  Many aggregate distributions of urban activities such as city sizes reveal
scaling but hardly any work exists on the properties of spatial distributions
within individual cities, notwithstanding considerable knowledge about their
fractal structure. We redress this here by examining scaling relationships in a
world city using data on the geometric properties of individual buildings. We
first summarise how power laws can be used to approximate the size
distributions of buildings, in analogy to city-size distributions which have
been widely studied as rank-size and lognormal distributions following Zipf and
Gibrat. We then extend this analysis to allometric relationships between
buildings in terms of their different geometric size properties. We present
some preliminary analysis of building heights from the Emporis database which
suggests very strong scaling in world cities. The data base for Greater London
is then introduced from which we extract 3.6 million buildings whose scaling
properties we explore. We examine key allometric relationships between these
different properties illustrating how building shape changes according to size,
and we extend this analysis to the classification of buildings according to
land use types. We conclude with an analysis of two-point correlation functions
of building geometries which supports our non-spatial analysis of scaling.
