Editing Fullerene (section)

===Topology===
[[Schlegel diagram]]s are often used to clarify the 3D structure of closed-shell fullerenes, as 2D projections are often not ideal in this sense.<ref name="iupacful">{{Cite journal |last1=Powell |first1=W. H. |last2=Cozzi |first2=F. |last3=Moss |first3=G. P. |last4=Thilgen |first4=C. |last5=Hwu |first5=R. J.-R. |last6=Yerin |first6=A. |display-authors=3 |date=2002 |title=Nomenclature for the C60-Ih and C70-D5h(6) Fullerenes (IUPAC Recommendations 2002) |url=http://doc.rero.ch/record/303076/files/pac200274040629.pdf |journal=Pure and Applied Chemistry |volume=74 |issue=4 |pages=629–695 |doi=10.1351/pac200274040629 |s2cid=93423610}}</ref>

In mathematical terms, the [[combinatorial topology]] (that is, the carbon atoms and the bonds between them, ignoring their positions and distances) of a closed-shell fullerene with a simple sphere-like mean surface ([[Orientability|orientable]], [[genus (topology)|genus]] zero) can be represented as a convex [[polyhedron]]; more precisely, its [[dimension (mathematics)|one-dimensional]] skeleton, consisting of its vertices and edges. The Schlegel diagram is a projection of that skeleton onto one of the faces of the polyhedron, through a point just outside that face; so that all other vertices project inside that face.
<gallery widths="160px" heights="160px" style="text-align:center;" caption="Schlegel diagrams of some fullerenes">
Graph of 20-fullerene w-nodes.svg|C20<br>(dodecahedron)
Graph of 26-fullerene 5-base w-nodes.svg|C26
Graph of 60-fullerene w-nodes.svg|C60<br/>(truncated icosahedron)
Graph of 70-fullerene w-nodes.svg|C70
</gallery>
The Schlegel diagram of a closed fullerene is a [[graph theory|graph]] that is [[planar graph|planar]] and [[regular graph|3-regular]] (or "cubic"; meaning that all vertices have [[degree (graph theory)|degree]] 3).

A closed fullerene with sphere-like shell must have at least some cycles that are pentagons or heptagons. More precisely, if all the faces have 5 or 6 sides, it follows from [[Euler characteristic|Euler's polyhedron formula]], ''V''−''E''+''F''=2 (where ''V'', ''E'', ''F'' are the numbers of vertices, edges, and faces), that ''V'' must be even, and that there must be exactly 12 pentagons and ''V''/2−10 hexagons. Similar constraints exist if the fullerene has heptagonal (seven-atom) cycles.<ref>[https://www.britannica.com/EBchecked/topic/221916/fullerene "Fullerene"], ''Encyclopædia Britannica'' on-line</ref>