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Duality of isosceles tetrahedra

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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 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.

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References

  1. Altshiller-Court, N.: The isosceles tetrahedron. In: Modern Pure Solid Geometry, pp. 94–101 and 300. Chelsea, New York (1979)

  2. Brandts, J., Korotov, S., Křížek, M., Šolc, J.: On nonobtuse simplicial partitions. SIAM Rev. 51, 317–335 (2009)

    Article  MathSciNet  Google Scholar 

  3. Brandts, J., Křížek, M.: Simplicial vertex-normal duality with applications to well-centered simplices. In: Radu, F.A., et al. (eds.) Proceedings of the 12th European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017, vols, 761–768 . Springer, Berlin (2019)

    Google Scholar 

  4. Edmonds, A.L.: The geometry of an equifacetal simplex. Mathematika 52, 31–45 (2005)

    Article  MathSciNet  Google Scholar 

  5. Edmonds, A.L., Hajja, M., Martini, H.: Coincidences of simplex centers and related facial structures. Beitr. Algebra Geom. 46, 491–512 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Fiedler, M.: Über qualitative Winkeleigenschaften der Simplexe. Czechoslov. Math. J. 7, 463–478 (1957)

    MATH  Google Scholar 

  7. Fiedler, M.: Matrices and Graphs in Euclidean Geometry. Dimatia, MFF UK, Praha (2001)

    MATH  Google Scholar 

  8. Gaddum, J.W.: Distance sums on a sphere and angle sums in a simplex. Am. Math. Mon. 63, 91–96 (1956)

    Article  MathSciNet  Google Scholar 

  9. Goldoni, G.: Problem 10993. Am. Math. Mon. 110, 155 (2003)

    Article  Google Scholar 

  10. Hošek, R.: Face-to-face partitions of 3D space with identical well-centered tetrahedra. Appl. Math. 60, 637–651 (2015)

    Article  MathSciNet  Google Scholar 

  11. Klee, V., Wagon, S.: Old and New Unsolved Problems in Plane Geometry and Number Theory. Mathematical Association of America, Washington (1991)

    MATH  Google Scholar 

  12. Křížek, M., Pradlová, J.: Nonobtuse tetrahedral partitions. Numer. Methods Partial Differ. Equ. 16, 327–334 (2000)

    Article  MathSciNet  Google Scholar 

  13. Leech, J.: Some properties of the isosceles tetrahedron. Math. Gaz. 34, 269–271 (1950)

    Article  MathSciNet  Google Scholar 

  14. Rajan, V.T.: Optimality of the Delaunay triangulation in \(R^d\). Discrete Comput. Geom. 12, 189–202 (1994)

  15. Rektorys, K.: Survey of Applicable Mathematics I. Kluwer, Dordrecht (1994)

    Book  Google Scholar 

  16. Sommerville, D.M.Y.: Space-filling tetrahedra in Euclidean space. Proc. Edinb. Math. Soc. 41, 49–57 (1923)

    Article  Google Scholar 

  17. VanderZee, E., Hirani, A.N., Guoy, D., Ramos, E.A.: Well-centered triangulation. SIAM J. Sci. Comput. 31, 4497–4523 (2010)

    Article  MathSciNet  Google Scholar 

  18. VanderZee, E., Hirani, A.N., Guoy, D., Zharnitsky, V., Ramos, E.A.: Geometric and combinatorial properties of well-centered triangulations in three and higher dimensions. Comput. Geom. 46, 700–724 (2013)

    Article  MathSciNet  Google Scholar 

  19. Vatne, J.E.: Simplices rarely contain their circumcenter in high dimensions. Appl. Math. 62, 213–223 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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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Correspondence to Michal Křížek.

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Dedicated to Dr. Milan Práger on his 90th birthday.

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Brandts, J., Křížek, M. Duality of isosceles tetrahedra. J. Geom. 110, 49 (2019). https://doi.org/10.1007/s00022-019-0506-y

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