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Icosadeltahedral Geometry of Fullerenes, Viruses and Geodesic Domes

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Icosadeltahedral Geometry of Fullerenes, Viruses and Geodesic Domes Icosadeltahedral geometry of fullerenes, viruses and geodesic domes Antonio Siberˇ 1,2, ∗ 1Department of Theoretical Physics, Joˇzef Stefan Institute, SI-1000 Ljubljana, Slovenia 2Institute of Physics, P.O. Box 304, 10001 Zagreb, Croatia I discuss the symmetry of fullerenes, viruses and geodesic domes within a unified framework of icosadeltahedral representation of these objects. The icosadeltahedral symmetry is explained in details by examination of all of these structures. Using Euler’s theorem on polyhedra, it is shown how to calculate the number of vertices, edges, and faces in domes, and number of atoms, bonds and pentagonal and hexagonal rings in fullerenes. Caspar-Klug classification of viruses is elaborated as a specific case of icosadeltahedral geometry. I. INTRODUCTION 1. Icosahedron (shown in the upper left corner of Fig. 1) is a platonic solid consisting of twenty equilateral tri- Although the fullerene molecules were previously pre- angles and twelve vertices. When the spherical surface dicted and discussed1, it was quite a surprise for largest is covered with triangles so that the icosahedral nature part of the scientific community when Kroto et al pub- of the triangulation is preserved, the twelve icosahedral lished a paper announcing their experimental discovery2. vertices become special points. These twelve points be- Some part of this surprise can surely be attributed to come the only ones that have five nearest neighboring amazingly symmetrical arrangement of carbon atoms in points - all the other points have six nearest neighbors. the most abundant of all the fullerene molecules - Buck- In Fig. 1 the neighboring points of the icosahedral ver- tices are outlined as thick pentagons. As Fig. 1 illus- minsterfullerene or C60. Kroto and colleagues named this molecule after Richard Buckminster Fuller who was an trates, there are many ways to triangulate a sphere so American designer, inventor and architect2 most recog- that there are twelve points that make the vertices of nized for popularizing geodesic domes - networks of inter- an icosahedron and have five nearest neighbors, all other connected struts forming a (hemi)spherical grid. Fuller points having six nearest neighboring points. All of these constructed domes as an alternative to architecture that triangulations are called icosadeltahedral. Deltahedron was dominant in 1960’s in USA. Most famous of them is a polyhedron whose faces are all equilateral triangles. was used to house American pavilion for the World Expo Strictly speaking the icosadeltahedral geodesic domes are exhibition in Montreal in year 1967. Geodesic domes not (icosa)deltahedra since all of the triangles that they that Fuller constructed were not chiral3, i.e. their mirror consist of are not equilateral. The requirement that all image retained their symmetry. Even before the discov- of the polyhedral faces be equilateral triangles necessarily ery of fullerenes it became clear that achiral domes that produces aspherical (spiky) polyhedra. Nevertheless, the Fuller constructed belong to a larger class of mathemati- icosadeltahedral geodesic domes have the same symmetry cal structures called icosadeltahedra. Extending Fuller’s as an icosadeltahedron which is a spiky shape shown in design ideas4, in year 1962 Donald Caspar and Aaron Fig. 1. In fact the domes can be considered as projections Klug constructed the first theory that explained the fea- of icosadeltahedra on their circumscribed spheres. In the tures of most of the so-called ”spherical” (icosahedral) following the word dome is used only for icosadeltahedral viruses known at the time5. geodesic dome. It is intriguing that objects so different in size Each of the domes can be characterized by two non- (fullerenes 1 nm, viruses 100 nm, and geodesic domes negative integers, denoted by m and n in the following. 100 m) share the same design and symmetry. The reason These can be thought of as numbers of ”jumps” through for this is that all these objects are built up of nearly the vertices of a dome that need to be performed in or- identical elements (carbon atoms in fullerenes, proteins arXiv:0711.3527v1 [physics.pop-ph] 22 Nov 2007 der to reach a center of a pentagon from its neighboring in viruses, and struts in geodesic domes) that have to ar- pentagon. Except for (m, 0) dome, the jumps need to range so to fully enclose the object interior i.e. to form be directed along two different spherical geodesics (the a cage-like structure. The aim of this paper is to explain shortest lines between two points on a sphere), m along the design principles and symmetries of a large subset of one of them, and n along another one, making an angle of these structures that has an icosadeltahedral symmetry. 60 degrees with the first one. To be definite, we need to specify whether the ”jumper” needs to turn left or right after the m jumps along the first spherical geodesic. In II. ICOSADELTAHEDRAL GEODESIC DOMES what follows, I shall assume the left turn and denote the symmetry of icosadeltahedral structures by (m,n). Icosadeltahedral geodesic domes can be mathemati- Were the other convention chosen, the (m,n) dome in cally described as triangulations of the spherical surface our convention would correspond to (n,m) dome in the (or a part of it) with the icosahedral ”backbone”. A alternative convention. The domes with m =6 n; m,n > 0 visual representation of this statement is shown in Fig. are chiral. This means that their mirror image has differ- 2 ent symmetry. A mirror image of (m,n) dome is (n,m) neighbors. This type of bonding is also present in the dome. This is illustrated in the upper-right corner of Fig. planes of carbon atoms in graphite (graphene planes). It 1 for (3, 2) dome. is called sp2 bonding, in contrast to sp3 bonding that From m and n one can calculate the number of trian- is characteristic of diamond (in diamond each of the gles in a dome. The number of triangles per one spheri- carbon atoms is bonded with three nearest neighboring cal segment bounded by three spherical geodesic passing atoms). There are many different structures that can through neighboring fivefold coordinated vertices (out- be made of carbon atoms connected with sp2 bonds, at lined on a (4, 4) dome in Fig. 1) is least conceptually (see e.g. Refs. 8,9). Quite a differ- 2 2 ent question is whether such structures can be exper- T = m + n + mn, (1) imentally obtained. The icosadeltahedral symmetry of so that the total number of triangles (or faces) in an geodesic domes is characteristic of a class of especially icosadeltahedron is symmetric fullerene molecules, sometimes called icosahe- dral fullerenes, or giant icosahedral fullerenes in case that f = 20T. (2) molecules contain more than about 100 carbon atoms. Buckminsterfullerene belongs to this class (its ”compan- T is called the triangulation number or simply the ion” molecule C that was discovered simultaneously2 T 70 T-number. It adopts special integer values, = does not, however). The symmetry of these carbon , , , , , , , ... m n 1 3 4 7 9 12 13 . Instead of and integers, the T- molecules can be obtained from icosadeltahedral domes number can be used to classify the icosadeltahedral sym- by placing carbon atoms in (bary)centers of every tri- metry. The problem with this choice is that it doesn’t angle in the dome. The newly obtained set of points m,n n,m discriminate between ( ) and ( ) domes. That is (carbon atoms) is then ordered so that each point is con- why the T-number is sometimes used in combination with nected with its three nearest neighbors, i.e. the carbon- laevo dextro words (left) and (right) to resolve this am- carbon bonds are established (this procedure is illus- , biguity. For example, (2 1) structure in our convention trated in the upper-right corner of Fig. 2). This ful- T laevo T l would be in this case denoted as = 7 or = 7 fills the basic chemical requirement for carbon atoms or simply T = 7, while (1, 2) structure would be denoted 2 7 in sp bonding electronic configuration. The thus ob- T dextro T d as = 7 or = 7 . From the known number of tained structure contains now twelve pentagonal carbon f polyhedron faces ( ) one can proceed to find the number rings (pentagons) and certain number of hexagonal car- of its vertices (v) and edges (e) by using Euler’s theo- 6 bon rings (hexagons), depending on the T-number of the rem on polyhedra which relates these nonzero integer dome. The starting dome and the fullerene-like poly- quantities as hedron that was obtained from it are called dual poly- v − e + f =2. (3) hedra. The (m,n) symmetry that was characteristic of the dome will also be characteristic of its dual fullerene- This equation is valid for polyhedra that are homeomor- like polyhedron, but now the ”jumping” that character- phic to the sphere, i.e. their topology is the same as izes icosadeltahedral symmetry is allowed only through that of a sphere, which is the case of interest to us. the centers of pentagons and hexagons (not along the In icosadeltahedral domes twelve vertices belong to five carbon-carbon bonds). I have used the term ”fullerene- edges - these are located at the vertices of an icosahe- like polyhedron” since the true fullerene molecules will in dron. All the other vertices belong to six edges, i.e. six general be different from the polyhedron obtained by a edges meet at those vertices. Each edge is bounded by simple mathematical dualization of the dome. The domes two vertices and all these fact together can be used to are icosadeltahedral triangulations of the sphere and this relate e and v as necessarily means that the lengths of triangular edges are not all the same.
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