Sphere triangulation

Thu98] W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, in Geometry and Topology Monographs I, The Epstein Birthday Schrift, ed. by J. Rivin, C. Rourke, and C. Series, Geom. Topol. Publ., 1998, 511-549. 

T. Tarnai. So what can I say about a triangulated sphere is that for a long time mathematicians believed that there was no such structure which had free motion which would constitute a mechanism. One of them proved that such polyhedra do not exist, but Bill Conolly (late 1970s) produced a triangulated sphere which has free motion. This meant that it had a mechanism involving finite motion so if you wish to consider it as a structure that can vibrate, it has motion that causes no stress. Such a structure exists. Later a German mathematician produced another, so we have some examples. 

Figure 1. The Goldberg/Coxeter construction of an icosahedral triangulation of the sphere. Twenty copies of the large equilateral triangle fit together to form the net of a master icosahedron, which yields an icosahedral fullerene on taking the dual. The Coxeter parameters in this particular example are a = 3 and b= i.e., to reach B from A, take 3 steps along the horizontal to the right, turn left through 60°, and take 1 step.

Fig. 30. The 19 protein neutron map of the 30S ribosomal subunit determined by label triangulation (right). The proteins are depicted as spheres whose volumes are to scale. For clarity, several spheres are drawn unfilled. The centre-to-centre distance between S13 and S17 is 17.3 nm [490]. The left view shows the electron microscopy model of the 30S subunit and the sites of the antigenic determinants from immune electron microscopy techniques [490]. Note that most of the protein is located to the top of the model as viewed, while the RNA is predominant in the lower half.

In the applieation of the boundary element method, it is emeial to seleet appropriate boundary surfaee for the solute eavity and to proeeed as aeeurate as possible tessellation (triangulation) of this surfaee. For instanee, it has been proposed that in the ease of the eavity formation from overlapping van-der-Waals spheres, the atomie van-der-Waals radii should be multiplied by a eoeffieient equal to 1.2. Other possibilities of the surfaee definition inelude the elosed envelope obtained by rolling a spherieal probe of adequate diameter on the van-der-Waals surfaee of the solute moleeule and the surfaee obtained from the positions of the eenter of sueh spherieal probe around the solute. 

In particular, Example 3.15(1) together with Theorem 3.26 shows that, independently of the triangulation, the homology groups of an n-dimensional sphere are given by Z) = 0 for i n, and Z) = Z. 

The simplest model of a tethered membrane is composed of purely repulsive spheres which are connected together to form a planar triangulated network. In MC simulations, the spheres are taken as hard spheres, while in MD simulations, they interact with a purely repulsive Lennard-Jones interaction, eq. (9.3). A variety of tethering potentials have been used, as discussed in Section 9.2, usually for a hexagonal sheet of size L containing N = 3L -L l)/4 monomers. These systems are often referred to as open since the perimeter is free. To minimize finite size effects, some simulations have been done on closed systems in which the monomers are connected to form a spherical shell. Abraham used periodic boundary conditions and a computational cell which was allowed to vary in size using a constant-pressure MD technique. More recently, simulations have been carried out for membranes in which linear chains of n monomers are... 

Fig. 22 Triangulated-network model of a fluctuating membrane. Ail vertices have short-range repulsive interactions symbolized by hard spheres. Bonds represent attractive interactions which imply a maximum separation of connected vertices. From [175]...

---