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This essay sheds new light on the meaning and measurement of compactness-one of the most intriguing and least-understood properties of geographic shapes. We articulate a unified theoretical foundation for the study of shape compactness that rests on two simple observations: First, that the circle is the most compact of shapes. And second, that there are 10-and possibly more-distinct geometrical properties of the circle that make it the most compact of shapes. We introduce these 10 properties, illustrate them with real-world examples and define indices associated with these properties that can be calculated using a geographic information system. Dix propriétés de compacité du cercle : La mesure de forme en géographie Cet essai apporte une perspective nouvelleà la signification età la mesure de la compacité spatialeune des propriétés parmi les plus fascinantes et les moins comprises des formes géographiques. Nous développons les fondations d'une théorie unifiée pour l'étude de la compacité géographique des formes qui reposent sur deux observations simples : premièrement, que le cercle est la plus compacte des formes, et deuxièmement, qu'il y a au moins dix propriétés géométriques distinctes du cercle qui font de celui-ci la forme la plus compacte. Nous décrivons ces dix propriétés, nous les illustrons par des exemples du monde réel et nous définissons des index associésà ces propriétés qui peuventêtre calculés en utilisant un logiciel de SIG (Système d'information géographique). Mots clés : cercle, compacité, indice de forme, morphologie The Canadian Geographer / Le Géographe canadien 54, no 4 (2010) 441-461

The fragmentation of urban landscapes − or the inter-penetration of the built-up areas of cities and the open spaces in and around them − is a key attribute of their spatial structure. Analyzing satellite images for 1990 and 2000 for a global sample of 120 cities, we find that cities typically contain or disturb vast quantities of open spaces equal in area, on average, to their built-up areas. We also find that fragmentation, defined as the relative share of open space in the urban landscape, is now in decline. Using multiple regression models, we find that larger cities are less fragmented, that higher-income cities are more fragmented, that cities with higher levels of car ownership are less fragmented, and that cities that constrain urban development are less fragmented. We recommend that making room for urban expansion in rapidly growing cities should take into account their expected fragmentation levels.KEYWORDS cities / fragmentation / infill / landscape metrics / leapfrogging / metropolitan areas / open space / sprawl / urban expansion

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