04/20/23 University of Virginia

Implications for Everyday Systems

Presented by

Selvin George

A New Kind of Science (Ch. 8)

By Stephen Wolfram

2/3/2003 University of Virginia

Overview Issues with traditional system modelling Mathematical models v/s cellular automata Study specific examples of everyday systems

Snowflakes shapes, crystallization Fluid Flow, eddies Branching pattern of leaves Stripes/spots on the skins of animals

Model most important features, patterns, shapes etc., using simple cellular automata

Critique

2/3/2003 University of Virginia

Traditional modelling A model is an idealization of a system We capture some aspects, ignore others Compare the behaviour generated by the

model to the system for significant similarities Behaviour is often characterised as metrics

(stability, hysteresis etc.,) based on mathematical derivations

A good model is simple, captures a large number of system features

2/3/2003 University of Virginia

Issues with modelling From traditional science: if the behavior of a

system is complex, then any model for the system must somehow be correspondingly complex

Often the models are as complicated as the phenomenon it purports to describe

Typically models are complicated and need to be “patched” when differing results are obtained

2/3/2003 University of Virginia

Mathematical v/s Cellular“In most cases, there have been in the past, never really been any

models that can even reproduce the most obvious features of the behaviour we see”

Mathematics models describe a system using equations. Numbers represent system behaviour

Best first step in assessing a model is not to look at these numbers but rather just to use one’s eyes

Easy to set up Cellular automata for most systems Growth-Inhibition is set up using the automaton rules Often Wolfram’s models have been extended

2/3/2003 University of Virginia

Snowflakes

2/3/2003 University of Virginia

Snowflakes using Cellular Automata

2/3/2003 University of Virginia

Breaking of Solids

2/3/2003 University of Virginia

Fluid Flow and eddies – (1)

2/3/2003 University of Virginia

Fluid Flow and eddies – (2)

2/3/2003 University of Virginia

Fluid Flow Model using Cellular Automata – (1)

2/3/2003 University of Virginia

Fluid Flow Model using Cellular Automata – (2)

2/3/2003 University of Virginia

Fluid Flow Model using Cellular Automata – (3)

2/3/2003 University of Virginia

Branching patterns

2/3/2003 University of Virginia

Branching patterns using Substitution Model – (1)

2/3/2003 University of Virginia

Branching patterns using Substitution Model – (2)

2/3/2003 University of Virginia

Mollusc shells

2/3/2003 University of Virginia

Mollusc shells using Substitution Models

2/3/2003 University of Virginia

Designs and Patterns on Animal Skin

2/3/2003 University of Virginia

Stripes using Cellular Automata

2/3/2003 University of Virginia

Wolfram’s Admissions No control over the underlying rules Must deduce them from phenomena Even his models may not capture

many features Some of the models described earlier

were found by trial and error

2/3/2003 University of Virginia

Critique – (1) System Modelling

Detail v/s Basic Behaviour Wolfram’s models capture the basic

mechanisms However he does not give a framework

Panning present-day models is unfair

Basic ModelLevel o

f D

eta

il

Detailed Model

2/3/2003 University of Virginia

Critique – (2) The rules of a cellular automata does not give

us an insight into the system behaviour On the other hand, mathematical models are

more descriptive in nature Unless we work at the lowest LOD, cellular

automata based models are prone to the same inefficiencies of current modelling methods

System modelling with cellular automata will be based more on trial and error rather than repeated refinement of models