1
Reprint from the magazine Design, London: Council of Industrial Design, N° 206, 1966]
A city is not a tree
By Christopher Alexander
The article that follows has won for itself, and its author - an architect and
mathematician - a special distinction among all that has been written about design during
the past few years. Together with a series of articles by Ada Louise Huxtable and two essays
by Lewis Mumford, it was selected as one of the 1965 Kaufmann International Design Awards.
The 1965 awards follow a series started in 1960, sponsored by the Edgar J. Kaufmann
Foundation and administered by the Institute of International Education. On previous
occasions they have gone to Charles and Ray Eames, Walter Gropius, and the Olivetti
Corporation, and subsequently have provided a series of research grants totalling $ 61,500.
The latest awards were given for "the most effective statements dealing with the field of
design, published in periodical or occasional form within the past five years", and represent
the first occasion on which the contribution of criticism to the development of design has
been overtly acknowledged.
The international jury which selected the 1965 awards (Richard Latham, Peter
Muller-Munk and David Strout from the USA, Finn Juhl from Denmark, and John E. Blake,
chairman, from Britain) studied some 200 articles and essays that had been submitted to
them by consultants in seven countries. Commenting on the selection, the judges' report
stated, "Essentially, three aspects of the problem emerged, all of which we considered to be
of equal importance. The first involved those statements which contribute to new thinking in
the field of design.... The second involved statements which, though possibly not containing
new thought, contribute to a wider understanding of known problems.... The third
concerned the quality of writing, for the effectiveness of any statement will depend on
clarity of expression, on the logical and economical presentation of an argument, on the
mastery of words.... The first and second aspects often overlapped in the same item, and the
third we considered to be essential to any statement which was to be selected for an award".
The brief to the jury had defined design as "pIanning that results in any visually
expressive, practical implementation of human occupations, ceremonies or play". This broad
definition created certain difficulties for the judges, but their report continues that the final
selection was ". . . a recognition that the detailed considerations of architecture and industrial
design paled into comparative insignificance when seen against the massive problems of
social planning and its expression in the structure and forms of the modern city. The three
awards, each in its own way, had tackled this problem. Each recognised that the evolution of
the modern city was reaching a point of crisis and that its solution was possibly the greatest
challenge facing the second half of the twentieth century. Each recognised that the city is a
system of vast complexity and in turn is part of a bigger system of social organisation whose
values and goals are being questioned."
In deciding to reprint A City is not a Tree, DESIGN is aware that it will seem to be
outside the range of subjects normally covered by the magazine. The judges' report
emphasised, however, that "The principles he [Dr Alexander] describes, and the analytical
methods he adopts, are applicable at all levels of design". It was felt that Dr Alexander's
thesis is as relevant to industrial designers, architects and engineers as it is to city planners,
for the city provides the context into which most buildings, products and services must fit.
And it is important that those of us who are primarily concerned with such things should
stand back, once in a while, to take in the broader view.
A City is not a Tree is reproduced here by kind permission of the American journal
Architectural Forum, where it originally appeared in two parts in April and May last year.
Since the article was first published some slight amendments have been made at the request
of the author.

2
Christopher Alexander, a member of
the faculty of the University of California
College of Environmental Design, is author
of Notes on the Synthesis of Form and co-
author with Serge Chermayeff of
Community and Privacy. He received his
bachelor's degree in architecture and
master's degree in mathematics from
Trinity College, Cambridge, and his
doctorate in architecture from Harvard.
***
The tree of my title is not a green tree
with leaves. It is the name of an abstract
structure. I shall contrast it with another,
more complex abstract structure called a
semi-lattice. The city is a semi-lattice, but it
is not a tree. In order to relate these
abstract structures to the nature of the city,
I must first make a simple distinction.
Natural and artificial cities
I want to call those cities which have
arisen more or less spontaneously over
many, many years "natural cities". And I
shall call those cities and parts of cities
which have been deliberately created by
designers and planners "artificial cities".
Siena, Liverpool, Kyoto, Manhattan are
examples of natural cities. Levittown,
Chandigarh, and the British New Towns
are examples of artificial cities.
It is more and more widely
recognised today that there is some
essential ingredient missing from artificial
cities. When compared with ancient cities
that have acquired the patina of life, our
modern attempts to create cities artificially
are, from a human point of view, entirely
unsuccessful.
Architects themselves admit more
and more freely that they really like living
in old buildings more than new ones. The
non-artloving public at large, instead of
being grateful to architects for what they
do, regards the onset of modern buildings
and modern cities everywhere as an
inevitable, rather sad piece of the larger
fact that the world is going to the dogs
It is much too easy to say that these
opinions represent only people's
unwillingness to forget the past, and their
determination to be traditional. For myself,
I trust this conservatism. People are
usually willing to move with the times.
Their growing reluctance to accept the
modern city evidently expresses a longing
for some real thing, something which for
the moment escapes our grasp.
The prospect that we may be turning
the world into a place peopled only by
little glass and concrete boxes has alarmed
many architects, too. To combat the glass
box future, many valiant protests and
designs have been put forward, all hoping
to recreate in modern form the various
characteristics of the natural city which
seem to give it life. But so far these designs
have only remade the old. They have not
been able to create the new.
Outrage, the Architectural Review's
campaign against the way in which new
construction and telegraph poles are
wrecking the English town, based its
remedies, essentially, on the idea that the
spatial sequence of buildings and open
spaces must be controlled if scale is to be
preserved - an idea that really derives from
Camillo Sitte's book about ancient squares
and piazzas.
Another kind of remedy, in protest
against the monotony of Levittown, tries to
recapture the richness of shape found in
the houses of a natural old town. Llewelyn
Davies' village at Rushbrooke in England
is an example - each cottage is slightly
different from its neighbour, the roofs jut
in and out at picturesque angles, the
shapes are 'interesting' and cute.
A third suggested remedy is to get
high density back into the city. The seems
to be that if the whole metropolis could
only be like Grand Central Station, with
lots and lots of layers and tunnels all over
the place, and enough people milling
around in them, maybe it would be human
again. The artificial urbanity of Victor
Gruen's schemes and of the LCC's scheme
for Hook New Town, both betray this
thought at work.
Another very brilliant critic of the
deadness which is everywhere is Jane
Jacobs. Her criticisms are excellent. But
when you read her concrete proposals for
what we should do instead, you get the
idea that she wants the great modern city

3
to be a sort of mixture between Greenwich
Village and some Italian hill town, full of
short blocks and people sitting in the
street.

The problem these designers have
tried to face is real. It is vital that we
discover the property of old towns which
gave them life, and get it back into our
own artificial cities. But we cannot do this
merely by remaking English villages,
Italian piazzas and Grand Central Stations.
Too many designers today seem to be
yearning for the physical and plastic
characteristics of the past, instead of
searching for the abstract ordering
principle which the towns of the past
happened to have, and which our modern
conceptions of the city have not yet found.
These designers fail to put new life into the
city, because they merely imitate the
appearance of the old, its concrete
substance: they fail to unearth its inner
nature.
What is the inner nature, the ordering
principle, which distinguishes the artificial
city from the natural city? You will have
guessed from the first paragraph what I
believe this ordering principle to be. I
believe that a natural city has the
organisation of a semi-lattice; but that
when we organise a city artificially, we
organise it as a tree.

Trees and semi-lattices
Both the tree and the semi-lattice are
ways of thinking about how a large
collection of many small systems goes to
"make up a large and complex system.
More generally, they are both names for
structures of sets.
In order to define such structures, let
me first define the concept of a set. A set is
a collection of elements which for some
reason we think of as belonging together.
Since, as designers, we are concerned with
the physical living city and its physical
backbone, we most naturally restrict
ourselves to considering sets which are
collections of material elements like
people, blades of grass, cars, bricks,
molecules, houses, gardens, water pipes,
the water molecules that run in them, etc.
When the elements of a set belong
together because they co-operate or work
together somehow, we call the set of
elements a system.

Here is an example. In Berkeley, at
the corner of and Euclid, there is a drug
store, and outside the drug store a traffic
light. In the entrance to the drug store
there is a newsrack where the day's papers
are displayed. When the light is red,
people who are waiting to cross the street
stand idly by the light; and since they
have nothing to do, they look at the
papers displayed on the newsrack which
they can see from where they stand. Some
of them just read the headlines, others
actually buy a paper while they wait.
This effect makes the newsrack and
the traffic light interdependent; the
newsrack, the newspapers on it, the money
going from people's pockets to the dime
slot, the people who stop at the light and
read papers, traffic light, the electric
impulses which make the lights change,
the sidewalk which the stand on form a
system - they all work together.
From the designer's point of view, the
physically unchanging part of this system
is of special interest. The news rack, the
traffic light, and the sidewalk between
them, related as they are, form the fixed
part of the system. It is the unchanging
receptacle in which the changing parts of
the system - people, newspapers, money,
and electrical impulses - can work
together. I define this fixed part as a unit of
the city. It derives its coherence as a unit
both from the forces which hold its own
elements together, and from the dynamic
coherence of the larger living system
which includes it as a fixed, invariant part.
Other examples of systems in the city
are: the set of particles which go to make
up a building; the set of particles which go
to make up a human body; the cars on the
freeway, plus the people in them, plus the
freeway they are driving on; two friends
on the phone, plus the telephones they
hold, plus the telephone line connecting

4
them; Telegraph Hill with all its buildings,
services and inhabitants; the chain of
Rexall drug stores; the physical elements
of San Francisco that fall under the
administrative authority of City Hall;
everything within the physical boundary
of San Francisco, plus all the people who
visit the city regularly and contribute to its
development (like Bob Hope or the
president of Arthur D. Little), plus all the
major sources of economic welfare which
supply the city with its wealth; the dog
next door, plus my garbage can, plus the
garbage out of my garbage can which he
lives on; the San Francisco chapter of the
John Birch Society.
Each one of these is a set of elements
made coherent and co-operative by some
sort of inner binding forces. And each one,
just like the traffic light - newsrack system,
has a physically fixed part which we think
of as a unit of the city.
Of the many, many fixed concrete
subsets of the city which are the
receptacles for its systems, and can
therefore be thought of as significant
physical units, we usually single out a few
for special consideration. In fact, I claim
that whatever picture of the city someone
has is defined precisely by the subsets he
sees as units.

Now, a collection of subsets which
goes to make up such a picture is not
merely an amorphous collection.
Automatically, merely because
relationships are established among the
subsets once the subsets are chosen, the
collection has a definite structure.
To understand this structure, let us
think abstractly for a moment, using
numbers as symbols. Instead of talking
about the real sets of millions of real
particles which occur in the city, let us
consider a simpler structure made of just
half a dozen elements. Label these
elements 1, 2, 3, 4, 5, 6. Not including the
full set (1, 2, 3, 4, 5, 6), the empty set (-),
and the one element sets (1), (2), (3), (4),
(5), (6), there are 56 different subsets we
can pick from these six elements.
Suppose we now pick out certain of
these 56 sets (just as we pick out certain
sets and call them units when we form our
picture of the city). Let us say, for example,
that we pick the following subsets: (123),
(34), (45), (234), (345), (12345), (3456).
What are the possible relationships
among these sets? Some sets will be
entirely part of larger sets, as (34) is part of
(345) and (3456). Some of the sets will
overlap, like (123) and (234). Some of the
sets will be disjoint - that is, contain no
elements in common, like (123) and (45).
We can see these relationships
displayed in two ways. In diagram a each
set chosen to be a unit has a line drawn
round it. In diagram b the chosen sets are
arranged in order of ascending magnitude,
so that whenever one set contains another
(as (345) contains (34) ), there is a vertical
path leading from one to the other. For the
sake of clarity and visual economy, it is
usual to draw lines only between sets
which have no further sets and lines
between them; thus the line between (34)
and (345), and the line between (345) and
(3456), make it unnecessary to draw a line
between (34) and (3456).
As we see from these two
representations, the choice of subsets alone
endows the subsets as a whole with an
overall structure. This is the structure
which we are concerned with here. When
the structure meets certain conditions, it is
called a semi-lattice. When it meets other
more restrictive conditions, it is called a
tree.
The semi-lattice axiom goes like this:
"A collection of sets forms a semi-lattice if
and only if, when two overlapping sets
belong to the collection, then the set of
elements common to both also belongs to
the collection".

5

The structure illustrated in diagram a
and b is a semi-lattice. It satisfies the axiom
since, for instance, (234) and (345) both
belong to the collection and their common
part, (34), also belongs to it. (As far as the
city is concerned, this axiom states merely
that wherever two units overlap, the area
of overlap is itself a recognisable entity
and hence a unit also. In the case of the
drug store example, one unit consists of
the newsrack, sidewalk, and traffic light.
Another unit consists of the drug store
itself, with its entry and the newsrack. The
two units overlap in the newsrack. Clearly
this area of overlap is itself a recognisable
unit, and so satisfies the axiom above
which defines the characteristics of a semi-
lattice).
The tree axiom states: "A
collection of sets forms a tree if, and
only if, for any two sets that belong to
the collection, either one is wholly
contained in the other, or else they are
wholly disjoint".
The structure illustrated in diagrams
c and d is a tree. Since this axiom excludes
the possibility of overlapping sets, there is
no way in which the semi-lattice axiom can
be violated, so that every tree is a trivially
simple semi-lattice.
However, in this paper we are not so
much concerned with the fact that a tree
happens to be a semi-lattice, but with the
difference between trees and those more
general semi-lattices which are not trees
because they do contain overlapping units.
We are concerned with the difference
between structures in which no overlap
occurs, and those structures in which
overlap does occur.

It is not merely the overlap which
makes the distinction between the two
important. Still more important is the fact
that the semi-lattice is potentially a much
more complex and subtle structure than a
tree. We may see just how much more
complex a semi-lattice can be than a tree in
the following fact: a tree based on 20
elements can contain at most 19 further
subsets of the 20, while a semi-lattice based
on the same 20 elements can contain more
than one million different subsets.
This enormously greater variety is an
index of the great structural complexity a
semi-lattice can have when compared with
the structural simplicity of a tree. It is this
lack of structural complexity, characteristic
of trees, which is crippling our conceptions
of the city.
Artificial cities which are trees

6
To demonstrate, let us look at some
modern conceptions of the city, each of
which I shall show to be essentially a tree.
It will perhaps be useful, while we look at
these plans, to have a little ditty in our
minds:
"Big fleas have little fleas
Upon their back to bite'em;
Little fleas have lesser fleas,
And so ad infinitum".
This rhyme expresses perfectly and
succinctly the structural principle of the
tree.

1 Columbia, Maryland, Community
Research and Development Inc.
Neighbourhoods, in clusters of five, form
villages. Transportation joins the villages
into a new town. The organisation is a tree.

2 Greenbelt, Maryland, Clarence Stein.
This 'garden city' has been broken down
into super-blocks. Each super-block
contains schools, parks, and a number of
subsidiary groups of houses built around
parking lots. The organisation is a tree.

3 Greater London plan (1943),
Abercrombie and Forshaw. The drawing
shows the structure conceived by
Abercrombie for London. It is made up of
a large number of communities, each
sharply separated from all adjacent
communities. Abercrombie writes, "The
proposal is to emphasise the identity of the
existing communities, to increase their
degree of segregation, and where
necessary to reorganise them as separate
and definite entities." And again, "The
communities themselves consist of a series
of sub-units, generally with their own
shops and schools, corresponding to
neighbourhood units." The city is
conceived as a tree with two principal

7
levels. The communities are the larger
units of the structure; the smaller sub-units
are neighbourhoods. There are no
overlapping units. The structure is a tree.

4 Tokyo plan, Kenzo Tange. This is a
beautiful example. The plan consists of a
series of loops stretched across the Tokyo
Bay. There are four major loops, each of
which contains three medium loops. In the
second major loop, one medium loop is
the railway station and another is the port.
Otherwise, each medium loop contains
three minor loops which are residential
neighbourhoods, except in the third major
loop where one contains government
offices and another industrial offices.

5 Mesa City, Paolo Soleri. The organic
shapes of Mesa City lead us, at a careless
glance, to believe that it is a richer
structure than our more obviously rigid
examples. But when we look at it in detail
we find precisely the same principle of
organisation. Take, particularly, the
university centre. Here we find the centre
of the city divided into a university and a
residential quarter, which is itself divided
into a number of villages (actually
apartment towers) for 4,000 inhabitants,
each again subdivided further and
surrounded by groups of still smaller
dwelling units.

8
6 Chandigarh (1951), by Le Corbusier. The
whole city is served by a commercial
centre in the middle, linked to the
administrative centre at the head. Two
subsidiary elongated, commercial cores
are strung out along the major arterial
roads, running north-south. Subsidiary to
these are further administrative
community and commercial centres, one
for each of the city's 20 sectors.

7 Brazilia, Lucio Costa. The entire
form pivots about the central axis, and
each of the two halves is served by a single
main artery. This main artery is in turn fed
by subsidiary arteries parallel to it. Finally,
these are fed by the roads which surround
the super-blocks themselves. The structure
is a tree.

8 Communitas, Percival and Paul
Goodman. Communitas is explicitly
organised as a tree: it is first divided into
four concentric major zones, the innermost
being a commercial centre, the next a
university, the third residential and
medical, and the fourth open country.
Each of these is further sub-divided: the
commercial centre is represented as great
cylindrical skyscraper, containing five
layers: airport, administration, light
manufacture, shopping and amusement;
and, at the bottom, railroads, buses, and
mechanical services. The university is
divided into eight sectors comprising
natural history, zoos and aquariums,
planetarium, science laboratories, plastic
arts, music and drama. The third
concentric ring is divided into
neighbourhoods of 4,000 people each, not
consisting of individual houses, but of
apartment blocks, each of these containing
further individual dwelling units. Finally,
the open country is divided into three
segments: forest preserves, agriculture,
and vacation-lands. The overall
organisation is a tree.

9

9 The most beautiful example of all I have
kept until last, because it symbolises the
problem perfectly. It appears in
Hilberseimer's book called The Nature of
Cities. He describes the fact that certain
Roman towns had their origin as military
camps, and then shows a picture of a
modern military encampment as a kind of
archetypal form for the city. It is not
possible to have a structure which is a
clearer tree. The symbol is apt for, of
course, the organisation of the army was
created precisely in order to create
discipline and rigidity. When a city is
endowed with a tree structure, this is what
happens to the city and its people. The
lower photo shows Hilberseimer's own
scheme for the commercial area of a city
based on the army camp archetype.

Each of these structures is a tree.

The units of which an artificial city is
made up are always organised to form a
tree. So that we get a really clear
understanding of what this means, let us
define a tree again:
Whenever we have a tree structure, it
means that within this structure no piece
of any unit is ever connected to other units,
except through the medium of that unit as
a whole.
The enormity of this restriction is
difficult to grasp. It is a little as though the
members of a family were not free to make
friends outside the family, except when the
family as a whole made a friendship.
The structural simplicity of trees is
like the compulsive desire for neatness and
order that insists that the candlesticks on a
mantelpiece be perfectly straight and
perfectly symmetrical about the centre.
The semi-lattice, by comparison, is the
structure of a complex fabric; it is the
structure of living things - of great
paintings and symphonies.
It must be emphasised, lest the
orderly mind shrink in horror from
anything that is not clearly articulated and
categorised in tree form, that the ideas of
overlap, ambiguity, multiplicity of aspect,

10
and the semi-lattice, are not less orderly
than the rigid tree, but more so. They
represent a thicker, tougher, more subtle
and more complex view of structure.
Let us now look at the ways in which
the natural city, when unconstrained by
artificial conceptions, shows itself to be a
semi-lattice.
A living city is and needs to be a
semi-lattice
Each unit in each tree that I have
described is the fixed, unchanging residue
of some system in the living city. A house,
for instance, is the physical residue of the
interactions between the members of a
family, their emotions and their
belongings. A freeway is the residue of
movement and commercial exchange. But
a tree contains only very few such units -
so that in a tree-like city only a few of its
systems can have a physical counterpart.
Thousands of important systems have no
physical counterpart.
In the worst trees, the units which do
appear fail to correspond to any living
reality; and those real systems, whose
existence actually makes the city live, have
been provided with no physical receptacle.

Neither the Columbia plan nor the
Stein plan, for example, corresponds to
social realities. The physical layout of the
plans, and the way they function, suggests
a hierarchy of stronger and stronger closed
social groups, ranging from the whole city
down to the family, each formed by
associational ties of different strength. Yet
this is entirely unreal.
In a traditional society, if we ask a
man to name his best friends and then ask
each of these in turn to name their best
friends, they will all name each other so
that they form a closed group. A village is
made of a number of separate closed
groups of this kind.
But today's social structure is utterly
different. If we ask a man to name his
friends and then ask them in turn to name
their friends, they will all name different
people, very likely unknown to the first
person; these people would again name
others, and so on outwards. There are
virtually no closed groups of people in
modern society. The reality of today's
social structure is thick with overlap - the
systems of friends and acquaintances form
a semi-lattice, not a tree, 10.
In the natural city, even the house on
a long street (not in some little cluster) is a
more accurate acknowledgment of the fact
that your friends live not next door, but far
away, and can only be reached by bus or
automobile. In this respect Manhattan has
more overlap in it than Greenbelt. And
though one can argue that in Greenbelt,
too, friends are only minutes away by car,
one must then ask: since certain groups
have been emphasised by the physical
units of the physical structure, why are
they socially irrelevant ones?
Another aspect of the city's social
structure which a tree can never mirror
properly is illustrated by Ruth Glass's
redevelopment plan for Middlesborough,
a city of 200,000, which she recommends
be broken into 29 separate
neighbourhoods. After picking her 29
neighbourhoods by determining where the
sharpest discontinuities of building type,
income, and job type occur, she asks
herself the question: "If we examine some
of the social systems which actually exist
for the people in such a neighbourhood, do
the physical units defined by these various
social systems all define the same spatial
neighbourhood?" Her own answer to this
question is "No, they do not."
Each of the social systems she
examines is a nodal system. It is made of
some sort of central node, plus the people
who use this centre. Specifically she takes
elementary schools, secondary schools,
youth clubs, adult clubs, post offices,
greengrocers, and grocers selling sugar.
Each of these centres draws its users from
a certain spatial area or spatial unit. This
spatial unit is the physical residue of the
social system as a whole, and is therefore a
unit in the terms of this paper. The units
corresponding to different kinds of centres
for a single neighbourhood, Waterloo
Road, are shown in 11.
The hard outline is the boundary of
the so-called neighbourhood itself. The

11
grey spot stands for the youth club, and
the small solid rings stand for areas where
its members live. The ringed spot is the
adult club, and the homes of its members
form the unit marked by dashed
boundaries. The white square is the post
office, and the dotted line marks the unit
which contains its users. The secondary
school is marked by the spot with a white
triangle in it. Together with its pupils, it
forms the system marked by the
dot-dashed line.
As you can see at once, the different
units do not coincide. Yet neither are they
disjoint. They overlap.
We cannot get an adequate picture of
what Middlesborough is, or of what it
ought to be, in terms of 29 large and
conveniently integral chunks called
neighbourhoods. When we describe the
city in terms of neighbourhoods, we
implicitly assume that the smaller
elements within any one of these
neighbourhoods belong together so tightly
that they only interact with elements in
other neighbourhoods through the
medium of the neighbourhood to which
they themselves belong. Ruth Glass herself
shows clearly that this is not the case.
12 and 13 are two pictures of the
Waterloo neighbourhood. For the sake of
argument, I have broken it into a number
of small areas. 12 shows how these pieces
stick together in fact, and 13 shows how
the redevelopment plan pretends they
stick together.
There is nothing in the nature of the
various centres which says that their
catchment areas should be the same. Their
natures are different. Therefore the units
they define are different. The natural city
of Middlesborough was faithful to the
semi-lattice structure they have. Only in
the artificial tree conception of the city are
their natural, proper and necessary
overlaps destroyed.

12
The same thing happens on a smaller
scale. Take the separation of pedestrians
from moving vehicles, a tree concept
proposed by Le Corbusier, Louis Kahn,
and many others. At a very crude level of
thought, this is obviously a good idea. It is
dangerous to have 60 mph cars in contact
with little children toddling. But it is not
always a good idea. There are times when
the ecology of a situation actually
demands the opposite. Imagine yourself
coming out of a Fifth Avenue store: you
have been shopping all afternoon; your
arms are full of parcels; you need a drink;
your wife is limping. Thank God for taxis!
Yet the urban taxi can function only
because pedestrians and vehicles are not
strictly separated. The prowling taxi needs
a fast stream of traffic so that it can cover a
large area to be sure of finding a
passenger. The pedestrian needs to be able
to hail the taxi from any point in the
pedestrian world, and to be able to get out
to any part of the pedestrian world to
which he wants to go. The system which
contains the taxicabs needs to overlap both
the fast vehicular traffic system and the
system of pedestrian circulation. In
Manhattan, pedestrians and vehicles do
share certain parts of the city, and the
necessary overlap is guaranteed, 14.

Another favourite concept of the
CIAM theorists and others is the
separation of recreation from everything
else. This has crystallised in our real cities
in the form of playgrounds. The
playground, asphalted and fenced in, is
nothing but a pictorial acknowledgement
of the fact that 'play' exists as an isolated
concept in our minds. It has nothing to do
with the life of play itself. Few
self-respecting children will even play in a
playground.
Play itself, the play that children
practise, goes on somewhere different
everyday. One day it may be indoors,
another day in a friendly gas station,
another day in a derelict building, another
day down by the river, another day on a
construction site which has been
abandoned for the weekend. Each of these
play activities, and the objects it requires,
forms a system. It is not true that these
systems exist in isolation, cut off from the
other systems in the city. The different
systems overlap one another, and they
overlap many other systems besides. The
units, the physical places recognised as
play places, must do the same.
In a natural city this is what happens.
Play takes place in a thousand places - it
fills the interstices of adult life. As they
play, children become full of their
surroundings. How can a child become
filled with his surroundings in a fenced
enclosure? He can't. In a semi-lattice, he
can; in a tree, he can't.

A similar kind of mistake occurs in
trees like those of Goodman's
Communitas, or Soleri's Mesa City, which
separate the university from the rest of the
city. Again, this has actually been realised
in the common American form of the
isolated campus.
What is the reason for drawing a line
in the city so that everything within the
boundary is university, and everything
outside is non-university? It is
conceptually clear. But does it correspond
to the realities of university life? Certainly
it is not the structure which occurs in
non-artificial university cities.
Take Cambridge University, for
instance. At certain points, Trinity Street is
physically almost indistinguishable from
Trinity College. One pedestrian crossover
in the street is literally part of the college.
The buildings on the street, though they
contain stores and coffee shops and banks
at ground level, contain undergraduates'
rooms in their upper stories. In many cases
the actual fabric of the street buildings
melts into the fabric of the old college
buildings so that one cannot be altered
without the other.
There will always be many systems
of activity where university life and city
life overlap: pub-crawling, coffee drinking,
the movies, walking from place to place. In
some cases whole departments may be
actively involved in the life of the city's
inhabitants (the hospital-cum-medical
school is an example). In Cambridge, a

13
natural city where university and city have
grown together gradually, the physical
units overlap because they are the physical
residues of city systems and university
systems which overlap, 15.

Let us look next at the hierarchy of
urban cores realised in Brazilia,
Chandigarh, the MARS plan for London
and, most recently, in the Manhattan
Lincoln Center, where various performing
arts serving the population of greater New
York have been gathered together to form
just one core.
Does a concert hall ask to be next to
an opera house? Can the two feed on one
another? Will anybody ever visit them
both, gluttonously, in a single evening, or
even buy tickets from one after going to a
concert in the other? In Vienna, London,
Paris, each of the performing arts has
found its own place. Each has created its
own familiar section of the city. In
Manhattan itself, Carnegie Hall and the
Metropolitan Opera House were not built
side by side. Each found its own place, and
now creates its own atmosphere. The
influence of each overlaps the parts of the
city which have been made unique by it.
The only reason that these functions
have all been brought together in the
Lincoln Center is that the concept of
performing art links them to one another.
But this tree, and the idea of a single
hierarchy of urban cores (which is its
parent), do not illuminate the relations
between art and city life. They are merely
born of the mania every simple-minded
person has for putting things with the
same name into the same basket.

The total separation of work from
housing, started by Tony Garnier in his
industrial city, then incorporated in the
1929 Athens Charter, is now found in
every artificial city and accepted
everywhere where zoning is enforced. Is
this a sound principle? It is easy to see how
bad conditions at the beginning of the
century prompted planners to try to get
the dirty factories out of residential areas.
But the separation misses a variety of
systems which require, for their
sustenance, little parts of both.
Jane Jacobs describes the growth of
backyard industries in Brooklyn. A man
who wants to start a small business needs
space, which he is very likely to have in his
own backyard. He also needs to establish
connections with larger going enterprises
and with their customers. This means that
the system of backyard industry needs to
belong both to the residential zone, and to
the industrial zone-these zones need to
overlap. In Brooklyn they do, 16. In a city
which is a tree, they can't.

Finally, let us examine the
subdivision of the city into isolated
communities. As we have seen in the
Abercrombie plan for London, this is itself
a tree structure. Yet the individual
communities have no reality as
functioning units. In London, as in any
great city, almost no one manages to find
work which suits him near his home.
People from one community work in the
factories which are in other communities.
There are, therefore, many hundreds
of thousands of worker-workplace
systems, each consisting of a man plus the
factory he works in, which cut across the
boundaries defined by Abercrombie's tree.
The existence of these units, and their
overlapping nature, indicates that the
living systems of London form a
semi-lattice. Only in the planner's mind
have they become a tree.
The fact that we have so far failed to
give this any physical expression has a
vital consequence. As things are, whenever
the worker and his workplace belong to
separately administered municipalities, the
community which contains the workplace
collects huge taxes and has relatively little
to spend it on, while the community where
the worker lives, if it is mainly residential,
collects only little in the way of taxes, and
yet has great additional burdens on its
purse in the shape of schools, hospitals,
etc. Clearly, to resolve this inequity, the
worker-workplace systems must be
anchored in physically recognisable units
of the city, which can then be taxed.

14
It might be argued that, even though
the individual communities of a great city
have no functional significance in the lives
of their inhabitants, they are still the most
convenient administrative units, and
should, therefore, be left in their present
tree organisation.
However, in the political complexity
of a modern city, even this is suspect.
Edward Banfield, in a recent book
called Political Influence, gives a detailed
account of the patterns of influence and
control that have actually led to decisions
in Chicago. He shows that although the
lines of administrative and executive
control have a formal structure which is a
tree, these formal chains of influence and
authority are entirely overshadowed by
the adhoc lines of control which arise
naturally as each new city problem
presents itself. These adhoc lines depend on
who is interested in the matter, who has
what at stake, who has what favours to
trade with whom.
This second structure, which is
informal, working within the framework
of the first, is what really controls public
action. It varies from week to week, even
from hour to hour, as one problem
replaces another. Nobody's sphere of
influence is entirely under the control of
any one superior; each person is under
different influences as the problems
change. Although the organisation chart in
the mayor's office is a tree, the actual
control and exercise of authority is
semi-lattice-like.
The origin of tree-like thought
The tree - though so neat and
beautiful as a mental device, though it
offers such a simple and clear way of
dividing a complex entity into units - does
not describe correctly the actual structure
of naturally occurring cities, and does not
describe the structure of the cities which
we need.
Now, why is it that so many
designers have conceived cities as trees
when the natural structure is in every case
a semi-lattice? Have they done so
deliberately, in the belief that a tree
structure will serve the people of the city
better? Or have they done it because they
cannot help it, because they are trapped by
a mental habit, perhaps even trapped by
the way the mind works; because they
cannot encompass the complexity of a
semi-lattice in any convenient mental
form; because the mind has an
overwhelming predisposition to see trees
wherever it looks and cannot escape the
tree conception?
I shall try to convince you that it is
for this second reason that trees are being
proposed and built as cities - that it is
because designers, limited as they must be
by the capacity of the mind to form
intuitively accessible structures, cannot
achieve the complexity of the semi-lattice
in a single mental act.

Let me begin with an example.
Suppose I ask you to remember the
following four objects: an orange, a
watermelon, a football and a tennis ball.
How will you keep them in your mind, in
your mind's eye? However you do it, you
will do it by grouping them. Some of you
will take the two fruits together, the
orange and the watermelon, and the two
sports balls together, the football and the
tennis ball. Those of you who tend to think
in terms of physical shape may group
them differently, taking the two small
spheres together - the orange and the
tennis ball, and the two larger and more
eggshaped objects - the watermelon and
the football. Some of you will be aware of
both groupings.
Either grouping taken by itself is a
tree structure. The two together are a
semi-lattice, 17. Now let us try and
visualise these groupings in the mind's
eye. 1 think you will find that you cannot
visualise all four sets simultaneously -
because they overlap. You can visualise
one pair of sets and then the other, and
you can alternate between the two pairs
extremely fast, so fast that you may
deceive yourself into thinking you can
visualise them all together. But in truth,
you cannot conceive all four sets at once in
a single mental act.

15

You cannot bring the semi-lattice
structure into a visualisable form for a
single mental act. In a single mental act
you can only visualise a tree.
This is the problem we face as
designers. While we are not, perhaps,
necessarily occupied with the problem of
total visualisation in a single mental act,
the principle is still the same. The tree is
accessible mentally, and easy to deal with.
The semi-lattice is hard to keep before the
mind's eye, and therefore hard to deal
with.

It is known today that grouping and
categorisation are among the most
primitive psychological processes. Modern
psychology treats thought as a process of
fitting new situations into existing slots
and pigeon holes in the mind. Just as you
cannot put a physical thing into more than
one physical pigeon hole at once, so, by
analogy, the processes of thought prevent
you from putting a mental construct into
more than one mental category at once.
Study of the origin of these processes
suggests that they stem essentially from
the organism's need to reduce the
complexity of its environment by
establishing barriers between the different
events which it encounters.
It is for this reason - because the
mind's first function is to reduce the
ambiguity and overlap in a confusing
situation, and because, to this end, it is
endowed with a basic intolerance for
ambiguity - that structures like the city,
which do require overlapping sets within
them, are nevertheless persistently
conceived as trees.
The same rigidity dogs even the
perception of physical patterns. In
experiments by Huggins and myself at
Harvard, we showed people patterns
whose internal units overlapped, and
found that they almost always invented a
way of seeing the pattern as a tree - even
when the semi-lattice view of the patterns
would have helped them perform the
experimental task.
The most startling proof that people
tend to conceive even physical patterns as

16
trees is found in some experiments of Sir
Frederick Bartlett's. He showed people a
pattern, for about 1/4 second, and then
asked them to draw what they had seen.
Many people, unable to grasp the full
complexity of the pattern they had seen,
simplified the patterns by cutting out the
overlap in them. 18 contains two fairly
typical redrawn versions of the pattern
shown above them. In the redrawn
versions the circles are separated from the
rest; the overlap between triangles and
circles disappears.
These experiments suggest strongly
that people have an underlying tendency,
when faced by a complex organisation, to
reorganise it mentally in terms of
non-overlapping units. The complexity of
the semi-lattice is replaced by the simpler
and more easily grasped tree form.

You are no doubt wondering, by
now, what a city looks like which is a
semi-lattice, but not a tree. I must confess
that I cannot yet show you plans or
sketches. It is not enough merely to make a
demonstration of overlap - the overlap
must be the right overlap. This is doubly
important, because it is so tempting to
make plans in which overlap occurs for its
own sake. That is essentially what happens
in the high density 'life-filled' city plans of
recent years. But overlap alone does not
give structure. It can also give chaos. A
garbage can is full of overlap. To have
structure, you must have the right overlap,
and this is for us almost certainly different
from the old overlap which we observe in
historic cities. As the relationships between
functions change, so the systems which
need to overlap in order to receive these,
relationships must also change. The
recreation of old kinds of overlap will be
inappropriate, and chaotic instead of
structured.
The work of trying to understand just
what overlap the modern city requires,
and trying to put this required overlap into
physical and plastic terms, is still going on.
Until the work is complete, there is no
point in presenting facile sketches of ill
thought out structure.
However, I can perhaps make the
physical consequences of overlap more
comprehensible by means of an image. The
painting illustrated is a recent work by
Simon Nicholson, 19. The fascination of
this painting lies in the fact that although it
is constructed of rather few simple
triangular elements, these elements unite
in many different ways to

17
form the larger units of the painting. If we
make a complete inventory of the
perceived units in the painting, we find
that each triangle enters into four or five
completely different kinds of unit, none
contained in the others, yet all overlapping
in that triangle. If we number the triangles
and pick out the sets of triangles which
appear as strong visual units, we get the
semi-lattice shown in 20.
Three and 5 form a unit because they
work together as a rectangle; 2 and 4
because they form a parallelogram; 5 and 6
because they are both dark and pointing
the same way; 6 and 7 because one is the
ghost of the other shifted sideways; 4 and 7
because they are symmetrical with one
another; 4 and 6 because they form another
rectangle; 4 and 5 because they form a sort
of Z; 2 and 3 because they form a rather
thinner kind of Z; 1 and 7 because they are
at opposite corners; 1 and 2 because they
are a rectangle; 3 and 4 because they point
the same way, like 5 and 6, and form a sort
of off-centre reflection of 5 and 6; 3 and 6
because they enclose 4 and 5; 1 and 5
because they enclose 2, 3, and 4.1 have
only listed the units of two triangles. The
larger units are even more complex. The
white is more complex still, and is not even
included in the diagram, because it is
harder to be sure of its elementary pieces.
The painting is significant, not so
much because it has overlap in it (many
paintings have overlap in them), but rather
because this painting has nothing else in it
except overlap. It is only the fact of the
overlap, and the resulting multiplicity of
aspects which the forms present, that
makes the painting fascinating. It seems
almost as though the painter had made an
explicit attempt, to single out overlap as a
vital generator of structure.
All the artificial cities I have
described have the structure of a tree
rather than the semi-lattice structure of the
Nicholson painting. Yet it is the painting,
and other images like it, which must be
our vehicles for thought. And when we
wish to be precise, the semi-lattice, being
part of a large branch of modern
mathematics, is a powerful way of
exploring the structure of these images. It
is the semi-lattice we must look for, not the
tree.

When we think in terms of trees we
are trading the humanity and richness of
the living city for a conceptual simplicity
which benefits only designers, planners,
administrators and developers. Every time
a piece of a city is torn out, and a tree
made to replace the semi-lattice that was
there before, the city takes a further step
toward dissociation.
In any organised object, extreme
compartmentalisation and the dissociation
of internal elements are the first signs of
coming destruction. In a society,
dissociation is anarchy. In a person,
dissociation is the mark of schizophrenia
and impending suicide. An ominous
example of city-wide dissociation is the
separation of retired people from the rest
of urban life, caused by the growth of
desert cities for the old, like Sun City,
Arizona. This separation is only possible
under the influence of tree-like thought.
It not only takes from the young the
company of those who have lived long, but
worse, it causes the same rift inside each
individual life. As you yourself pass into
Sun City, and into old age, your ties with
your own past will be unacknowledged,
lost, and therefore broken. Your youth will
no longer be alive in your old age - the two
will be dissociated; your own life will be
cut in two.
For the human mind, the tree is the
easiest vehicle for complex thoughts. But
the city is not, cannot, and must not be a
tree. The city is a receptacle for life. If the
receptacle severs the overlap of the strands
of life within it, because it is a tree, it will
be like a bowl full of razor blades on edge,
ready to cut up whatever is entrusted to it.
In such a receptacle life will be cut to
pieces. If we make cities which are trees,
they will cut our life within to pieces.