Centre for Advanced Spatial Analysis (CASA), UCL, Centre for Advanced Spatial Analysis (CASA), UCL, 1-19 Torrington Place, London WC1E 6BT, UK1-19 Torrington Place, London WC1E 6BT, UKemail m.batty@ucl.ac.uk web http://www.casa.ucl.ac.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 Fractals : The New Science of CitiesCities

Michael BattyMichael Batty

Outline of the TalkOutline of the Talk

1. Introduction: What are fractals ?1. Introduction: What are fractals ?2. Fractal and Geometry 2. Fractal and Geometry 4. Deterministic and Stochastic Fractals4. Deterministic and Stochastic Fractals3. Coastlines and Skylines3. Coastlines and Skylines4. Regular Patterns: 4. Regular Patterns: 5. Less Regular Form: Tree-like Structure5. Less Regular Form: Tree-like Structure

6. Diffusion-Limited Aggregation6. Diffusion-Limited Aggregation7. Examples of DLA - Urban Models7. Examples of DLA - Urban Models8. Fractal Urban Growth8. Fractal Urban Growth9. Fractal Objects: Agents and Simulation9. Fractal Objects: Agents and Simulation

All this is a massive digression from my talk but in way it All this is a massive digression from my talk but in way it serves to show that we are interested in scaling, and the serves to show that we are interested in scaling, and the physics of systems and we want to apply these to citiesphysics of systems and we want to apply these to cities

In fact, I have not told you about most of the work that has 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 gone on in this field – it is in land use and traffic modelling where the gravitational hypothesis – scaling again – has where the gravitational hypothesis – scaling again – has been used to model traffic flow.been used to model traffic flow.

Here at UCL there has not been that much work in this area Here at UCL there has not been that much work in this area – although in the 1950s and 1960 Smeed and Wardrop – although in the 1950s and 1960 Smeed and Wardrop developed traffic theory here in the Centre for Transport developed traffic theory here in the Centre for Transport StudiesStudies

In fact there is much more now, and it is growing – the In fact there is much more now, and it is growing – the establishment of my own unit – CASA – one of Derek establishment of my own unit – CASA – one of Derek Roberts initiativesRoberts initiatives

Outline of the TalkOutline of the Talk

1. Introduction: What are fractals ?1. Introduction: What are fractals ?Scale, self-similarity, invariance, the local-global Scale, self-similarity, invariance, the local-global continuum, emergence, self-organisation, regularitycontinuum, emergence, self-organisation, regularity

2. Fractal and Geometry 2. Fractal and Geometry Euclidean dimensions and fractal dimensions - points, Euclidean dimensions and fractal dimensions - points, lines, areas, volumes and on into the n’th dimensionlines, areas, volumes and on into the n’th dimension

3. Deterministic and Stochastic Fractals3. Deterministic and Stochastic FractalsRegular and statistically regular forms, natural and man-Regular and statistically regular forms, natural and man-made forms, hard and soft forms, physical and abstractmade forms, hard and soft forms, physical and abstract

4. Coastlines and Skylines4. Coastlines and Skylines

5. Regular Patterns: 5. Regular Patterns: generating fractal forms from repetitive operations - generating fractal forms from repetitive operations - cellular automatacellular automata

6. Less Regular Form: Tree-like Structure6. Less Regular Form: Tree-like Structurecoastlines and trees - the Koch curve, Barnsley’s fern coastlines and trees - the Koch curve, Barnsley’s fern (IFS systems), road systems at local and urban scales(IFS systems), road systems at local and urban scales

7. Diffusion-Limited Aggregation7. Diffusion-Limited Aggregationcell-like constructions, and physical processes like cell-like constructions, and physical processes like crystal growthcrystal growth

8. Examples of DLA - Urban Models8. Examples of DLA - Urban Models

9. Fractal Urban Growth 9. Fractal Urban Growth different scales - cellular modelsdifferent scales - cellular models

10. Fractal Objects: Agents and 10. Fractal Objects: Agents and SimulationSimulationnew ideas for simulating objects new ideas for simulating objects at the most local scale at the most local scale

Fractal Lines: Coastlines: Skylines: Manhattan

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

The Koch snowflake curve: How you generate tree-like fractals from lines, putting lines together to from areas

like islands and cities

Regular fractals from cellular operations

Planting many seeds in a plane and

then operating the repetitive

rules over and over gain to

grow a cellular landscape

A classic landscape which is

skewed into a regular

triangular shape called a

Sierpinski gasket

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

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

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

The fractal route structure of a small English town centre:

Wolverhampton

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

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

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

http://www.casa.ucl.ac.uk/fractals.ppt

http://www.geosimulation.com/http://www.geosimulation.com/

Fractal Britain

Scale and fractals: physics gets in on the act

How to Model Fractals – CellularHow to Model Fractals – CellularAutomataAutomataChange in CA models takes place whenChange in CA models takes place whensome decision in a local neighborhoodsome decision in a local neighborhoodactivates some rule which in turn leads to activates some rule which in turn leads to development of some sort.development of some sort.

The cellular space is usually a grid and theThe cellular space is usually a grid and theneighborhoods neighborhoods usuallyusually based on cells based on cells which are adjacent to the one in questionwhich are adjacent to the one in question

NeighborhoodsNeighborhoods

(a) Moore(a) Moore

(b) von Neumann(b) von Neumann

(c) Extended Moore(c) Extended Moore von Neumannvon Neumann

Rules: for any cell {x,y} in theRules: for any cell {x,y} in theneighborhoodneighborhood

if any neighborhood cell other than {x,y}if any neighborhood cell other than {x,y}is already developed,is already developed,

then cell {x,y} is developedthen cell {x,y} is developed

{note what ‘developed’ means}{note what ‘developed’ means}

Fields: apply related quantitiesFields: apply related quantitiesto the cells in a neighborhoodto the cells in a neighborhood

if any neighborhood cell other than {x,y}if any neighborhood cell other than {x,y}is already developed,is already developed,then the field value p {x,y} is setthen the field value p {x,y} is set&&if p {x,y} > some threshold valueif p {x,y} > some threshold valuethen the cell {x,y} is developedthen the cell {x,y} is developed

Fields again: fields are often Fields again: fields are often thought about as density orthought about as density orprobability fieldsprobability fields

any cell on a Moore grid with a simple ruleany cell on a Moore grid with a simple rule

one cell on a Moore grid with a simple ruleone cell on a Moore grid with a simple rule

one cell in a von N grid with a simple ruleone cell in a von N grid with a simple rule

one cells in a Moore-vN with a simple ruleone cells in a Moore-vN with a simple rule

any cell with random field in a Moore neiany cell with random field in a Moore nei

Diffusion Limited Aggregation (DLA-DBM)Diffusion Limited Aggregation (DLA-DBM)

DLA Potential FieldDLA Potential Field

DLA on a 10000 x 10000 LatticeDLA on a 10000 x 10000 Lattice

Symmetric Crystal from central seedSymmetric Crystal from central seed

DLA as Crystal GrowthDLA as Crystal Growth

Temperature Map of DLA as Crystal GrowthTemperature Map of DLA as Crystal Growth

Many Seeds - simple rule based growthMany Seeds - simple rule based growth

Many Seeds - field based growthMany Seeds - field based growth

Many Seeds - DLA crystal growthMany Seeds - DLA crystal growth

Many Seeds - DLA crystal growthMany Seeds - DLA crystal growth