Skip to main content
SpringerLink
Log in

Multidimensional triangle-truncated simplexes

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We consider the convex polytopes, called triangle-truncated simplexes. From the point of view of a constructive object in four and higher dimensions vector space such polytopes are multidimensional analogs of one classical semi-regular polytopes, namely, truncated tetrahedron. We present results of investigations of inner geometrical structure and combinatorial characteristics of the complete assemblage of faces of triangle-truncated simplexes in vector spaces of arbitrary dimension. We formulate and prove a theorem about the volumes of multidimensional truncated simplex of generalized kind in Euclidean space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Reznikova, Yu. S. “Mid-Truncated Simplexes in Affine Spaces of Arbitrary Dimension”, Proc. of Intern. Geometry Center 3, No. 2, 13–24 (2010).

    Google Scholar 

  2. Coxeter, H. S. M. Regular Polytopes. 3rd Ed. (Dover Publications Inc., New York, 1973; Methuen and Co. Ltd., London, 1948).

    MATH  Google Scholar 

  3. Deza, M., Grishuhin, V. P., Shtogrin, M. I. Isometric Polyhedral Subgraphs in Hypercubes and Cube Lattices (MTsMO, Moscow, 2007) [in Russian].

    Google Scholar 

  4. Gosset, T. “On the Regular and Semiregular Figures in Spaces of n Dimensions”, Messenger of Mathematics 29, 43–48 (1899).

    Google Scholar 

  5. Kettler, Paul C. “Vertex and Path Attributes of the Generalized Hypercube with Simplex Cuts”, Pure Mathematics e-Print Series (Department of Mathematics, University of Oslo, 2008), Vol. 7.

    Google Scholar 

  6. Ziegler, Gunter M. Lectures on Polytopes, Graduate Texts in Mathematics (Springer-Verlag, New York, 1995), Vol. 152.

    Book  Google Scholar 

  7. Berger, M. Geometry. I (Springer-Verlag, Berlin, 1987).

    MATH  Google Scholar 

  8. Emelichev, V. A., Kovalev, M. M., Kravtsov, M. K. Polytopes, Graphs, Optimization (Nauka, Moscow, 1981) [in Russian].

    MATH  Google Scholar 

  9. Rozenfel’d B.A. Multidimensional Spaces (Nauka, Moscow, 1966) [in Russian].

    Google Scholar 

  10. Smirnov, E. Yu. Reflection Groups and Regular Polytopes (MTsNMO, Moscow, 2009) [in Russian].

    Google Scholar 

  11. Stringham W.I. “Regular figures in n-dimensional space”, UspekhiMat. Nauk X, 22–33 (1944) [Russian translation].

    Google Scholar 

  12. D’Andrea, C., Sombra, M. “The Cayley–Menger Determinant is Irreducible for n ≥ 3”, Sib.Math. J. 46, No. 1, 71–76 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  13. Nevskii, M. B. Geometric Estimates in Polynomial Interpretation (Yaroslavl’ State Univ. Press, Yaroslavl’, 2012) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Scientific Research Institute of Labour and Employment of the Population of the Ministry of Social Policy of Ukraine and NAS of Ukraine, ul. Ivana Franka 15B, Kiev, 01030, Ukraine

    Yu. S. Reznikova

Authors
  1. Yu. S. Reznikova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. S. Reznikova.

Additional information

Original Russian Text © Yu.S. Reznikova, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 7, pp. 92–100.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reznikova, Y.S. Multidimensional triangle-truncated simplexes. Russ Math. 60, 79–86 (2016). https://doi.org/10.3103/S1066369X16070100

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X16070100

Keywords

Search

Navigation