The energetic costs of widening the fullerene definition to include carbon cages with heptagonal as well as pentagonal and hexagonal faces are investigated theoretically. Relative energies of all 426 hypothetical C₄₀ cages that can be assembled from pentagonal, hexagonal and heptagonal faces are calculated within two independent semi-empirical models. All isomers are found to lie in local minima on the potential surface. The QCFF/PI (quantum consistent force field/π) and DFTB (density functional tight binding) approaches agree in predicting that no cage with one or more heptagons is of lower energy than the best classical C₄₀ fullerene, but that many such cases are more stable than many C₄₀ fullerenes. All one-heptagon C₄₀ cages are predicted to lie within the range of energies spanned by the classical fullerene isomers. Energy penalties of 90–150 kJ mol^–1 per heptagon are suggested by the DFTB calculations, and penalties about half as large again by the QCFF/PI model. The energy variation across the range of fullerenes and pseudo-fullerenes is rationalised by an extension of the isolated-pentagon rule: when heptagons are present, structures of low energy are those that maximise the number of pentagon–heptagon contacts subject to prior minimisation of the number of pentagon–pentagon adjacencies.