Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Two-faced maps
- 3 Fullerenes as tilings of surfaces
- 4 Polycycles
- 5 Polycycles with given boundary
- 6 Symmetries of polycycles
- 7 Elementary polycycles
- 8 Applications of elementary decompositions to (r, q)-polycycles
- 9 Strictly face-regular spheres and tori
- 10 Parabolic weakly face-regular spheres
- 11 General properties of 3-valent face-regular maps
- 12 Spheres and tori that are aRi
- 13 Frank-Kasper spheres and tori
- 14 Spheres and tori that are bR1
- 15 Spheres and tori that are bR2
- 16 Spheres and tori that are bR3
- 17 Spheres and tori that are bR4
- 18 Spheres and tori that are bRj for j ≥ 5
- 19 Icosahedral fulleroids
- References
- Index
19 - Icosahedral fulleroids
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Two-faced maps
- 3 Fullerenes as tilings of surfaces
- 4 Polycycles
- 5 Polycycles with given boundary
- 6 Symmetries of polycycles
- 7 Elementary polycycles
- 8 Applications of elementary decompositions to (r, q)-polycycles
- 9 Strictly face-regular spheres and tori
- 10 Parabolic weakly face-regular spheres
- 11 General properties of 3-valent face-regular maps
- 12 Spheres and tori that are aRi
- 13 Frank-Kasper spheres and tori
- 14 Spheres and tori that are bR1
- 15 Spheres and tori that are bR2
- 16 Spheres and tori that are bR3
- 17 Spheres and tori that are bR4
- 18 Spheres and tori that are bRj for j ≥ 5
- 19 Icosahedral fulleroids
- References
- Index
Summary
In this chapter, which is an adaptation of, are considered icosahedral fulleroidsy (or I -fulleroidsy, or, more precisely, I (5, b)-fulleroidsy, i.e. ({5, b}, 3)- spheres of symmetry I or Ih). For some values of b, the smallest such fulleroids are indicated and their unicity is proved. Also, several infinite series of them are presented.
The case b=6 is the classical fullerene case. Theorem 2.2.2 gives that all I (5, 6)-fulleroids, i.e. fullerenes of icosahedral symmetry, are of the form GCk,l (Dodecahedrony). See on Figure 19.1 the first three of the following smallest icosahedral fullerenes besides Dodecahedron:
C60(Ih), buckminsterfullerene,
C80(Ih), chamfered Dodecahedron,
C140(I), smallest chiral one,
C180(Ih).
Here Cν(G) stands for a ({5, 6}, 3)-sphere with ν vertices and symmetry group G. Although this notation is not generally nique, it will suffice for our purpose.
Both smallest I (5, 7)-fulleroids are described in; see them on Figure 19.2.
All I -fulleroids known so far and simple ways to describe them are given in Section 19.1; based on that some infinite series are introduced. In Section 19.2, a necessary condition for the p-vectors, which implies that five of the new I -fulleroids are minimal for their respective values of ν, is derived.
Table 19.1 shows the smallest possible p-vectors for 7 ≤ b ≤ 20, accordingly see Lemma 19.2.3 below. The first four columns of the table show the quantities b, p5, pb (the number of 5-, b-gons) and the number of vertices ν. The invariants m5, mb, κ2, κ3, and κ5 are described in Section 19.2.
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- Geometry of Chemical GraphsPolycycles and Two-faced Maps, pp. 284 - 294Publisher: Cambridge University PressPrint publication year: 2008