Christaller’s central place theory is as much famous as it was controversial for a long time. At present, it is largely agreed that Christaller’s system is too rigid to have any chance of representing reality, but the theory is an outstanding creation, offering great insights about capability of economic factors to shape spatial systems, without ever simulating them in any detailed respect. At the bases of this judgment, there is the tacit axiom that spatial structures are ordered (namely, preferential sizes of centres and preferential spacing among these centres), even if this order is intrinsically complex and highly perturbed by specific local peculiarities, so that it is almost impossible to trace back it to the economic causing factors. Against this viewpoint, firstly, in a perspective of spatial systems as very complex disorganized processes (so quasi random structures), we show the incompatibility of Central Place Theory with the Power Law (free scale distribution of urban sizes), a very well experimentally tested rule: in this context, we point out a logical fault in Beckmann’s theorem assessing the compatibility. Second, as far as spacing among centres is concerned, we evidence no signs of difference from a random spatial distribution of centres (power law distributed), in a number of case-studies including the case of southern Germany in the early 1930s, the Christaller’s original one. We support the thesis that we are facing a “pareidolia” (illusion of order), i.e. apparent spatial schemes (such as embedded hexagons or other regular geometries) seemingly emergent from the mere mean distance of centres of different size in a totally random configuration. Being unquestioned the greatness of the German geographer (and the importance of the contributions of a number of scholar in his footsteps) the conclusion is that today the presentation (e.g. in teaching) of central place theory must be put in its historical context, leaving space to a description of human (individuals and organizations) behaviours (extremely diversified, often far from economic rationality, etc.) and theirs multiple “complex” interactions (among them and with the physical, technical, natural environments) leading to random (or almost random) spatial settlements, as far as their size and spacing is concerned.

1. NO HEXAGONS. THE CENTRAL PLACE THEORY ILLUSION REVEALED
Matteo CAGLIONI UMR ESPACE, University of Nice Sophia Antipolis
Giovanni RABINO DAStU, Polytechnic of Milan
European Colloquium of Theoretical & Quantitative Geography September 5-9th 2013, Dourdan, France

2. INTRODUCTION
• The Central Place Theory (or presumed theory), together with
the power law and the spatial interaction, is considered as
one of the fundamental laws of geography.
“ Christaller’s system is too rigid to have any chance of
representing reality, but…
the theory is an outstanding creation, offering great insights
about capability of economic factors to shape spatial systems,
without ever simulating them in any detailed respect ”
A. Wilson

3. INTRODUCTION
• Underneath this Wilson’s consideration there are 2 tacit
assumptions:
1. Spatial Structures of city systems are substantially ordered.
2. Classical economic theories are able to explain this order.

4. INTRODUCTION
In this work we want to show that those assumptions should be
rejected:
• On the theoretical side
 CPT is not compatible with the power law
 Critics from Science of Complexity and Neuroeconomics
• On the operational side
 Empirical spatial structures show no order in hierarchy and distances
 Power law is well suited to show a phenomenon called pareidolia

5. POWER LAW with negative exponent
Power
Law
80 / 20 rule
Lorenz
Pareto’s
distribution
Lotka’s law
Noise 1 / f
Rank-size
rule
Zipf’s law
Fractal law

6. POWER LAW in geography
• Auerbach (1913): Pr = C r -1
• Zipf (1941): Pr = C r -β
• Pareto (1896): F(P) = rP = C P -1/β
• Power Law: f(P) = C P -(1+1/β)
 For β = 1 all of them are the same.
 Integral of a power law is still a power law.

7. POWER LAW in geography
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
200 1.400 2.600 3.800 5.000 6.200 7.400 8.600 9.800 11.000 12.200
popolazione
frequenza
Popolazione 2001 PowerLaw LogNormale
Population sample (Italy 2001) follows a Log-Normal distribution with power law tail

8. BECKMANN’S MODEL
• Several authors referred to models of Beckmann (1958),
Beckmann-McPherson (1970) and Parr (1969) in order to
explain relation between city hierarchy and Zipf’s law
(empirically verified).
• Can Beckmann’s model validate the Central Place Theory?

9. y = 3E+06x
-1,062
R
2
= 0,9882
1.000
10.000
100.000
1.000.000
10.000.000
1 10 100 1.000 10.000
rango
popolazione
BECKMANN’S MODEL
• We cannot observe mean values of city size (neither classes)
• It is rare to find two cities with the same size
• Christaller and Zipf are antithetic (degeneration)
It seems we have a
perfect correspondence
between a rank-size rule
and Christaller’s model
but…
rank
population

10. SPATIAL HIERARCHY
• Geographers are fascinated by spatial hierarchy.
• Christaller (1933), Lösch (1940), Isard (1956), Skinner (1964)
reported spatial hierarchies formed by systems of cities.
• Reality = deformation of theoretical model.
• Is a spatial hierarchy just an illusion ?
It is possible to find Christaller’s spatial hierarchy in a random
configuration (Okabe & Sadahiro, 1994).

11. DISTANCE ANALYSIS
• SHP files from ISTAT, INSEE, EUROSTAT
• For each municipality we considered its barycentre
• Point Pattern Analysis – Nearest Neighbour Distance

12. DISTANCE ANALYSIS

13. DISTANCE ANALYSIS
8.101 Italian Municipalities in 2001 – Real and Simulated

14. DISTANCE ANALYSIS
36.565 French Municipalities in 1999 – Real and Simulated

15. DISTANCE ANALYSIS
23.011 French Agglomerations in 1999 – Real and Simulated

16. DISTANCE ANALYSIS
0,00
0,05
0,10
0,15
0,20
0,25
0 10 20 30 40 50 60 70 80 90 100 110
distanza [km]
frequenza
P-P MP-MP M-M MG-MG G-G
P-P sim MP-MP sim M-M sim MG-MG sim G-G sim
P-P Chr MP-MP Chr M-M Chr MG-MG Chr G-G Chr

17. PAREIDOLIA
• The natural and instinctive tendency of men in willing to
recognise ordered patterns, also where they does not exist,
and finding familiar forms in vague or random stimulus (e.g.
images, sounds, …) is called pareidolia*.
* It looks like, but it is not.

18. PAREIDOLIA in geography
• Continuous search of ordered patterns in the settlements
location, for us, it is a typical case of pareidolia: we can find a
order just because we believe in its existence, also if it does
not exist at all.
• An example in the classical location theory is Christaller’s
model of the central places, which organise cities in an
extremely ordered way on the territory.

19. PAREIDOLIA in geography
Constant distribution of population

20. PAREIDOLIA in geography
Linear distribution of population

21. PAREIDOLIA in geography
Power Law distribution of population

22. PAREIDOLIA in geography
Power Law distribution of population

23. PAREIDOLIA in geography
Thiessen’s polygons with 10 and 7 main cities.

24. PAREIDOLIA
• “Our brains misperceive evenness as random,
and wrongly assume that groupings are deliberate.”
Charlie Eppes
NUMB3RS 3x05

25. A.
C.
B.

27. CONCLUSIONS
• Spatial structures are not random… they are pseudo-random.
• Christaller’s and CPT’s contribution has been really important
in development of a geographical theoretic and quantitative
thought.
• Nowadays we need to recognise its erroneousness and teach
to young people how to recognise spatial structure taking into
account the spatial complexity and the new economical
theories.

28. THANKS FOR YOUR ATTENTION!
Matteo CAGLIONI matteo.caglioni@unice.fr
Giovanni RABINO giovanni.rabino@polimi.it
European Colloquium of Theoretical & Quantitative Geography September 5-9th 2013, Dourdan, France

29. POWER LAW DISTRIBUTION
• A continuous stochastic variable X is distributed like a power
law with negative exponent, if it presents this probability
density function:
fX(x) = C x - a
• Frequency of an event x is inversely proportional to his
dimension.

30. CENTRAL PLACE THEORY
• Marketing principle
• Transport/Traffic principle
• Administrative principle

31. CENTRAL PLACE THEORY
Località Popolazione Numero Descrizione
R Reichshauptstadt 4.000.000 1 località internazionali
RT Reichsteil 1.000.000 2 località parte dello stato
L Landeszentrale 500.000 6 capoluoghi di regione
P Provinzialhauptorte 100.000 18 capoluoghi di province più ampie
G Gaubezirk 30.000 54 capoluoghi di provincia
B Bezirkshauptorte 10.000 162 capoluoghi di distretti
K Kreisstädtchen 4.000 486 capoluoghi
A Amtsstädtchen 2.000 1.458 città amministrative
M Marktorte 1.200 4.374 borgo-mercato
H Hilfszentrale Orte 800 13.122 località centrali ausiliarie
• The number of cities of each class is an arithmetic progression
• Centrality / Importance expressed in term of population