1. Centre for Advanced Spatial Analysis (CASA), UCL, 1-19 Torrington Place, London WC1E 6BT, UK email m.batty@ucl.ac.uk web http://www.casa.ucl.ac.uk/ Talk to the MSc in VE in the Bartlett Fractals : The New Science of Cities Michael Batty

2. Outline of the Talk 1. Introduction: What are fractals ? 2. Fractal and Geometry 4. Deterministic and Stochastic Fractals 3. Coastlines and Skylines 4. Regular Patterns: 5. Less Regular Form: Tree-like Structure

3. 6. Diffusion-Limited Aggregation 7. Examples of DLA - Urban Models 8. Fractal Urban Growth 9. Fractal Objects: Agents and Simulation

4. All this is a massive digression from my talk but in way it serves to show that we are interested in scaling, and the physics of systems and we want to apply these to cities In fact, I have not told you about most of the work that has gone on in this field – it is in land use and traffic modelling where the gravitational hypothesis – scaling again – has been used to model traffic flow. Here at UCL there has not been that much work in this area – although in the 1950s and 1960 Smeed and Wardrop developed traffic theory here in the Centre for Transport Studies In fact there is much more now, and it is growing – the establishment of my own unit – CASA – one of Derek Roberts initiatives

5. Outline of the Talk 1. Introduction: What are fractals ? Scale, self-similarity, invariance, the local-global continuum, emergence, self-organisation, regularity 2. Fractal and Geometry Euclidean dimensions and fractal dimensions - points, lines, areas, volumes and on into the n’th dimension

6. 3. Deterministic and Stochastic Fractals Regular and statistically regular forms, natural and man- made forms, hard and soft forms, physical and abstract 4. Coastlines and Skylines 5. Regular Patterns: generating fractal forms from repetitive operations - cellular automata 6. Less Regular Form: Tree-like Structure coastlines and trees - the Koch curve, Barnsley’s fern (IFS systems), road systems at local and urban scales

7. 7. Diffusion-Limited Aggregation cell-like constructions, and physical processes like crystal growth 8. Examples of DLA - Urban Models 9. Fractal Urban Growth different scales - cellular models 10. Fractal Objects: Agents and Simulation new ideas for simulating objects at the most local scale

8. Fractal Lines: Coastlines: Skylines: Manhattan

9. The skyline is outlined and the volume divided into squares: the number of squares at each level of scale is then counted and the number versus the measure of the scale related from which the fractal dimension is computed

10. The Koch snowflake curve: How you generate tree-like fractals from lines, putting lines together to from areas like islands and cities

11. Regular fractals from cellular operations

12. Planting many seeds in a plane and then operating the repetitive rules over and over gain to grow a cellular landscape

13. A classic landscape which is skewed into a regular triangular shape called a Sierpinski gasket

14. Barnsley’s fern, from his book Fractals Everywhere which is generated by a rather sophisticated mathematical systems of routine and repetitive transformations called the Iterated Function System

15. Computer graphics depends upon fractals ! At least for natural forms such as trees

16. Networks at the local level: pedestrian segregation - the Radburn Layout

17. The fractal route structure of a small English town centre: Wolverhampton

18. Radial-concentric structure of the South East of England based on fractal like growth of road systems

19. Some of the most interesting work is being done in virtual space - in cyberspace not in real space. Here is an example of such a network

20. DLA - Diffusion-Limited Aggregation: everything from lightening to road structures to trees to lungs to life

21. http://www.casa.ucl.ac.uk/fractals.ppt http://www.geosimulation.com/

23. Fractal Britain

24. Scale and fractals: physics gets in on the act

26. How to Model Fractals – Cellular Automata Change in CA models takes place when some decision in a local neighborhood activates some rule which in turn leads to development of some sort. The cellular space is usually a grid and the neighborhoods usually based on cells which are adjacent to the one in question

27. Neighborhoods (a) Moore (b) von Neumann (c) Extended Moore von Neumann von Neumann

28. Rules: for any cell {x,y} in the neighborhood if any neighborhood cell other than {x,y} is already developed, then cell {x,y} is developed {note what ‘developed’ means}

29. Fields: apply related quantities to the cells in a neighborhood if any neighborhood cell other than {x,y} is already developed, then the field value p {x,y} is set & if p {x,y} > some threshold value then the cell {x,y} is developed

30. Fields again: fields are often thought about as density or probability fields

31. any cell on a Moore grid with a simple rule

32. one cell on a Moore grid with a simple rule

33. one cell in a von N grid with a simple rule

34. one cells in a Moore-vN with a simple rule

35. any cell with random field in a Moore nei

36. Diffusion Limited Aggregation (DLA-DBM)

37. DLA Potential Field

38. DLA on a 10000 x 10000 Lattice

39. Symmetric Crystal from central seed

40. DLA as Crystal Growth

41. Temperature Map of DLA as Crystal Growth

42. Many Seeds - simple rule based growth

43. Many Seeds - field based growth

44. Many Seeds - DLA crystal growth