This simulation visualises C60 buckminsterfullerene as a rotating 3D truncated icosahedron — 60 sp² carbon atoms forming 12 pentagons and 20 hexagons with full Ih symmetry. The geometry is built by truncating an icosahedron's edges at one-third points, then bonding each atom to its three nearest neighbours. Shorter hexagon–hexagon bonds (more double-bond character) and longer pentagon–hexagon bonds are drawn in distinct colours, with a painter's algorithm ordering by depth.
Beyond C60, the tool maps single-walled carbon nanotubes on a graphene lattice. The chiral vector Ch = n·a₁ + m·a₂ sets the tube's diameter and whether it is metallic or semiconducting. Sliders adjust rotation speed and the (n,m) indices; view buttons switch between the C60 model, the chirality map, and a bar chart comparing bandgaps of graphene, several nanotubes, C60, C70 and diamond. Such nanostructures underpin sensors, transistors, organic photovoltaics and high-strength composites.
What is buckminsterfullerene (C60)?
C60 is a molecule of 60 carbon atoms arranged like a football, a truncated icosahedron with 12 pentagonal and 20 hexagonal faces. Every carbon is sp²-hybridised and bonded to three neighbours. It was discovered in 1985 and named after the geodesic domes of Buckminster Fuller.
How is the 3D C60 model built in this simulation?
It starts from the 12 vertices of an icosahedron defined using the golden ratio. Each of the 30 edges is truncated at its one-third and two-thirds points to give the 60 carbon positions. Each atom is then linked to its three nearest neighbours, producing the 90 bonds you can rotate by dragging the canvas.
What do the (n,m) sliders control?
They set the chiral indices of a single-walled carbon nanotube. The chiral vector Ch = n·a₁ + m·a₂ describes how a graphene sheet is rolled up. In the chirality view this vector is drawn on the hexagonal lattice, and the live stats update the tube's diameter, type and bandgap.
A nanotube is metallic when (n minus m) is divisible by three, and semiconducting otherwise. Armchair tubes (n equals m) are always metallic, zigzag tubes have m equal to zero, and any other combination is chiral. The simulation labels each case and colours metallic entries differently.
The diameter is d = a√(n² + nm + m²) / π, where a is the graphene lattice constant, about 0.246 nm, derived from a carbon–carbon bond length of 0.142 nm. For the default (10,10) armchair tube this gives roughly 1.36 nm, matching the value shown in the stats panel.
For semiconducting tubes the model uses the standard approximation Eg = 2γ₀aCC / d, with a hopping energy γ₀ of about 2.9 eV and a carbon–carbon distance of 0.142 nm. The bandgap therefore scales inversely with diameter, so thinner semiconducting tubes have larger gaps. Metallic tubes are assigned a gap of zero.
These names describe the pattern of carbon rings around the tube circumference. Armchair tubes (n,n) and zigzag tubes (n,0) are achiral with mirror symmetry, while general (n,m) tubes are chiral and come in left- and right-handed forms. Chirality strongly affects electronic and optical behaviour.
The closed cage confines the π-electron system, splitting the highest occupied and lowest unoccupied molecular orbitals by roughly 1.9 eV. This makes C60 a molecular semiconductor that absorbs visible and ultraviolet light and readily accepts electrons, which is why it is used as an acceptor in organic solar cells.
The geometry, symmetry and the zone-folding formulas for diameter and metallicity are textbook-correct, and the bandgap follows the widely used tight-binding approximation. The values are first-order estimates, so they capture the trends well but omit curvature effects, many-body corrections and substrate influences present in real measurements.
C60 and C70 serve as electron acceptors in organic photovoltaics and as additives in lubricants and medicine. Carbon nanotubes are used in conductive composites, field-effect transistors, gas and biosensors, and reinforced materials. The properties view contrasts these allotropes alongside graphene and diamond to show how structure dictates function.