Abstract
A well-centered simplex is characterized by the position of its circumcenter in the interior of the simplex. For triangles, i.e., two-dimensional simplices, well-centeredness complies with the property of acuteness. In higher dimensions these two properties are distinct. Both well-centered and acute tilings are of interest for their applications in numerical mathematics. While it has been already proven that there is no acute face-to-face simplicial tiling of five-dimensional Euclidean space, the question of the existence of well-centered tiling of five-dimensional Euclidean space remained open. In this paper, we deliver a positive answer. The proof of the main assertion is constructive. It is based on Sommerville’s construction of tetrahedral tilings, generalized for arbitrary dimension. Beside the main result, we also construct well-centered face-to-face tilings of four-dimensional Euclidean space formed by congruent copies of a single 4-simplex.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brandts, J., Korotov, S., Křížek, M., Šolc, J.: On nonobtuse simplicial partitions. SIAM Rev. 51(2), 317–335 (2009)
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis, vol. 7, pp. 713–1020. North-Holland, Amsterdam (2000)
Feireisl, E., Karper, T., Michálek, M.: Convergence of a numerical method for the compressible Navier–Stokes system on general domains. Numer. Math. 134(4), 667–704 (2016)
Feireisl, E., Karper, T., Novotný, A.: A convergent numerical method for the Navier–Stokes–Fourier system. IMA J. Numer. Anal. 36(4), 1477–1535 (2016)
Herbin, R., Gallouët, T., Latché, J.-C., Therme, N.: Consistent internal energy based schemes for the compressible Euler equations (2019). arXiv:1906.11648
Hošek, R.: Face-to-face partition of 3D space with identical well-centered tetrahedra. Appl. Math. 60(6), 637–651 (2015)
Hošek, R.: Strongly regular family of boundary-fitted tetrahedral meshes of bounded \(C^2\) domains. Appl. Math. 61(3), 233–251 (2016)
Hošek, R.: Construction and shape optimization of simplicial meshes in \(d\)-dimensional space. Discrete Comput. Geom. 59(4), 972–989 (2018)
Hošek, R., Volek, J.: Discrete advection-diffusion equations on graphs: maximum principle and finite volumes. Appl. Math. Comput. 361, 630–644 (2019)
Kopczyński, E., Pak, I., Przytycki, P.: Acute triangulations of polyhedra and \(\mathbb{R}^N\). Combinatorica 32(1), 85–110 (2012)
Křížek, M.: An equilibrium finite element method in three-dimensional elasticity. Apl. Mat. 27(1), 46–75 (1982)
Křížek, M.: There is no face-to-face partition of \({{\mathbf{R}}}^5\) into acute simplices. Discrete Comput. Geom. 36(2), 381–390 (2006)
Křížek, M.: Erratum to: There is no face-to-face partition of \({{\mathbf{R}}}^5\) into acute simplices. Discrete Comput. Geom. 44(2), 484–485 (2010)
Křížek, M., Pradlová, J.: Nonobtuse tetrahedral partitions. Numer. Methods Partial Differ. Equ. 16(3), 327–334 (2000)
Naylor, D.J.: Filling space with tetrahedra. Int. J. Numer. Methods Eng. 44(10), 1383–1395 (1999)
Nicolaides, R.: Direct discretization of planar div-curl problems. SIAM J. Numer. Anal. 29(1), 32–56 (1992)
Sazonov, I., Hassan, O., Morgan, K., Weatherill, N.P.: Yee’s scheme for the integration of Maxwell’s equation on unstructured meshes. In: Proceedings of the European Conference on Computational Fluid Dynamics. Delft University of Technology (2006)
Senechal, M.: Which tetrahedra fill space? Math. Mag. 54(5), 227–243 (1981)
Sommerville, D.M.Y.: Space-filling tetrahedra in Euclidean space. Proc. Edinb. Math. Soc. 41, 49–57 (1923)
VanderZee, E., Hirani, A.N., Guoy, D.: Triangulation of simple 3D shapes with well-centered tetrahedra. In: Proceedings of the 17th International Meshing Roundtable, pp. 19–35. Springer, Berlin (2008)
VanderZee, E., Hirani, A.N., Guoy, D., Ramos, E.A.: Well-centered triangulation. SIAM J. Sci. Comput. 31(6), 4497–4523 (2009/10)
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(6), 700–724 (2013)
Vatne, J.E.: Simplices rarely contain their circumcenter in high dimensions. Appl. Math. 62(3), 213–223 (2017)
Acknowledgements
The author would like to thank Jonáš Volek and Vladimír Švígler for careful reading of the manuscript and help with the illustrations.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that he has no conflict of interest.
Additional information
Editor in Charge: Kenneth Clarkson
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hošek, R. Existence of Well-Centered Simplicial Face-to-Face Tiling of Five-Dimensional Space. Discrete Comput Geom 66, 1168–1189 (2021). https://doi.org/10.1007/s00454-020-00195-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-020-00195-y