Seminar of Spatial Information Science, 12th December 2007

Scaling, Rank Size, & Allometry in theGeometry of Large Cities
Michael Batty, Rui Carvalho, Andy Hudson-Smith, Richard Milton,Duncan Smith and Philip Steadman

University College London

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Collaborators: Rui Carvalho, Andy Hudson-Smith,Richard Milton, Duncan Smith and Philip Steadman

MikeRui

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AndyRichard

DuncanPhil

Lecture Outline• A 10 Minute Primer on Distributions and Scaling• Distribution Shapes: Heights of Skyscrapers• Growth, Evolution, Isometry and Allometry• A Massive Geometric Data Base: Virtual London• All London Buildings
Rank Size of Heights, Areas, Perimeters and Volumes• Rank-Size of Buildings by Land Use• Allometry of Areas, Volumes and Building Skins• What Next? How Big is a City? Space & Energy?
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A 5 Minute Primer on Distributions and Scaling
This talk is about how measuring cities as buildings, notpopulations or other activities. It is about geometry too.
We could argue for a long time about whether we shouldmeasure the geometry of cities as streets or buildings
but a couple of things are clear:
1. buildings are less ambiguous in terms of their definition
than streets, despite some caveats.
2. other physical characteristics of cities such as energy
use and traffic generation are much more associatedwith buildings than streets.
3. Land use (viz socio-econ activity) is tagged to buildings.
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As a prelude to several larger projects we are involved insuch as measuring the heat island effect (LUCID) inLondon and looking at climate change in the Thames
Gateway (Tyndall Cities Project), this project is aboutmeasuring the properties of the distributions of buildingsin London. I am not going to talk about applications.
To anticipate the conclusions which we already know fromcommonsense, these properties involved scaling anddistortion which are very widely studied in the statistics
of extreme distributions, in fractal geometry, in allometry.

Let us begin with some simple ideas about distributions –we will first look at the normal distribution – the so-called
Bell Shaped curve and show why it has little relevance tothe distribution of geometric objects in cities.
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The Normal Distribution

The Lognormal Distribution

frequency f(x)frequency f(x)average ormean xsize (x)You can approximate the fat
tail of the normal distribution
with a power function
The Dan Montello question?

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Themean x is not definedsize (x)inverse power function –the mean is not definedfrequency – number of adults

Let us look at the distribution of adult heights – which
follows the normal – this is not an extreme distribution

height of adults

Centre for Advanced Spatial Analysisfrequency – number of buildings

But now look at the distribution ofbuilding heights –this is an extreme
distribution with a lot of little buildings
and very few big ones

height ofbuildings

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Now many, many distributions in life are extreme – income
is an excellent example – how many Bill Gates are there
Talib says for objects or events that are distributed normally:
who have assets of $80 billion? Nick Talib has a wonderfulexamplein hisrecentbook
ThenoBlack
Swan,willandI quote.
“…. whenyoursample
is large,single instantchangetheaggregate or total”.

We can repeat these types of examples ad nausea butAddingfall
or takingout samea person
3 meters
highstreet
will notlengths
change asbuildings
into theclass.
So doanything
Rui and Alan showed. So do city sizes as many people
fromWhen
Paretoto Zipf
have shownover the last 150 years.
eventsare distributed
extremely:
Some of this is included in my book Cities and“…. inequalities
such 2005).
that one single observation canComplexity(MIT arePress,disproportionately impact the aggregate or the total”.

WhatAdding
I wantdo isyou 400
a fewof theseor totakingoutshow
a building
meterstall willproperties
change the but
first world
we need to learn a little mathematical trick to convert
our lognormal into a power function and to convert
frequency to what we call rank. Here goes.
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





Frequency transforms
from the vertical axis toRank on the horizontal

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The reverse of cumulative
frequency transforms to rankand swaps axis while theheight simply swaps axis

Height (size) moves
to the vertical fromthe horizontal axis

Height (size)

This is rank-size and theinverse power law which is
implied by this shape is therank-size rule, the termcoined by Zipf in 1949.

Rank

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Log of height (size)

And when we take logs we
get ………….

Log of rank
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So this is it. We have to rank the object in the distribution
from largest to smallest and number them from 1 to n, takelogs to get rid of the nonlinearity and then to see how wellthis distribution fits a curve shaped like a power law, weuse some statistical technique to see how well it fits
This is not explanation, It is simply seeing what shape ourdistribution is – it doesn’t explain anything – Explanation
might come from various processes that generate such
distributions and there are many of these based oncompetitive random growth all the way to elaborate models
of how demand and supply play themselves out in markets.
But to convince you that we have something interestingand perhaps surprising, let’s look at some data- on
skyscraper heights
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Distribution Shapes: Heights of Skyscrapers
There are some good data bases out there on the web andfrom the Skyscraperpage http://skyscraperpage.com/ ,there is a list of the top 200 buildings world wide by height.
There is also the Emporis data http://www.skyscrapers.com
which has a large number of top building heights for many
cities world wide as well classifying these data by land use.
There is some nice graphics too as each of these buildingson the Skyscaperpage is illustrated.

We will start with the top 200. Here is the rank size relation:

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Log of Height in meters

Log of Rank

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Correlatio n (r 2 )  0.996;
Parameter ( slope)  0.158

Log of Height in meters

New York CityTop 2424 Buildings
LondonTop 1302 Buildings
TokyoTop 1036 Buildings

Log of Rank

London (all ) : r 2  0.979; Para  0.288; (Top 300) r 2  0.990; Para  0.278;
NYC (all ) : r 2  0.919; Para  0.478; (Top 200) r 2  0.982; Para  0.170;
Tokyo (all ) : r 2  0.827; Para  0.377; (Top 444) r 2  0.970; Para  0.232;

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LondonNew York CityTokyo

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We also have the global
data for five land uses;
the top 100 buildingheights for commercial,residential and hotel
and the top 50 for healthand education. The
distributions are as
follows:
Commercial

Residential

Hotel

Health

Education

Parameter-0.175-0.144-0.207-0.163-0.260

Correlation r 20.988

0.968

0.978

0.983

0.965

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Growth, Evolution, Isometry and Allometry
We are going to look at the scaling using rank-size for fourgeometric properties that are very relevant for size effects
in cities – namely height as we have been doing but more
importantly perimeter, area, and volume (or mass) of theirassociated distributions.
Now we will expect all of these to scale: and we can evenget some sense of how consistent the parameters are foreach geometric attribute such as these are related in termsof Euclidean geometry: i.e. area is perimeter squared – the2square of the perimeter of a circle 2 r is proportional to  r
and so on, but the key to all this how buildings get distorted
as they get bigger, and they do, let me assure you.
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As the volume of a building grows, then the area needed tofeed it light – i.e. its surface area does not grow in thesame ratio; it can’t because strictly if it is a solid block, its
volume grows as the cube of its linear dimension but its
surface area only grows as its square; so area grows to thepower of 2/3 the volume. Not enough.
This is best seen in animals. As we grow bigger, we needmore surface area because that is the thing that feeds us,
and if you look at our lungs and our guts, they are all foldedin terms of getting food from the air to support our mass.
Compare the lungs and gut of a mouse (or a baby) with aman. This business of increasing the surface area as theobject grows at more than the geometric rate of increase iscalled allometry. And to do this, always requires a changein shape. This is the key – shape:
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This is Phil Steadman territory: read his book The Evolutionof Designs (CUP, 1978) and reissued by Routledge – soon!
A nice example for Alasdair and Ozlem as they will
experience this change in shape almost daily at present is …

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Ok. We need to get some formal bits out of the way and thenwe can turn in earnest to our London data with all its millionsof pieces and really see what the shape of the city is about.
If the surface area of a building were to increase at the samerate as its volume, then this would be isometry. It doesn’t as
in simple Euclidean space it increase as 2/3rds the volume,and this is a form of allometry. But it needs to increase
faster so we might expect the surface area to increase fasterthan 2/3rds the power of volume. Indeed it does, Ranko Bon
found it did in 1972; Phil did in his work with Frank Brown –Frank is here somewhere – and Phil has also demonstrated
it in his archetypal building project.
Buildings achieve this through courtyards, skylights and soon. Here are the equations!
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We say that the rate of change of one attribute of an object
has the same fixed ratio of growth as another and if this is
the case, the growth is allometric:
dy
 y
dtdy

dt y
dy
dt yandanddx
 x
dtdx

dt x

dx

dt x dy x
dx y 
Thereforey  Kx

which simplifies to the key allometric relationy  Kx 

We can apply this to all attributes: for perimeter, area, heightand volume and all interrelations between them. Here we go:
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A Massive Geometric Data Base: Virtual London
The data base is composed of OS Master Map landparcels/footprints to 1 meter accuracy of which there aresome 3,595,689 (or ~ 3.6 million) for the 1579 square
kilometers that comprise the 33 boroughs that make up
Greater London.
The OS data in layer 2 has about 1000 land use types
coded to address points or TOIDS. Note that some 42% ofbuilding footprints are unclassified – do not have an addresspoint. We have used a ‘nearest spatial join’ method tomerge these into nearest addressable parcels and wereckon that after some dissolving of adjacent boundaries too,some 2.5% of footprints are misclassified.
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The vertical height data is taken from InfoTerra LIDAR data
which is to 1 meter accuracy. We can therefore computecourts, light-wells and many other detailed geometries. We
have not yet done so simply using average – mean – heightas the appropriate measure.
So from this data base we compute perimeter and area
which are simple ArcGIS functions for each of the buildingparcels and by applying average height to area we computevolume: we define the following:
perimeter p; area a: surface area or skin s; height h; andvolume v. We have not yet defined surface area or skin but
will do so at the next stage. Let us first look at some data:

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Education
Mixed
RetailResidential

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Would you like to see some of the data in 3D?

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All London Buildings
Rank Size of Heights, Areas, Perimeters and Volumes
The rank-size relations are first estimated as:

h j (r )  K 2 r

0.223p j (r )  K1r

0.432a j ( r )  K 3r
v j (r )  K 4 r

0.684
0.799I can’t find the correlations – we have done them but theZipf Plots are as follows showing remarkable scaling:
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Rank-Size of Buildings by Land Use
We will quickly show the land use classified into 8 types
residential : 3320579 (92.34%)
retail : 77075 (2.14%)
service : 33949.000000 (0.94%)
industrial : 67270.000000 (1.87%)
educational : 16257.000000 (0.45%)
hotel : 3936.000000 (0.11%)
transport : 1002.000000 (0.03%)
gencomm : 122874.000000 (3.41%)
And we show the rank size relations as follows where wehave collapsed the plots onto one another by normalising.
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Allometry of Areas, Volumes & Building Skins
There are many many results we are generating and suchremarkable scaling over many orders of magnitude isextremely encouraging for our proposed work on deriving
the energy associated with building use. Note also the factthat in the London case there is barely any evidence oflognormality in the distributions.
The allometry is less regular as we would expect for there
is massive variation with respect to building size and shapeand as yet we have not controlled for the detail that wewould need for accurate work on relating building surfaces
to volume. We show they key relations between volume,
perimeter and area as follows:
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a j ~ p 2j  a j ~ p1j.833
v j ~ p 3j  v j ~ p 2j .386
v j ~ a 3j / 2  v j ~ a1j.296

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
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
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servicesCentre for Advanced Spatial Analysis

GeneralCommerce

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Industrial

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Office

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Residential

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RetailingCentre for Advanced Spatial Analysis

Bringing Space Back In
The two point correlation function which computes thecorrelation between any two buildings in the system withrespect to their distance from each other. If this is highthen it means that there is a power law relating todistance with respect to the property being measured, e.g.
height or volume.
We will do this for the top 100, 000 buildings by height. Herethey areCentre for Advanced Spatial Analysis

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What Next? How Big is a City? Space & Energy?
A long presentation does not warrant a long conclusion. Letme list six key issues:
1. Better measurements of all geometric attributes, with aview to generating building skins and defining floor areasfrom storey heights.
2. Generating much more relevant allometries for different
building types and for land uses
3. Embedding physical data about streets from OS Mastermap street centre line data into this nexus. We havealready looked at the 65000 street centre lines for themain streets in the data and found the usual scaling: hereis the data.
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4. Linking socio-economic data to the geometric data: asking
the questions how much built form per capita
5. Doing the same analysis for NYC, Tokyo and other places
where virtual city models exist. We can get the data from
Kyoto easily enough and probably Tokyo from the CAD
centre there with whom we have good relations; maybe
from Michael Kwartler in NYC too.
6. Generating a more complete theory of urban allometry
linking this to cities at different scales – already we see
big differences in the cities in the Skyscrapers data
7. Linking this kind of geometric scaling to Geoff West andhis SFI group’s work on urban infrastructure scaling …..
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I would be pleased to answer any questions andrespond to any comments.

Questions?
http://www.casa.ucl.ac.uk/CASA Working Paper 126
Scaling and Allometry in the Building Geometries of Greater London
Michael Batty, Rui Carvalho, Andrew Hudson-Smith, Richard Milton, Duncan Smith, Philip Steadman

http://www.casa.ucl.ac.uk/publications/workingPaperDetail.asp?ID=126

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