A class of expandable polyhedral structures
Introduction
Certain viruses having the shape of a truncated icosahedron expand under the effect of pH change in biological media (Speir et al., 1995). The pentamers and hexamers move apart from each other, rotating but remaining in contact through protein links. The discovery of this phenomenon calls attention to the problem of motion of expandable polyhedra.
The characteristics of the virus motion differ from those of another expanding object, the popular toy known as the Hoberman SphereTM (Hoberman, 1991). This transforming globe defines an underlying polyhedron, whose faces preserve their orientation as they expand. In the virus, however, the faces essentially preserve their size but each is subjected to a translation and rotation along its symmetry axes and, as the virus expands, interstices appear between the faces. Thus, the swelling motion of the virus, although complex and still incompletely understood, seems to be more similar to a special type of deformation that has been studied for honeycombs (Kollár and Hegedűs, 1985), and to the motion exhibited by Fuller’s (1975) jitterbug.
In order to gain a better insight into the virus problem, Kovács and Tarnai (2000) initiated a study of the motion of a simpler object: an expandable dodecahedron (Fig. 1). A physical model exhibited the breathing expansion mode that was expected on intuitive grounds, although paradoxically it turns out that this mode is not accounted for by mobility analysis using standard techniques.
The expandable dodecahedron suggests a class of similar objects, the expandohedra, which will also be studied in this paper. These expandable polyhedra have analogues in fullerene chemistry; they model the leapfrog transformation (Fowler and Steer, 1987), which produces fullerenes with ideal π-electronic structures. A fullerene, and in fact any convex polyhedron, can be converted into its leapfrog by a two-stage procedure. If an original fullerene polyhedron is capped on every face, a deltahedron is obtained, and if this deltahedron is then converted to its dual, a new fullerene polyhedron with three times as many vertices is obtained. Leapfrogging is a discontinuous process which jumps from one fullerene to another over the intervening deltahedron. For example, the leapfrog transformation converts a dodecahedron to a truncated icosahedron. It turns out that this particular transformation can alternatively be described as a continuous process using Verheyen’s (1989) dipolygonid composed of 30 digons and 12 pentagons, which he denotes by the symbol 30{2}+12{5}|31.717474°. The expandohedra provide an alternative continuous description for this, and for certain other, leapfrog transformations.
The expandohedron can also be considered as a deployable structure providing a large volume expansion. The fully open expandohedron has a number of mechanisms, see below, but the addition of e.g. locking hinges (Seffen et al., 1999) would give rigidity to the deployed structure.
This paper has therefore two aims. The first is to provide a detailed analysis of the mobility of the expandable dodecahedron, and the second is to place the dodecahedron within a general class of expandable polyhedra, providing general symmetry theorems for their mobility. The paper brings together a number of different techniques to analyse the kinematics of these novel deployable systems. New features arise at each level of analysis. Section 2 contains a geometric analysis of the finite breathing motion in a particular case, the expanding dodecahedron, and identifies by geometric arguments additional sets of possible infinitesimal motions. Section 3 uses recently developed symmetry techniques (Kangwai and Guest, 1999, Kangwai and Guest, 2000; Fowler and Guest, 2000) to find and classify the possible motions of several general classes of expandohedra; the finite breathing mode is shown to be symmetrically distinct from the mechanisms that can be found by simple counting. Section 4 reports the results of detailed numerical analysis of open and closed configurations of particular expandohedra, giving a complete account of the kinematics, and confirming the results of the more general symmetry analysis. This combination of techniques proves to be an effective route to understanding the behaviour of complex kinematic systems.
Section snippets
Structure
The expandable polyhedral viruses have the basic property that all pentamers and hexamers are situated in icosahedral symmetry at any stage of swelling. To simulate the expansion of these viruses, a dodecahedral assembly producing this symmetrical motion was constructed. The cardboard model in Fig. 1 represents a mechanism composed of rigid bodies with hinged connections.
The dodecahedron cardboard model was designed under the following constraints:
- (i)
the structure consists entirely of equal rigid
The mobility of an expandohedron
In the language of polyhedra and related combinatorics, the general construction can be described as follows. Call the parent polyhedron P. The expandohedron E is constructed from the elements of P as shown schematically in Fig. 9. First, the faces of P are separated: E contains a distinct rigid prism for each face of P, the same in size and shape as the original (Fig. 9(a)). Each edge of P is thereby doubled, with the edge ab, that was common to two faces now replaced by edges a′b′ and a″b″ of
Description of the model
This section will describe a numerical model of the expandohedra based on the dodecahedron, tetrahedron and the cube that confirms the analysis in the previous sections. It shows that in general configurations there are no additional infinitesimal mechanisms beyond those predicted by (31). In the fully open configuration there are no additional infinitesimal mechanisms beyond those predicted by (35).
A numerical model could be set up most straightforwardly using a simple bar-and-joint assembly
Conclusions
This paper has used a combination of geometry, symmetry and numerics to provide a detailed understanding of the kinematics of expandohedra; each of these techniques makes an essential contribution to the paper.
The original expandohedron (Kovács and Tarnai, 2000) was designed to exhibit a finite breathing motion as a physical model of the experimentally observed swelling of viruses. This key feature escapes elementary counting of mobility, but is revealed by the symmetry analysis to be a totally
Acknowledgements
The research reported here was done within the framework of the Hungarian–British Intergovernmental Science and Technology Cooperation Programme with the support of OMFB and the British Council, TéT Grant No. GB-15/98. Partial support by OTKA Grant No. T031931 and FKFP Grant No. 0177/2001 is also gratefully acknowledged. SDG acknowledges support from the Leverhulme Trust.
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