Roll a sheet of graphene into a tube, or curve it round into a ball, and you have two more forms of pure carbon: nanotubes and buckyballs.
Nanotubes
A carbon nanotube is a sheet of graphene rolled up so its edges join, with no seam. Every carbon atom in the wall holds 3 neighbours, just as in graphene. Tubes with a single wall are about a nanometre across, tens of thousands of times thinner than a hair, and they can be thousands of times longer than they are wide.
In 1991 Sumio Iijima, working for the electronics company NEC in Japan, reported tubes of carbon in the deposit left by an electric arc between two graphite rods. His tubes had several walls, one tube nested inside the next. Tubes with a single wall followed two years later.
Buckyballs
In 1985 Harold Kroto, Robert Curl and Richard Smalley, working with students at Rice University in Texas, fired a laser at graphite. Under the right conditions, one molecule formed far more often than any other: 60 carbon atoms, C₆₀. They worked out that 60 carbons can close into a hollow ball of 12 five-sided rings and 20 six-sided rings, the pattern of a football. They named it buckminsterfullerene, after the architect Richard Buckminster Fuller, whose geodesic domes it resembles. Most people call it a buckyball. The three of them won the Nobel Prize in Chemistry in 1996.
The five-sided rings do the folding. A sheet of hexagons stays flat, and each five-sided ring bends the sheet a little. Twelve of them close it into a ball. Every carbon still holds 3 neighbours. The ball is about 0.7 nm across, and it is a molecule: C₆₀ dissolves in methylbenzene (benzene carrying a -CH₃ group) and turns it purple.