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In this paper, we define measures of urban diversity, density and segregation using newdata and software systems based on GIS. These allow us to visualise the meaning of themultifunctional city. We begin with a discussion of how cities have become moresegregated in their land uses and activities during the last 200 years and how the currentfocus is on reversing this trend through limiting urban sprawl and bringing new lifeback to the inner and central city. We define various indices which show how diversityand density manifest themselves spatially. We argue that multifunctionalism is a relativeconcept, dependent upon the spatial and temporal scale that we use to think about themixing and concentration of urban land uses. We present three examples using spatiallysmoothed indicators of diversity: for a world city ? London, for a highly controlledpolycentric urban region ? Randstad Holland, and for a much more diffusely populatedsemi-urban region ? Venice-Padua-Teviso. We conclude by illustrating that urbandiversity varies as people engage in different activities associated with different landuses throughout the day, as well as through the vertical, third dimension of the city. Thisimpresses the point that we need to understand multifunctional cities in all theirdimensions of space and time.