A cellular automaton is a discrete model studied in computer science, mathematics, physics, complexity science, theoretical biology and microstructure modeling. This presentation is a basic introduction.

1. CPAD|KirthiBalakrishnan|SimrithiSKumar

2. A cellular automaton consists of a regular grid of
cells, each in one of a finite number of states, such as
on and off (in contrast to a coupled map lattice).
The grid can be in any finite number of dimensions.
For each cell, a set of cells called its neighborhood is
defined relative to the specified cell. An initial state
(time t = 0) is selected by assigning a state for each
cell.
A new generation is created (advancing t by 1),
according to some fixed rule (generally, a
mathematical function)[3] that determines the new
state of each cell in terms of the current state of the
cell and the states of the cells in its neighborhood.
What is ?
2
CA in a dodeca grid

3. To illustrate how CA works, we first define
● a grid of cells, ( or it could be irregular but to simplify we will assume a
square grid)
● a neighbourhood around each cell which is composed of the nearest cells,
● a set of rules as to how what happens in the neighbourhood affects the
development of the cell in question
● a set of states that each cell can take on – i.e. developed or not developed
● an assumption of universality that all these features operate uniformly and
universally

4. Each box stands for a student wearing (black) or not wearing (white) a hat. Let us make
the two following assumptions:
● Hat rule: a student will wear the hat in the following class if one or the
other—but not both—of the two classmates sitting immediately on her left and on
her right has the hat in the current class (if nobody wears a hat, a hat is out of
fashion; but if both neighbors wear it, a hat is now too popular to be trendy).
● Initial class: during the first class in the morning, only one student in the middle
shows up with a hat.

5. 5
Consecutive rows represent the evolution in time through
subsequent classes.
The evolutionary pattern displayed contrasts with the
simplicity of the underlying law (the “Hat rule”) and
ontology (for in terms of object and properties, we only
need to take into account simple cells and two states).
The global, emergent behavior of the system supervenes
upon its local, simple features, at least in the following
sense: the scale at which the decision to wear the hat is
made (immediate neighbors) is not the scale at which the
interesting patterns become manifest.

6. Even perfect knowledge
of individual decision
rules does not always
allow us to predict
macroscopic structure.
We get macro-surprises
despite complete
micro-knowledge. ”
—Epstein (1999: 48)

7. There’s an amazing
diversity of forms.
And, yes, they’re often
complicated.
But because they’re
based on simple
underlying rules, they
always have a certain
logic to them: in a
sense each of them
tells a definite
“algorithmic story”.

8. ● Essentially CA models developed in the late 1980s early 1990s from at least three
sources: bottom up thinking about systems in contrast to top down, concepts of
emergence in particular related to morphology, GIS and raster based
representation of activity layers.
● These models have found favour in rapidly growing systems which are characterised
by urban sprawl, like Phoenix. They have been quite inappropriately applied to non ‐
rapid growth cities where the focus is on redistribution.
● They have not been widely applied by municipalities as they do not contain explicit
mechanisms for generating numerical forecasts that are demographically or
economically based.

9. (a)
The
neighbourhood is
composed of 8
cells around the
central cell
How a CA works defined on
a grid of cells with two states – not
developed & developed
9
(b)
Place the neighbourhood over each
cell on the grid. The rule says that if
there is one or more cells developed
(black) in the neighbourhood, then
the cell is developed.
(c)
If you keep on doing
this for every cell,
you get the diffusion
from the central cell
shown below.

10. MOORE VON NEUMANN
EXTENDED MOOR
VON NEUMANN
composed of different
combinations of cells in strictly
deterministic CA models

11. For example, for any cell {x,y},
● if only one
neighborhood cell
either NW, SE, NE, or
SW other than {x,y} is
already developed,
● then cell {x,y} is
developed according
to the following
neighborhood
switching rule

13. For probabilistic rules, we can generate statistically self‐similar structures which look more like
real city morphologies. For example,
● 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

14. Over and over again we will
see the same kind of thing:
that even though the
underlying rules for a
system are simple, and
even though the system is
started from simple initial
conditions, the behavior
that the system shows can
nevertheless be highly
complex. ”
—Wolfram (2002: 28)

15. 15
Any questions?
SOURCES:
● Stanford Plato Entry:
Cellular Automata
● Steve Wolfram: Rule 30 in
CA
● CA Lecture Notes: UCL