek
Dedicated to Dr. Milan Pr´ager on his 90th birthday
Abstract. In this paper we define a so-called dual simplex of an n-simplex
and prove that the dual of each simplex contains its circumcenter, which
means that it is well-centered. For triangles and tetrahedra Swe inves-
tigate when the dual of S, or the dual of the dual of S, is similar to
S, respectively. This investigation encompasses the study of the iterative
application of taking the dual. For triangles, this iteration converges to
an equilateral triangle for any starting triangle. For tetrahedra we study
the limit points of period two, which are known as isosceles or equifacetal
tetrahedra.
Keywords. Well-centered simplices, Dual simplices, Isosceles tetrahedra,
Circumcenter, Circumradius.
1. Preliminaries
Let m≥nbe positive integers. Recall that an n-simplex Sin an m-dimensional
Euclidean space Rmis the convex hull of n+ 1 points that do not lie in a
hyperplane of an n-dimensional space. These points are called the vertices of
the simplex S.Itsfacets are (n−1)-dimensional simplices formed by any n
vertices of the simplex S. Throughout the paper we shall identify vertices with
vectors. The center of the circumscribed sphere of Sis called the circumcenter
and its radius circumradius.
Definition 1. An n-simplex Sis said to be well-centered if its circumcenter lies
in the interior of S.
The next theorem gives an interesting characterization of well-centered sim-
plices.
Theorem 1. An n-simplex is well-centered if and only if its vertices do not lie
on a hemisphere of the circumscribed sphere.
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