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The kinematics of a framework presented by H. Harborth and M. Möller

  • P. Fazekas
  • O. RöschelEmail author
  • B. Servatius
Original Paper
  • 77 Downloads

Abstract

We consider a model of interlinked tetrahedra which has been described by Harborth and Möller (Geombinatorics XVII(2):53–56, 2007). It consists of 16 congruent regular tetrahedra connected via 32 spherical joints (in the vertices of the tetrahedra). In this arrangement they define a saturated packing. Every vertex of a tetrahedron is linked to a vertex of another tetrahedron. As the classical Grübler–Kutzbach–Chebyshev formula gives a theoretical degree of freedom f = −6 for this kinematic chain, the model is supposed to be rigid. However, we can demonstrate that this mechanism still admits at least a two-parametric self-motion in the general position. Further on we consider a special, degenerate case of this model, which again admits a two-parametric self-motion. This motion contains geometrically interesting positions regarding the cross-section of possible prismatic channels through the model. These channels occur as empty space through the model if the tetrahedra are considered to be solid bodies. We present positions with vanishing channels and with a cross section of area a 2 = 1 for tetrahedra with edge length a := 1.

Keywords

Saturated packings of regular tetrahedra Overconstrained mechanisms Zeolite-models Flexibility of packings 

Mathematics Subject Classification (2000)

53A17 52C25 

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References

  1. Harborth H., Möller M.: Saturated vertex-to-vertex packings of unit tetrahedra. Geombinatorics XVII(2), 53–56 (2007)Google Scholar
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  3. Mortimer, Ch.: Chemie Das Basiswissen der Chemie. 6. Auflage, Thieme, Stuttgart (1996)Google Scholar
  4. Servatius, B., Servatius, H., Thorpe, M.F.: Zeolites: Geometry and Combinatorics. Preprint (2010)Google Scholar
  5. Stachel H.: On the tetrahedra in the dodecahedron. KoG 4, 5–10 (1999)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Managing Editors 2011

Authors and Affiliations

  1. 1.Institute for Geometry, Graz University of TechnologyGrazAustria

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