Journal of Geometry

, 110:49 | Cite as

Duality of isosceles tetrahedra

  • Jan Brandts
  • Michal KřížekEmail author


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 S we investigate 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.


Well-centered simplices Dual simplices Isosceles tetrahedra Circumcenter Circumradius 



The authors are indebted to Antonín Slavík and Tomáš Vejchodský for useful suggestions. Research of Michal Křížek was supported by RVO 67985840 and the Grant No. 18-09628S of the Grant Agency of the Czech Republic.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Institute of MathematicsCzech Academy of SciencesPrague 1Czech Republic

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