1. Pianificare la città frattale
Defence plea for using fractal geometry in town planning
Dr Cécile Tannier
(University of Franche-Comté, National Centre for Scientific Research)
Gilles Vuidel
(University of Paris VI)
FRACTALS AND THE CITY (La città frattale)
Martedì 10 dicembre 2002, Facoltà di Ingegneria, Lecco (Italia)

2. Urban development policy
Guiding principles
Urban development policy
Planning rules
Laws, plans

3. Requirement of a best understanding of the urban development process
Relations between spatial behaviour of people and location of activities (residential, industrial, retailing…)
Relations between town planning policy and the urban patterns resulting
Respective rules of individual behaviours and town planning policy in the urban development

4. Framework: using fractal geometry in order to best understand the urban development process
implies
Concentrate on morphological aspects of the process
Deal with geometrical problems

5. Deal with geometrical problems
Two possibilities:
Euclidean geometry
Fractal geometry

6. Morphological aspects of the urban development process
Which forms (circle, square, Sierpinski carpet…) to answer which planning arrangements?
Which processes generate which forms?

7. Morphological aspects of the urban development process
Three action possibilities:
Describe the morphological characteristics of urban patterns
Traduce planning rules in a geometrical way
Simulate urban growth

8. 1. Describe the morphological characteristics of urban patterns
Two main measures:
Density measures
Fractal measures

9. 1. Describe the morphological characteristics of urban patterns
Density measures
A ratio: quantity per surface unit
A threshold
… Which refers to the ecologic capacity of socio-economic systems
A geometrical reference: homogeneous spatial distribution of elements
An intrinsically static indicator
Fractal measures
Complementary properties

10. 2. Traduce planning rules in a geometrical way
Urban development plans, land-use plans
Geometrical reference
Euclidean geometry
Fractal geometry

11. Principles to generate fractal forms
3. Simulate urban growth
Fractal geometry
Control parameters
Principles to generate fractal forms

12. Possible contributions of fractals in the field of town planning
Why use fractal measures?
Or...
…Because land use patterns don’t fit with the laws of Euclidean geometry
non homogeneity
characteristics of the relation between perimeter and surface
urban patterns are fragmented, shredded
internal organisation principle of the urban patterns is often hierarchical

13. Possible contributions of fractals in the field of town planning
Or...
Why use fractal measures?
…Because land use patterns don’t fit with the laws of Euclidean geometry
…Because the properties of fractal objects are the same as the properties of urban patterns
Spatial distribution of elements follows a hierarchical law
Same distribution principle of elements at a multitude of scales
Existence of clusters at each scale
The homogeneity exists as limit case of the fractal geometry

14. Morphological aspects of the urban development process
Which properties of the urban patterns could be revealed by which measures of fractal dimensions?
What traduce these properties in terms of individual behaviours?

15. Different methods for measuring the fractality of cities
The main goals of fractal measures:
Verification of the existence of a hierarchy
Identification of spatial thresholds
Determination of the degree of heterogeneousness

16. Different methods for measuring the fractality of cities
A software: Fractalyse

17. Principles to calculate fractal dimension
Input data: a black and white image of a city

18. Besançon

19. Principles to calculate fractal dimension
Counting the number of black pixels at each iteration step
As result, two elements varying according to the iteration step:
- the number of counted elements
- the size of the counting window

21. Principles to calculate fractal dimension
Make fit the obtained curve (i.e. the empirical curve) with an estimated curve
A constraint:
the estimated curve must correspond to a fractal law (y = xD).

23. Principles to calculate fractal dimension
The fractal dimension corresponds to the exponent D of the fractal law.
y = xD
Fractal dimension

24. Different method for measuring fractal dimensions
Fractal dimension of surfaces
correlation analysis
dilation analysis
Fractal dimension of borders
gaussian convolution

25. Correlation analysis

26. Dilation analysis