Abstract
This paper investigates the duality of the five Platonic polyhedrons via the ray and axis coordinates of Line geometry. Then, based on the five Platonic polyhedrons, regular deployable polyhedral mechanisms are constructed by implanting the plane-symmetric eight-bar linkages into the edges, faces and vertices of the polyhedrons. Due to the duality of the five regular Platonic polyhedrons, it is revealed in this paper that the dual regular deployable polyhedral mechanisms constructed based on the dual Platonic polyhedrons are isomorphic and the isomorphism of the dual regular deployable polyhedral mechanisms is then for the first time presented via topology graphs and their corresponding adjacency matrices.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Coxeter HSM (1963) Regular polytopes. Macmillan, New York (Dover reprint, 1973)
Holden A (1991) Shapes, space and symmetry. Dover, New York
Wohlhart K (2008) New polyhedral star linkages. In: Proceedings of the 10th international conference on the theory of machines and mechanisms, Liberec, Czech Republic
Agrawal SK, Kumar S, Yim M (2002) Polyhedral single degree-of freedom expanding structures: design and prototypes. ASME J Mech Des 124(9):473–478
Gosselin CM, Gagnon-Lachance D (2006) Expandable polyhedral mechanisms based on polygonal one-degree-of-freedom faces. J Mech Eng Sci 220:1011–1018
Wei G, Dai JS (2011) Geometry and kinematics of a plane-symmetric spatial eight-bar linkage with exact straight-line motion. DETC2011-48281. In: Proceedings on ASME 2011 international design engineering and technical conferences, Washington, USA, 28–31 Aug 2011
Klein F (1924) Elementary mathematics from an advanced standpoint: geometry. Dover, New York (Reprinted in 1939)
He PR, Zhang WJ, Li Q (2002) Eigenvalues and eigenvectors information of graph and their validity in identification of graph isomorphism. In: Proceedings of ASME IDETC 2002, Montreal, Canada
Acknowledgments
The authors gratefully acknowledge the support from the EU FP7 project TOMSY under Grant No. 270436, the EU FP7 project ECHORD DEXDEB under Grant No. 231143 and the Engineering and Physical Science Research Council (EPSRC) of the United Kingdom under grant number of EP/F031394/1.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag London
About this paper
Cite this paper
Wei, G., Dai, J.S. (2012). Duality of the Platonic Polyhedrons and Isomorphism of the Regular Deployable Polyhedral Mechanisms (DPMs) . In: Dai, J., Zoppi, M., Kong, X. (eds) Advances in Reconfigurable Mechanisms and Robots I. Springer, London. https://doi.org/10.1007/978-1-4471-4141-9_68
Download citation
DOI: https://doi.org/10.1007/978-1-4471-4141-9_68
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4140-2
Online ISBN: 978-1-4471-4141-9
eBook Packages: EngineeringEngineering (R0)