Lines are then drawn from through each of the polyhedron's vertices. The intersection of each line with P defines the 2D coordinate of the vertex v. If is chosen sufficiently close to the face, and if the points of the polyhedron lie on a sphere, the resulting 2D embedding will be planar.
While Schlegel only proved that the projection produces a planar layout when the vertices lie on a sphere, it still works in many cases where this is not the case, and the method is widely used to the extent that planar graph embeddings sometimes incorrectly are called Schlegel diagrams. However, Schlegel diagrams are specifically the embeddings that result from the Schlegel projection.
The Schlegel projection is illustrated in Figure 5 a. In the Tutte embedding, one fixes the coordinates of a single, outer face, and solves a sparse linear system 4 such that in the result, every vertex is placed in the barycenter of its neighbors. While the Tutte embedding is guaranteed to be planar, the solution to the linear system results in exponential crowding of the vertices when embedding large graphs, making the result difficult to interpret.
Hence, it is useful to follow the Tutte embedding by an optimization to make the face sizes more even. This is, however, three orders of magnitude too large, but the proof is nontrivial.
Rovibrational quantum state resolution of the C60 fullerene
The triangulations with N vertices or less correspond to the lattice points inside the intersection between a certain cone and a ball of radius in this space. Hence their number is roughly proportional to N 10 , and the number of triangulations with exactly N vertices is roughly proportional to 5 or, in short,. Because they are the largest class of the finitely many triangulation classes with vertex degree six or less, the fullerenes have the same asymptotic count. Hence, the number of C N isomers grows as.
We note in passing that the connection of fullerenes to combinatorial manifolds is a deep one and gives rise to many of the beautiful mathematical properties exhibited by fullerenes.
The high order polynomial growth of fullerene isomer counts makes it difficult to establish a complete database for fullerene graphs up to high vertex numbers. One of the first methods for encoding fullerene graphs was the face spiral algorithm by Manolopoulos et al. More specifically, starting with a sequence of three mutually adjacent faces, new faces are added to the string such that the next face is adjacent to the previous one and the one that was added to the string earliest, and that has neighboring faces left which are not part of the spiral string yet. There are 6 N possible spiral starts resulting in up to 6 N spirals per fullerene graph.
In most fullerenes, some spirals cannot be completed because the two rules for choosing the next face cannot be both fulfilled. The lexicographically smallest of all successful spirals is the canonical spiral representation of the graph. While it was initially conjectured that every fullerene can be unwound into a face spiral 58 for good reason: the first counterexample comes after over 10 12 isomers there is an exceedingly small proportion of fullerene graphs 59 but infinitely many in total that do not admit a face spiral, 60 that is, where all 6 N spiral starts fail.
See Figure 6 for a case where a face spiral fails. Two of those are shown in Figure 7. Both Brinkmann 65 and Fowler et al.
Fowler's algorithm extends the two above mentioned rules such that the subgraph that has not been added to the spiral yet never gets disconnected, and as a result the spiral never gets stuck. With the extension to general spirals, all fullerene isomers with any vertex number can be generated. This makes it rather convenient for database storage as the face spiral uniquely defines the fullerene.
For fullerenes with no vertex spirals the scheme can be extended to general spirals with jumps similar to general face spirals. In order to explore chemical, physical, or graph theoretical properties for a wide range of fullerenes, it is important to have access to a list of stored fullerene graphs.
For this, one requires an exhaustive and efficient generator for all fullerene isomers of a given vertex number N. The general face spiral algorithm is well suited for compactly storing and recreating specific fullerene graphs. It also allows for sorting all isomers of C N for a given vertex number N according to their FSPIs including information about jumps if required.
It is, however, extremely inefficient to generate all isomers C N directly through a face spiral algorithm.
A different approach to generating fullerene graphs is by adding faces to an existing graph, while considering different sites for addition at each step. Liu et al. Brinkmann et al. These patches can be obtained by subdividing existing fullerene graphs. Further development in efficient graph generation came from transforming an existing fullerene graph into a new, larger one by adding faces.
However, three graphs were not accessible by the applied set of transformations. To cure this shortcoming, Hasheminezhad et al. The algorithm prunes the recursion tree in such a way that the only one representative of each isomorphism class is ever considered. Without such a scheme, exhaustively generating all fullerene graphs would succumb to combinatorial explosion.
For this, one may start with one specific isomer C N and derive others by certain transformations. This will be discussed in the following section. A patch is a set of faces that is bounded by a simple cycle, 82 , 85 that is, a cycle that traverses no vertex or edge twice.
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This definition includes patches that have bridges, and is used in the complete characterization of fullerenes by Hasheminezhad et al. Two patches can be exchanged if they share the same boundary code. The boundary code of a patch is the sequence of free valencies of the vertices that lie on the boundary in case of cubic graphs, a sequence of zeros and ones.
A few patch replacements are shown in Figure 8. Replacing patches of equal size can be understood as a formal isomerization, while replacement by larger or smaller patches, referred to as vertex insertion or deletion, or growth operations, formally derives a molecular graph of a different size. Two fullerene graph patches with the same boundary code contain the same number of pentagons, as can be seen from Euler's polyhedron formula. Astakhova et al. Growth operations can be classified according to the number of pentagons and the number of vertices that are added.
There are no growth operations for fullerene graphs that involve no or only one pentagon. Additional noteworthy examples are the addition of four and six vertices at a patch that contains three pentagons and has C 3 symmetry. The three classes of patch replacements defined by Hasheminezhad et al. With respect to the polyhedral representation of a fullerene graph, growth operations can be understood as the capping of a domain by additional vertices.
As a result the pentagons in that domain move toward each other. Patches in which all pentagons are fused cannot be capped. Conversely, the inverse of a growth operation corresponds to the truncation of a domain of high curvature: The distances between the affected pentagons increase. While patch replacements are useful from a graph theoretical point of view to obtain new fullerene isomers, the EK C 2 insertion and the SW transformation have also been suggested to resemble viable reaction pathways.
For a discussion of different mechanisms see Ref For every suggested pathway, however, the activation barrier is so high that SW transformations are only feasible at very high temperatures. Endo and Kroto proposed a concerted mechanism for what has since been known as the EK C 2 addition. The construction used by Goldberg was discovered independently, and applied to the shapes of vira, by Caspar and Klug in the s, and later popularized by Coxeter. While originally only defined for C 20 , yielding exactly all the fullerenes of icosahedral symmetry, Dutour and Deza have shown that it is well defined for all cubic planar graphs.
The general construction by Dutour and Deza is quite algebraically heavy handed, and is not easily understood, nor lends itself easily to implementation. In the case of fullerenes, however, it is quite easy to give a procedural description of the transformation in terms of the fullerene dual: Because a fullerene dual is a triangulation of the sphere with no vertices of degree more than 6, it can be unfolded onto the plane of equilateral triangles, called the Eisenstein plane.
The unfolded surface forms a polygon in the Eisenstein plane with every degree 5 vertex on the polygon periphery, and with each edge on the periphery appearing twice, once in each direction. The procedure is illustrated in Figure Details for how to efficiently perform the fold and unfold operations are given in Avery unpublished manuscript. Notice, that if one were to cut out the two diagrams in Figure 10 , gluing together the edges so that the numbers on the vertices match, one obtains the three dimensional structure of the given C 32 and C fullerenes.
The reader is invited to do so. Many of the beautiful properties of fullerenes derive from their relation to algebraic and differential geometry. This relation is mostly out of scope for this review, but in this section we will touch on the subject informally. The subject is treated in great depth by Thurston 19 and others.
An important quantity for understanding the geometry and shapes of fullerenes is the Gaussian curvature. The Gaussian curvature is the product of the two principal curvatures, which for each point on the surface are the maximal and minimal curvatures in any direction through that point.
Figure 12 illustrates three categories of surfaces with zero, positive, and negative Gaussian curvature respectively. If the Gaussian curvature is zero in a point, the surface only bends in one direction around that point. Positive Gaussian curvature bends the same way in all directions.
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A positive curvature surface can be cut open and unwrapped onto a plane. If one were to make a cut in a negative curvature surface, it could only be flattened out by allowing parts of it to overlap itself. The Gaussian curvature only depends on the topology of the surface, and is independent of how it is isometrically embedded in space. In the same way, the surface metric or Riemann metric , which determines the geodesics and distances between points along the surface, can be derived directly from the graph Avery et al.
Surprisingly, this is simply due to them having faces no larger than hexagons! It is most easy to understand the geometry of fullerenes when considering their duals. These are equilateral triangulations of a closed surface. The equilateral triangle plane, also called the Eisenstein plane, is the dual of a hexagonal mesh. What happens now if we set the degree to 5 of a single vertex in the plane? Unfolding again along the 12 cuts would result in a polygon similar to the diagrams shown in Figure The positions at which we placed the 12 pentagons determine where and how strongly the surface bends, and through that the natural shape of the fullerene, giving rise to the many interesting polyhedral shapes shown in Figure 2.