ENERGY BASIS FOR HIERARCHIES IN URBAN
AND REGIONAL SYSTEMS1
Mark T. Brown2
Abstract.-Understanding the hierarchical patterns of
energy flow in landscapes is a major objective in the scien-
ces of environment and human settlement. Data on regional
and national patterns of landscape organization are used to
test theories of energy flow control of hierarchies. Simu-
lation models are developed to quantitatively relate ideas
of mechanism and energetic to hierarchical structure,
spatial pattern, and spectral distribution observed in sys-
tems of humanity and nature.
Complex systems such as ecosystems, indus-
trial processes, and networks of cities in the
landscape appear to be organized in webs of ener-
gy flow with multiple levels of components (fig.
la). These may be visualized in simplified form
with diagrams as in figure lb. The patterns have
spatial manifestations, with many small units
converging energy to a few larger ones.
Theories developed to account for these
hierarchical patterns may be based, in part, on
the theory that systems compete for power and
survive by developing a structure of energy flows
that maximizes useful power. The maximum power
principle was enunciated by Lotka (1922) and
additional corollaries were proposed by Odum
(1967, 1971, 1975) and Odum and Odum (1976). The
type and form of web adapting to different com-
binations of energy from the environment produces
different spectral distributions and spatial
patterns, which may be predictable from simple
An Energy Basis for Hierarchies
Given in figures la and lb are simplified
energy circuit models (Odum 1971) that depict
paper presented at the international sym-
posium Energy and Ecological Modelling, sponsored
by the International Society for Ecological Mod-
elling. (Louisville, Kentucky, April 20-23, 1981.)
Mark T. Brown is Assistant Research Scien-
tist at the Center for Wetlands and Adjunct
Assistant Professor, Department of Urban and Re-
gioQal Plapning, University of Florida, Gaines-
ville, Florida U.S.A.
energy flow and storage in hierarchically organized
systems. These diagrams show energy flow and con-
trol action feedbacks in five-compartment (level)
hierarchies and are the basic configuration for the
organization of data in this investigation of an
energy theory of hierarchically organized systems.
The following concepts and theories about the
relationship of energy, its spatial distribution,
and resulting hierarchies are postulates and are
the basis for examining data on systems of Florida
and the nation.
Systems operate under the constraints of the
First and Second Laws of Thermodynamics and Lotka's
Maximum Power Principle (Lotka 1922) and corrollar-
ies as proposed by Odum (1975) and Odum and Odum
(1976); and are organized in a manner to remain
competitive and stable, increasing inflowing energy
when excess energy is available.
Energy Quality and Embodied Energy
Odum (1976, 1977, 1978a, 1978b) and Odum and
Odum (1976) suggest that there is a quality to
energy, which is a measure of its ability to do
work. Quality of energy is related to the degree
to which it is concentrated; with dilute energies
like sunlight, winds, waves, and other natural
energies having lower quality than the more concen-
trated energies of fossil fuels.
Energy Quality and Frequency of Energy Sources
Recently, Odum (1981) and, in earlier studies
of the cycles of order and disorder, Alexander
(1978) have suggested that the quality of an energy
is related to its frequency in the time domain.
Others (see Simon 1973) have suggested that fre-
quency and place in hierarchy are related to the
extent that high frequency is associated with low
place in hierarchical order and low frequency
with high hierarchical place.
Energy Quality and Power Density
One measure of the intensity of energy util-
ization in the landscape is power density (Odum,
Brown, and Costanza 1976), or the rate of energy
flow per unit area (Cal/acre'year). In this
manner, the energy intensity of one area can be
compared on a relative scale with others. In
urban systems, power density is considered to be
the rate of embodied energy consumption per unit
area and in natural ecological systems of the
landscape, power density is the rate at which
energy is fixed, as measured by gross primary
Previous Studies of hierarclies
Many theories for hierarchical organization
of systems and the resulting dlstrlbutorns of
components, dating from antiquity, are prevalent
in the literature. As the amount of published
scientific literature has grown exponentially in
the last 80 years, so has the number of scien-
tists using the concept of hierarchy In the anal-
ysis of physical, biological, and social systems.
Most notable in recent years are Woodger (1929,
1937), Whyte (1949, 1969), von Bertalanffy (1933,
1968), Simon (1962, 1973), Wilson (1969), Bunge
(1969), Weiss (1971), and Laszlo (1972, 1973).
Hierarchy in social systems is investigated by
numerous authors (i.e., Aldrich 1979; Blau 1972,
1974; Burgess and Park 1924; Emery and Triat
1973; Classman 1973; Landau 1969; Honane 1967;
Thompson 1967; Weber 1946, 1947).
The hierarchy associated with the landscape
of cities in regions was first enunciated in 1933
by Christaller (1966) and further developed by
Losch (1954). Many authors have applied central
place theory to regions and developed the theory
further (see Dokmeci 1973; Henderson 1972; Purver
1975). Others have been critical, finding at
least four specific weaknesses (see Beckman 1955;
Henderson 1972; Tinbergen 1968; Van Boventer
1969). A number of authors have used gravity
models and equations of diffusion for allocating
regional influence of centers and calculating the
spread of innovation (see Beckman 1956, 1958, 1970; .
Berry 1972; Hagerstrand 1966; Isard and Peck
1954; Mansfield 1963).
Zipf (1941) and later Steward (1947) have
suggested there is a mathematical relationship
between rank of cities and population size. Mac-
Arthur (1957) has suggested rank-abundance curves
for the study of the structure of animal comuni-
ties, and Odum, Cantlon, and Kornicker (1960)
have postulated a hierarchical organization of
ecological communities using a cumulative loga-
rithmic species-diversity index.
Few previous studies of hierarchy have dealt
with the energy control of landscape organization,
but many with economic aspects and some with the
physical constraints of hierarchical organization.
SMICREASING QUALITY OF ENERGY
EAsN Q1T Of ENERGY
FIgure 1. The web of energy flow and compartments of complus
systems. Shobn above is a hypothetical energy web, and
below a stIpllfled form organized as a hierarchy.
Plan of Study 4", ;'
In this study the hierarchical organization '
of the landscape and resulting energy spectrum of
energy storage and flow were investigated at thf :
levels: the regional level of ecosystems aqnl4
urban land uses, the organization of cities in the
landscape, and the organization of land uses
within cities. In addition, relationship ofj .
intensity of development to the spatial srs 6f .
influence was investigated at different Ivelsl. .'
organization in the nation, the state, and ) b ';-, i
districts of the state. The specific plan, Ol tb* '
analysis of regions, districts, and subdi d~ ttt
is as follows.
IMCCREAS*4 W OF COMPONETS
DECREASING kUMSE OF COlamONNMI
First, energy spectra for many different
types and sizes of systems were constructed to
understand general trends of energy flow and
storage in observed hierarchies.
Second, maps were made of ecosystems and land
uses at two levels of study; the regions of Flor-
ida and cities within these regions.
Third, generalized models of each urban land
use type were evaluated and spectra and energy
storage and energy flow were calculated.
Fourth, specific analysis of the external
energy requirements of areas of different sizes
and an energetic evaluation of cities of different
sizes were conducted.
And fifth, a series of theoretical models of
hierarchical organization were simulated on analog
and digital.computers to explore different energy
flow and storage characteristics under different
organizations and pathway configurations. And
then data from an aquatic ecosystem of Florida
were used to test theories of hierarchical distri-
bution and energy control.
General Methods and Definitions
A graphic language is used throughout this
paper to describe energy flow and interaction in
complex systems. The language is a graphic means
of depicting systems as Nth order differential
equations, since each symbol represents a mathe-
matical relationship of either energy flow, inter-
action, or storage relative to time. For a com-
plete description of the language and its develop-
ment see Odum (1960, 1967, 1971, 1973, 1976a).
Evaluation of Observed Hierarchies
The trends of hierarchical organization and
energy spectra were graphed semilogarithmically
for systems of differing scales and complexity.
Data were gathered from various sources in liter-
ature and from various local, state, and federal
agencies in published reports and in some cases
Land Use Maps of Region and Cities
Three regional areas of differing character
and size were analyzed for total energy budgets,
land use, and resulting hierarchic organization:
the Kissimmee Everglades Basin in south Florida,
a subtropical region of relatively intense urban
development; the St. Johns River Basin, a region
on the coast of central Florida dominated by a
major river and agricultural lands with moderate
urban development; and Lee County, Florida, an
area in southwest Florida that is a coastal
county with extensive tourism and an agriculture
o s lom .**
Figure 2. Map of Florlds shoving three regional study areas and
two urban study area.
based inland. The county has experienced recent
very rapid growth (fig. 2).
Two urban areas were analyzed in detail for
total energy budgets and power densities, energy
budgets and power densities of land uses within
the urban areas, and resulting hierarchic organi-
zation of the urban landscape: Ft. Myers, Flor-
ida, an urban area on the southwestern coast of
Florida whose main economic inputs are from tour-
ism, and is a government center; and Cainesville,
Florida, an urban area in central Florida that is
a governmental center and center of university
education (fig. 2).
The average direct power density for each of
the land use classifications was calculated by
first selecting a representative sample of struc-
tures in each of the land use types (approxi-
mately 10% sample size), and then obtaining
yearly energy consumption data from local utility
records for each of the selected structures.
Indirect power density was calculated for
the land use categories in the Ft. Myers study
area only. A detailed model of energy flow
between the main sectors of the local economy for
1973 was evaluated (Brown 1980) to obtain the
energy embodied in goods and services that were
consumed in the residential, commercial, indus-
trial, and construction sectors of the economy.
Evaluation of the flows of dollars among sectors
of the economy was used and converted to embodied
energies using a conversion factor of 21,000 Cal
The structure associated with each land use
was determined from property tax records of both
cities where total area of structure for each of
the sample structures was used. Volume of struc-
ture was calculated by multiplying area by aver-
age height of buildings.
Land Use Maps
Land use maps were constructed from false
color infrared and black and white infrared photo-
graphs taken in 1973 and 1974. The land use map
for the south Florida region was prepared during
previous studies (Odum and Brown 1976). Areas of
each land use were determined by cutting different
land uses from the map and weighing them on an
analytical balance. A conversion factor of grams/
acre was used to convert from weight to area.
Urban Land Use Power Densities
Power density is a measure of energy flow per
unit of time per unit of area. In this study,
power density if expressed in units of Cal CE/acre
*year. Power density is expressed as the addi-
tion of energy consumption of fuels and electri-
city per unit area (referred to as direct power
density), and consumption of the energy embodied
in goods and services per unit area (referred to
as indirect power density). Total power density
results from the addition of both of these types
of input energies.
Classification of Cities
by Average Power Density
An average city power density was determined
for all cities within each region by using aver-
ages derived in detailed studies of the two urban
areas, Gainesville and Ft. Myers, Florida. The
area of each city is not necessarily the actual
area within legal city limits, rather it is the
area that when viewed from aerial photographs is
within the major concentration of urban land uses.
In some cases this area may be smaller than actual
city limits and in other cases, where suburban
sprawl is evident, the area may be considerably
Development Density and Imports/Exports
One measure of production and consumption in
regional systems is gross domestic product (CDP)
as determined from the total flow of dollars with-
in a regional economy. While domestic product is
not always available for regions, it may be deter-
mined from employment data and averages for
productivity per employee in each economic
CDP for counties in Florida, states, and
nations were determined from employment data, and
"development density" was calculated by dividing
CDP by land area of each county.
Export multipliers for each county and state
were determined in the following manner: employ-
ment data for eight broad economic sectors (agri-
culture, manufacturing, wholesale and retail
trade, government, services, transportation and
public utilities, banking and finance, and con-
struction) were obtained and compared to the same
data from the U.S. economy. Positive departures
from the U.S. percent employment were considered
to indicate that portion of each economic sector
that was export employment (for a detailed dis-
cussion of location quotients and methods see
Exports were determined by multiplying num-
ber of export employees in each economic sector
by the productivity per employee for that sector.
It was assumed that local differences In employee
productivity were negligible.
GDP and exports for 21 selected countries
were obtained from United Nations (1978). Calcu-
lations of export multipliers and GDP were not
necessary since published data are available
Simulation Models of Hierarchical
Organization and Energy Spectra
A series of theoretical models were simula-
ted on both digital and analog computers to test
hypothesis and evaluate structure and character-
istic properties of systems organized in hier-
archical fashion. As the models grew in complex-
ity and insight gained, a final model, which was
a synthesis of previous models, was simulated
using data from Fontaine (1918) for an aquatic
Models were drawn using the energy circuitL., ,:-
language, and computer programs were written
directly from the graphic model. The facilities" '.
of the Northeast Regional Data Center on the car-
pus of the University of Florida were used for
digital computer simulation, and DYNAMO simula- '
tion language (Pugh 1970) was used. Some mdels .
were simulated on an EAl Hinlac analog computer.
The simulation models had one thing common to ,
all-each is a chain of five autocatalytic com- -
ponents connected in series. Differences in the.
successive models are in the kinetics of the com-
nections between components; with the first md --
els having simple linear flows between compl-',-, A
cents, and later models being more complex. "", _.
Similarities of Differing '
Systems and Scales .
Empirical evidence of hierarchica tm.
large-scale, complex systems of the landsscai .
presented as energy spectra in graph form, eb it -
the number of units in each level of bte hi,--.:..
archy is graphed on the vertical axis, andm
power per unit (or power density per unit)' li -
S.... -"r "
graphed on the horizontal axis. The energy spec-
tra presented here are a few examples of the many
Figure 3 is an energy spectrum of cities in
Florida. Zipf (1941), using population and rank
of cities, described a frequency distribution that
existed for cities in the United States and other
countries. He empirically reasoned that all rank-
Figure 3. (a) Energy spectrum of cities In Florida graphed sel-
logarlthmlcally, showing the trend of frequent occurrence
of laow power cities and less frequent occurrence of the
very high power cities In the landscape. (b) Log-log plot
of the energy spectrum of cities In Florida, after Zlpf
(1941). showing a negative slope of approaliately I.
Notes to Fig. 3.
Oats on population of Incorporated cities of the State of
Florida are front the Bureau of Econonc end Buslness
Research, Unlversity of Florida (1978). The distribution
of city power density was done graphically, where cities
wlth similar power densities were grouped together and
assigned a weighted average power density.
INFLOWIN ENERGY (xlO'Co CE/oacryr)
Figure 4. Energy spectrum of embodIed energy In natural land
production and urban land use power densities In the south
Florida area, showing Ith spatial character of production
of both natural and urban lends.
size distributions of cities should be a straight
line when plotted as a log-log graph. Countries
not exhibiting a straight log-log distribution
were suffering from some form of disunity, and
would tend toward unity (as described by his
straight line log-log plot) if the forces causing
disunity were removed. In a growing region like
Florida, the forces of population growth may well
account for the departure from a straight line
distribution shown in figure 3a. The.departure
from the ideal distribution that occurs with the
smallest cities may be a function of data, since
many small cities are not incorporated and there-
fore not included in the statistical census of
The composite energy spectra for south Flor-
ida in'figure 4 is a graph of the power density of
all input energies versus the number of acres of
that particular power density. The graph is based
on the spatial distribution of incoming energy,
both-natural energies and fossil fuel derived
The energy spectrum in figure 5 shows the
energy chain of increasing quality of energy as it
flows from natural lands through the urban areas
of the St. Johns River Basin and the Kissimmee-
Everglades Basin. This graph depicts the spatial
character of energy chains, as there are large
areas of low-energy lands to concentrate dilute
low-quality energies and pass them up to the
smaller yet very high-energy lands of the cities.
The Landscape of Cities within Regions
When an average power density was calculated
for cities, there was a tendency for the cities to
fall into the five classes of cities listed in
100 "--- ST. JOHNS
\0 NATURAL LANDS
a AGRICULTURAL LANDS
a COMMERCIAL a INDUSTRIAL LANDS
0 POWER PLANTS
o 8o 1600 2400 200 4000
POWER DENSITY (K ICalCE/alemp)
Figure E. Energy spectrum of Incoing energies to south Flor-
Ida, showing the spatial distribution of both renewable
and fossil fuel sources. All energies are expressed In
Cal CE. Data on Incominlg natural energies (sunlight,
rain, winds, tides, end waves) are from Costanza (1975).
Data on Incoming fossil fuel derived energies are from
Odum and Brown (1976). The graph depicts the spatial
character of Inflowlng energies. Because of the nature of
the landscape and the Inflowing energies, there are areas
of concentration and areas of relatively sparse Inflowing
natural energy. Fossil fuel energies are somewhat point
sources having very small areas of concentration. There-
fore, the acreage of end use as used as the spatial
Figure 6. Generalized map of Jacksonville, Florida, a class I
city; showing the three land use categories: commercial
(black area) Industrial (cross-hatched area), and residen-
tial (stippled area).
table 1. Because of the large area of wetland
ecosystems in the south Florida study area, the
majority of cities of all sizes were sampled from
the St. Johns area. It was felt that due to the
extent of this wetlands coverage, and that devel-
opable land is confined to a very narrow strip
along both coasts, the south Florida region repre-
sents a unique situation from a physical stand-
point, and this inhibits the development of a
complete array of city sizes.
Shown in figure 6 is an example of a class I
city (Jacksonville, Florida). Three types of
urban land use are indicated and were used to cal-
culate average embodied energy power densities.
The percentage of each land use for each of
the city types were compared and are given in
table 1. The extent that a city serves as a ceo-
tral place is indicated by the data as the per-
cent of industrial and commercial land use
increases. Thus, the class 1 cities (which
includes Jacksonville and Miami) have a higher
percentage of total land area in commercial and --
industrial uses than the other classes. '
The areas of land use in each of the cate-
gories of commercial, industrial, and other u .ss
are listed in table 2 with corresponding embodied
energy power density. The percent of the total _
Table I. Urban land uses and population for 5 classes of cities In Florida.
Mean Area Mean Area Mean Area
Number Mean Total Commrcial Industrial All Other Mean
City Class of Cities Aree acres ) Uses (acres) Uses (acres) Ues (acres) Population
Class I I 988813.7 8003.9 8142.2 82667.6 504,265
Class 2 2 62677.5 4024.2 3521.1 55332.2 99,006
Class 3 5 13714.4 771.1 519.7 12423.6 37,177
Class 4 21 4384.1 244.9 151.6 4187.5 12,957
Class 5 116 692.9 29.7 9.6 653.6 1,754
Table 2. Areas of urban land use, bodied energy power density, and Ittal
enbodled energy flo for five classes of cities in Florida.
Commercial Lend Use Industrial Land Use All Other Land Uses
Average Embodied Average Embodled Average Ebodied Total
Energy Power Energy Power Energy Power Embodied
Density Density 0Dnsity Energy Flow
Area (x 109 Cal Area (x 109 Cal Area (x 109 Cal (x 10 l al
City Class (acres) CE/acr yr) (acres) CE/acre yr) (acres) CE/acre yr) CE/yr)
Class I 8004 11.1 8142 4.4 82668 0.7 182.4
Class 2 4024 8.7 3521 4.4 55332 0.7 89.0
Class 3 771 7.5 519 4.4 12423 0.7 16.7
Class 4 245 5.4 152 4.4 4188 0.7 4.9
Class 5 30 3.9 to 4.4 654 0.7 0.6
land area of each of the uses is also listed, and
when compared for each class of city, indicates
the extent that each city type serves as a cen-
tral place. The percent of land use in commer-
cial and industrial uses is highest for class I
cities and decreases with each class.
The Flows of Energy in a Regional
Hierarchy: Lee County
The flows of energy through, and the stor-
ages of energy within a regional landscape, while
somewhat web-like in their organization, can be
grouped by quality of energy and a hierarchy
emerges. Given in figure 7 is an energy model of
Lee County, Florida, organized as a regional
Figure 7 Is a "heat energy" diagram, where
all flows and storage of energy are evaluated in
their chemical potential energy, or heat energy
The regional hierarchy has two energy sour-
ces inflowing from the outside. The first is
natural renewable energies that are the sum of
all natural energies inflowing, including: sun-
light, chemical potential energy associated with
the purity of rainwater, potential energy asso-
ciated with runoff of rains due to their eleva-
tion as they flow to sea level, potential energy
associated with winds, potential energy in waves
at the coastal margins, and the potential energy
of tides over the estuarine areas. The second is
the sum of fossil fuel energies inflowing and
The renewable energies are cascaded through
the regional economy and embodied in natural
structure, agricultural structure, and urban
structure, directly and indirectly into the higher
quality components of governmental and educational
structure and humans. Some of this embodied nat-
ural energy is exported in locally harvested and
The inflowing fossil fuel energies and goods
are the primary sources of energy for the urban
structure and higher quality components. Much of
this energy outflows as used heat energy, but is
likewise embodied in the structure of the regional
Figure 7. Energy circuit model of Lee County, Florida, organized as a
regional hierarchy of components. Numbers are flows and storage of
ctual energy (heat equivalents). See notes for details of calcula-
Sometimes referred to as a "first law dia-
gram," because the inflowing energies equal the
degraded energies or heat losses from the system,
figure 7 shows the sum of inputs equal to the out-
flows for each component as well as for the
regional system as a whole.
Flows of energy decrease from right to left
as more and more energy is dissipated as dispersed
heat from components. In general, there are five
orders of magnitude difference in energy flows
from the inflows of natural energy to those of the
feedbacks of human work; supporting the notion
that there is a constant percent decrease from one
component to the next in hierarchically organized
When "heat" energies are converted to embod-
ied energy Calories of coal equivalent, the values
in figure 8 result. Figure 8 is an embodied
energy diagram of the regional hierarchy, thus
there is no energy outflowing as dispersed heat,
but it is embodied in the next level components as
energy is "concentrated" through the system. Com-
parison of figure 7 with figure 8 shows the very
large flows of energy from natural sources, when
expressed in coal equivalent Calories of embodied
energy, as having nearly the same order of magni-
tude as those of fossil fuel sources.
Development Density and Imports/Exports
Development density (GDP/sq mi) was evaluated
for various counties in the State of Florida, var-
ious states in the nation, and various countries. '
Then exports are evaluated using an export multi-
plier method for counties and states, and exports
for countries were obtained from the literature
directly. Development density was related to .:
exports in a series of nomographs for the three ,
different sized regions and are summarized in
The nomograph in figure 9a is a log-log -lgt o
of development density versus export for the ca ,..'
bined data from counties, states, and countries. '
Assuming a linear relationship between developmt'
density and exports and plotting an arithmetic'
scale gives the graphs in figure 9b. Statistlc
analysis using least squares regression gives tI*h
following equations for each set of data (R .' ..
Counties: Ex 0.21 x Dev. + 5.75
States: Ex 0.13 x Dev. 16.04
Countries: Ex 0.286 x Dev. 1.86.-
where, Ex Exports/sq mi, and
Dev. Development density (GDP/Pq i ,
Flgur 8. Energy circuit model of Lee County, Florida, evaluated In coal
equivalent Calories of embodled energy.
When a regression equation is fitted to the com-
bined set of data, the following equation is
given for the line: (R 0.83)
Combined Data: Ex 0.968 x Dev. + 23.84. (4)
Energy Flow and Structure in Urban Systems
The energy flow and structural characteris-
tics of land uses were analyzed using 1973 data
for two urban areas of Florida: Ft. Myers in
southwest Florida and Gainesville in central
Florida, and the data are summarized in table 3.
In table 3, the second column headed "fossil
fuel power density" is defined as the power den-
sity that is from the direct use of electricity
and other fossil fuels; and the column headed
"power density of embodied energy in goods and
services" is defined as the power density of the
embodied energy that is consumed indirectly in
the use of goods and services. All energy flows
are expressed as a density function on a yearly
basis; in this case, Cal/acre'year, rather than
on a housing or commercial unit basis. The vol-
ume of enclosed space occupied by built struc-
Generally, the land uses are arranged in
order of increasing power density from low den-
sity residential to the central business district
(CBD). The volume of structure per acre
Increases with increasing power density as might
be expected, with the exception of mobile home
land uses, where energy use is high as compared
with the volume of structure. In this land use
category, living units tend to be small (from 600
to 850 sq ft), while the energy demands of the
inhabitants are approximately equivalent to those
of other residential land use types.
Industrial land uses are not very energy
intensive on the average in the Florida urban
landscape in comparison to other industrialized
areas of the nation. For example, an average
value for fossil fuel power density of industrial
land uses for the nation derived from the Council
on Environmental Quality (1979) is equal to
approximately 4,600 x 106 Cal/acre'yr, or
about 6 times that computed for the Florida indus-
trial land uses. This is due primarily to the
"light industrial" nature of Florida industry, and
also to the fact that warehouse districts were
included in this classification.
Two different densities of CBD were evalu-
ated: those areas with an average height of two
stories and those with an average height of four
stories. Since energy use and therefore power
density is strongly related to the volume of
structure associated with a land use, it seems
apparent that the power densities of these land
uses will differ significantly from the very
urbanized areas of the nation where CBD's might
have heights as much as 10 times greater than
those experienced in Florida.
Table 3. Power density and total volume of structure for selected land
uses In Florida.
FosslI Fuel In Goods And Total Power3
Power Density Services Density Total Volume4
(x 106 Cal (x 106 Cal (x 106 Cal Of Structure
Land Use Type CE/acre yr) CE/acre yr) CE/acre yr) (x 103 ft/acr)
Low density 70 328 398 42.0
Medium density 90 411 501 70.0
High density 110 463 373 85.8
Low rise (2 stores) 340 1557 1897 302.0
High rise (4 stories) 570 2468 308 664.4
Medium density 122 597 719 30.6
High density 230 1086 1316 54.4
Coenrcial strtp 680 441 1121 150.3
Comerclal mall 3280 2032 5332 141.6
Industrial 760 548 1308 167.2
Central business district
Average 2 stories 2380 1525 3905 528.5
Average 4 stories 4320 2789 7109 1102.8
1Energy consumption data from billing records of Florida Power and Light, Ft. Myers office for
1973. In general, a 10% sample size of each land use classification was used.
2Goods and services consumed by each sector are from an Input/output analysis of the Lee County
analysis that gave total end use of goods and service by sector. Then that mount that was
attributable to each separate land use within sectors was apportioned according to the same percentage
of fossil fuel energies consumed by sector.
3Additlon of column 2 and 3.
4Volume of structure is calculated by multiplying the square feet of structural area (obalned
from property tam records) by average heights of buildings.
Simulation of Models of Hierarchical
Organization and Energy Spectra
Results of the study of several hierarch-
ically organized models are presented in this sec-
tion, starting with a simple model and progressing
to more complex examples. Differential equations
to describe the behavior of each are presented
along with time simulations of each model.
In general, the models are five compartment
systems (having 5 state variables) and differ in
kinetics of interaction between compartments as
the models become more complex. The final model
simulated is an aquatic food chain organized as a
hierarchic system of energy flow, using data from
Fontaine (1978) to evaluate each state variable
and pathways of energy flow between variables.
Presented in figures 10, 11, and 12 are
simulation results of a simple hierarchic chain of
energy flow without interacting feedback pathways.
The steady state simulation results are given
in figure 10 and then the results of various per- P'-'
turbations of the model are presented (figs. II
and 12). In all cases, pathway coefficientsare -
held constant in each simulation run, changing
only those coefficients indicated In the models in
the figures. When the initial conditions for
state variables are set low and the system allowed
to grow to steady state values (fig. 11), damped *
oscillation is exhibited by compartments QIland
Q2, with less noticeable oscillation in *upstrOeam;
In a final simulation of the simple chain
model, pathway coefficients were adjusted.so'that-
turnover times for all compartments were equa"''r:
The simulation results are given In figure 12..-,'
Without the dampening effect of increasing ''* '
turnover times for each compartment (as wn beL"h'
case in the simulation presented in fig. 11). -'i
increasing oscillatory behavior is exhiblilt
each component. The energy source has'l" n e'."
increased twofold, acting as stimulus'to'I thII
A feedback pathway is added between compart-
ments acting as a multiplicative interaction in
the next simulation, which is shown in figure 13.
The model has the added feature of a second energy
source that is multiplicatively interacted with
compartment Q3, using a switching function, so
that the model first runs in a steady state and
then at time 10, the second source is turned on.
The interaction of the second source changes the
distribution of energy within the system, with Ql
attaining a lower overall value, and Q2 and Q3
higher values. The graphs in figure 14 give the
energy spectral distributions for the steady state
solution, and as a result of the second source.
Aquatic Food Chain
Given in figure 15 is an evaluated aquatic
food chain organized as a hierarchy of energy
flow. The data used in the model given are a sum-
mary of Fontaine's (1978) data for Lake Convay
SIM0. o9maDOATIO mIDO*t
2000 4000 4000 0O
DEVELOPMENT OISTY (GOn
Figure 9. Graphs of the relationship of reports to gross do-
estic product when expressed as spatial functions for the
combined data of counties in Florida, states In the United
-tatas, and countries. (a) Log-lg plot as a nmograph;
(b) artlmnatic plot showing regression lines for each at
0 I0 to 30 40 50
Figure 10. Model and steady state slmulatloe results for a
simple hierarchic system of compertaents, lith no feedback
between compertments. Values In perentheses above e ch
compartment are turnover times, values Ia each storage
tank are steady state values, and differently equations
are as follows:
JR Jo/C 4( 1)
Qi .* IJRIt-Cq2JRQtlC3I-C3 Qi2
Q3 Cgq2A5-C 10Q2Q3-C I3-C 129Q4
Q4 c133Q4-c14 3Q4-c15 416Q4Q5
in central Florida. Compartments were summed
together into trophic levels based oon primary
energy source in the following manner: Ql -
Phytoplankton, Macrophytes, and Epipilec Algae;
Q2 Zooplankton and Benthic Invertebrates; Q3 -
Primary and Secondary Level Fish; Q4 Tertiary
Level Fish. A fifth compartment was added as a
top carnivore, and values of storage and flows
The major differences between the aquatic
food chain hierarchy and the previous models are
the addition of a sixth compartment that repre-
sents a pool of detritus and nutrient storage
that is recycled from the other five compart-
ments, the additive pathways of energy flow up
the food chain, and the additive feedback path-
S (2 5)
101 I0'" "'i ?.1 to
0 to 10 So 40 So
Figure 11. Simulation results of the simple chain when Initial
conditions are set low, as Indicated above each campart-
ment. Note that there is a difference In the vertical
scale froa the graph In figure 10.
(,) (0, 1) () (1)
mr of 02
SOURCE C oE4S
0 10 0 so 40 50
Figure 12. Model and simulation results when the turnover
times of each compartment are adjusted so that they are
equal. Numbers in parentheses above each compartment are
turnover times. Adjustment of turnover times was achieved
by recalling Inflows and outflows for each compartment so
that they were equal to the steady state value.
loll (O) )10) u (16) (25)
at 0 \ 02
4, a a
2 4 :)
Figure 13. Model and simulation results or a hierarchical
energy chain with Interactive feedback an a Secori energ
source. The simulation is first run in steady state con-
ditions until times 10, then the second source Is
switched on. Differential equations re as follows:
JR Jo/f +K(Q1+Q2)i
Q3 C992SHQQ3Q4-C 10I2SHQQ3Q4-C 1 C I 5-23045-C 22 19293
$4 C13Q3?451-C14Q34Q5-C I5Q4-16049-C 21A2SHI0Q3Q4
ways of control action flow from higher level
compartments to lower ones.
The model given in figure 16 shows the ener..
gy source as a smooth sine wave that represents
the variation in sunlight from summer to winter.
The graphs in figure 16 are the steady state
solution, where the effects of dampening of the
fluctuations in energy source are obvious. Since 5
Q2 draws most of its energy from the large "table
pool of detritus, very little oscillation I -
observed. Q3, on the other hand, draws much of
its energy from the first compartment (Ql) and
shows oscillation, but damped from that of Ql.
Yearly variation in standing crop in the hl4gter.
compartments (Q4 and Q5) is relatively small "
to the dampening of upstream compartments.,e '
The model in figure 17 was simulated' t e st
the effect of control actions by the highest 14'
level compartment. This last compartment has an
additional input pathway driven by a sine wave
with amplitude varying from +1 to -1, and fre-
quency equal to the turnover time of the compart-
ment (5 yr). The input pathway acts as both
positive input and drain, causing the compartment
The oscillation set up in compartment Q5 is
passed on to downstream compartments through
feedback pathways C28 and C29, and is passed fur-
ther downstream through feedback pathways C23,
C24, C17, C18, and C11. Because of differences
in scaling on the graph of the vertical axis for
each compartment, the differences in the magnitude
of oscillation of each compartment are not appar-
ent. The percent change from minimum to maximum
values for each compartment is given in figure
17. A general trend of decreasing magnitude of
oscillation from the highest level compartment to
the lowest level compartment is observed.
In all, many simulations of the various mod-
els reported here were conducted; testing various
organizations and perturbations to each model.
Different kinetic organization, energy sources
and magnitudes were tested for the theoretical
models. The aquatic food chain was simulated
testing different sources (constant and pulsing),
effects of harvesting of each compartment, and
the effects of a secondary energy source as a
stocking function. These simulation results are
reported in Brown (1980). Reported here are some
of the highlights of those initial investigations.
0, 02 O0 04 Os
0, 02 03 Q4 0Q
Figu-r 14. Graphs of the energy spectral distributions of com-
Partents In the simulation model In figure 13 where a
secondary energy source was Introduced Into the system.
(a) The distribution achieved wlth single law quality
souce; (b) the distribution achieved lwit the addition of
a Secondary high quality energy source.
Analysis of regions, urban systems, and
smaller ecosystems showed many manifestations of
hierarchy that were related to supporting energy
flows. Simulated models were able to duplicate
many of the observed features of hierarchical
organization. This evidence supports a theory of
energy control of the organization of systems of
man and nature.
Pathways of energy flow were shown to be
greatest, hierarchically, in the highest compon-
ents of urban and regional landscapes. Here com-
ponents were found to be largest in size, fewest
in number, to have larger time constants, and have
greatest potential control of the overall systems
Hierarchical Principles of
From the measurements and models, principles
may be formulated for relating parts to whole
landscapes and as guidelines for regional plan-
Energy Quality and Spatial Effect
While the number of components decreased in
successive levels of hierarchies, the spatial
area over which their effect was spread increased.
The nomographs of development density versus
exports suggested this relationship, for as the
density of human activity increased, the total
exports to other regions increased.
Energy Convergence in Landscape Hierarchies
The evaluated models of regional and ecolog-
ical systems and data on regional land uses and
spatial distribution of incoming energies showed
the pattern of convergence of energy flows to
higher and higher quality components in smaller
STORAGE. qc n5
Figure 15. Model of an aquatic food web organized In a hierarchic fashion.
Data are from Fontaine (1978). Numbers In parentheses are turnover
times of the state variables.
5 10 05 0 5
Figure 16. Model and steady state simulation results of mdefl
In figure 15, where the energy source is generated ine
wave representing seasonal fluctuation of sunlight. The
vertical axis on the graph Is the natural log of the valve
In compartments. Differential equations are as follows$
JR sine wave
QI CI2JI (Q2+3)96C2JRQI (Q2+93)Q6-C3Ql-C4(Q49Q6)Q2
(Q2+Q 1)Q3(Q4+Q5)-C 10 (2+Q3)Q405-CI JRQI (Q240)Q
93 C 12(Q2+Q1 )Q3(Q4+Q5)-C 13(Q2+Q1)Q3(Q44+5)-CIQ-CIS
(Q2+3)Q4Q5-C 16 (Q3+Q4 )Q5-C17JRQI (2+93)Q6-C 8 (Qt46)
94 C 19(0Q2+3)Q4Q5-C20(Q2+Q3)Q4Q5-C21Q4-C22(Q44+Q3
95 C25(Q3+Q4)Q5-C26(03+44)5-C2705-C28(Q1*92)P3t I
06 C30(C3Q4Q24C2C14Q34C21Q44C27Q5)-C31(Qf 4Ql9)WrQ )
spatial extent. Incoming energies, low in quality
and spatially dilute, were transformed into higher
quality energies, many as storage, as energy
flows converge from many low quality components to
fewer and fewer high quality ones.
Energy Divergence (Dispersion)
in Landscape Hierarchies
Energy is not only concentrated and con-
verged in landscape processes, but much is fed
back in dispersing actions of recycle and con-
trol. Spectral distributions of incoming energies
0 10 20 30 40 so
Figure 17. Model and simulation results of the aquatic food
chain when compartment Q5 Is caused to oscillate from an
Input pathway driven by a sine wave with amplitude +1 to
-1. The oscillation is passed onto lower level compart-
ents with decreasing amplitude (decreasing amplitude Is
not graphically apparent due to different vertical scales
for each compartment on the gaph). The percent change
from minimum to maximum values for each compartment is as
94 = 42%
Differentlal equations are the same as In figure 16 except
for the Sine Is nput to canpartment Q5. Th equation
for Q5 Is as follows:
and power density of land uses, as well as the
distributions of cities within regional landscapes
indicated that dispersion of high quality energy
from centralized sources follows a hierarchical
Control Actions of High-Quality Components
Evaluation and simulation ot hierarchically
organized models suggested that the highest qual-
ity pathways were those associated with the high-
est level component in the hierarchy, and sugges-
ted a general principle of hierarchic organiza-
tion and control action feedbacks. The greatest
control effect was achieved with the highest qual-
ity pathways, since their overall cost was large
and their effect must at least be equal their
cost. In the simulations of the aquatic ecosys-
tem, greatest overall effect to the system was
achieved when the final compartment was perturbed.
Primary and Secondary Energy Sources
and Their Effect on Hierarchies
The landscape is a mosaic of natural lands,
agricultural lands, roads, cities, and people
related through pathways of energy flow and
exchange. When viewed as a whole system of pro-
cesses, it was seen that energy sources inflowing
support the processes of the entire system. Sys-
tems that are sustained by one energy source such
as the evaluated aquatic ecosystem, develop rela-
tively smooth distributions of energy between com-
ponents as energy is cascaded up the hierarchy
supporting fewer and fewer components. Secondary
energy sources of higher quality inflow at levels
in the hierarchy where their quality nearly
matches, with additional energies for support at
these levels, greater structure was developed and
the hierarcy was shifted somewhat in the relative
distribution of energy between levels.
Stability Through the "Filtering"
Actions of Hierarchies
The term stability has been applied to a
number of concepts. Orians (1975) distinguishes
seven different concepts of stability. When
models of hierarchies were simulated, stability
was greatly enhanced if turnover times were
adjusted so that turnover increased with each
level in the energy chain. When the turnover
times were set equal in all compartments, oscilla-
tion was exhibited, since any perturbation in one
compartment was passed on to the next.
A Theory of Regional Boundaries Derived
from Place in the Landscape Hierarchy
A method that may have significance in deter-
mining the regional boundaries of systems was sug-
gested by the results of the nomographs of exports
versus development density of regions. The nomo-
graphs showed that the propensity to export (and
therefore to import to maintain balance of pay-
ments) was greater as the density of development
increased for regional systems of all sizes. In
essence, this relationship suggests that the
greater the density of development the more an
area relies on external areas for sources of pri-
mary goods and energies. And, to carry it one
step further, since primary goods require large
areas of the landscape for their production (i.e.,
they are low in quality and occupy large spatial
area), the greater the density of development, the
greater the size of the region required for
The author wishes to acknowledge Dr. H. T.
Odum, whose reviews of the research in progress
and previous drafts was invaluable. Happing of
the south Florida region and Lee County was sup-
ported under joint contract to the Division of
State Planning, Department of Administration,
State of Florida, and the National Park Service,
U.S. Department of the Interior, H. T. Odum prin-
cipal investigator. Mapping of the St. Johns
region was supported under joint contract to the
Jacksonville Area Planning Board and the St. Johns
River Water Management District, H. T. Odum prin-
cipal investigator. Initial studies of the Ft.
Myers urban area were funded by Lee County Board
of County Commissioners, the author and G. V.
Genova principal investigators. Preliminary stud-
ies of the Gainesville urban area were funded by
the Division of State Planning, Department of
Administration, State of Florida, J. F. Alexander
Support was also received from the Department
of Energy on contract EY-76-5-05-4398 entitled
"Energy Models of the United States" with the
Department of Environmental Engineering Sciences,
H. T. Odum principal investigator.
Aldrich, H. E. 1979. Organization and environ-
ments. Prentice Hall, Inc., Englewood
Cliffs, New Jersey.
Alexander, J. F., Jr. 1978. Energy basis of
disasters and the cycles of order and dis-
order. Ph.D. dissertation, University of
Beckmann, H. 1955. Some reflections on Losch's
theory of location. Papers, Regional Sci-
ence Association 1:139-48.
Beckmann, M. J. 1956. Distance and the pattern
of intra-European trade. Review of Econom-
ics and Statistics 38:31-40.
Beckmann, M. J. 1958. City hierarchies and the
distribution of city size. Economic Devel-
opment and Cultural Change 6:234-48.
Beckmann, J. 1970. The analysis of spatial
diffusion processes. Papers, Regional Sci-
ence Association 25:109-17.
Berry, B. J. L. 1972. Hierarchical diffusion:
The basis of developmental filtering and
spread in a system of growth centers. Pages
108-38 in N. M. Hanson (ed.), Growth centers
in regional economic development. Free
Press, New York.
Blau, P. M. 1972. Interdependence and hierarchy
in organization. Social Science Research
Blau, P. M. 1974. On the nature of organiza-
tions. John Wiley and Sons, New York.
Brown, M. T. 1980. Energy basis for hierarchies
in urban and regional landscapes. Ph.D.
dissertation, University of Florida, Gaines-
Bunge, M. 1969. The metaphysics, epistomology,
and methodology of levels in hierarchical
structures. Pages 17-26 in L. L. Whyte, A.
C. Wilson, and P. Wilson Teds.), Hierarchi-
cal structures. American Elsevier, New ,
Bureau of Economics and Business Research,
College of Business Administration, Univer-
sity of Florida. 1978. Florida statistical *
abstract 78. University of Florida Press,
Burgess, E. W., and R. E. Park. 1924. Introduc-
tion to the science of sociology. Univer-
sity of Chicago Press, Chicago.
Christaller, W. 1966. Central places in south-
ern Germany. (Transl. by G. C. W. Baskln.)
Prentice Hall, Inc., Englewood Cliffs, New
Costanza, R. 1975. The spatial distribution of
land use subsystems, incoming energy and
energy use in south Florida from 1900 to
1973. Master's terminal project. Depart-
ment of Architecture, University of Florida,
Council on Environmental Quality. 1979. Envi-
ronmental statistics, 1978. National Tech-
nical Information Service. U.S. Department
of Commerce, Springfield, Virginia.
Dokmeci, V. 1973. An optimization model for a
hierarchical spatial system. Journal of
Regional Science 13:439-51.
Emery, F. E., and E. L. Trist. 1973. Towards a
social ecology: Contextual appreciation of
the future in the present. Plenum, New
Fontaine, T. D. 1978. Comunity metabolism
patterns and a simulation model of a lake In ,
central Florida. Ph.D. dissertation, Uni- t:
versity of Florida, Gainesvllle.
Glassman, R. 1973. Persistence and loose
coupling. Behavioral Science
Hagerstrand, T. 1966. Aspects of the spatial
structure of social communication and dif-
fusion of information. Papers, Regional
Science Association 16:27-42.
Henderson, S. 1972. Hierarchy models of city
size: An economic evaluation. Journal of
Regional Science 12:435-41.
Hielbrun, S. 1974. Urban economics and public
policy. St. Martin's Press, New York.
Isard, W., and M. Peck. 1954. Location theory
and international and interregional trade.
Quarterly Journal of Economics 68:97-114.
Landau, M. 1969. Redundancy, rationality, and
the problem of duplication and overlap.
Public Administration Review 29:346-58.
Laszlo, E. 1972. The systems view of the world.
George Braziller, New York.
Laszlo, E. 1973. Introduction to systems phil-
osophy. Harper and Row, New York.
Losch, A. 1954. The economics of location.
(Transl. by U. Waglom and W. F. Stalpor.)
Yale University Press, New Haven, Connect-
Lotka, A. J. 1922. A contribution to the ener-
getics of evolution. Proceedings of the
National Academy of Science 8:147-55.
MacArthur, R. H. 1957. On the relative abun-
dance of bird species. Proceedings of the
National Academy of Science 43:293-95.
Mansfield, E. 1963. The speed of response of
firms to new techniques. Quarterly Journal
of Economics 77:290-311.
Honane, J. H. 1967. A sociology of human sys-
tems. Appleton Century, Crofts, New York.
Odum, H. T. 1960. Ecological potential and ana-
logue circuits for the ecosystem. American
Odum, H. T. 1967. Biological circuits and the
marine systems of Texas. Pages 99-157 in F.
J. Burgess and T. A. Olson (eds.), Pollution
and marine ecology. John Wiley and Sons,
Odum, H. T. 1971. Environment, power, and
society. John Wiley and Sons, New York.
Odum, H. T. 1973. Energy, ecology, and econom-
ics. Ambio 2(6):220-27. Swedish Royal
Academy of Science, Stockholm.
Odum, H. T. 1975. Marine ecosystems with energy
circuit diagrams. Pages 127-51 in J. C. J.
Nihoul (ed.), Modeling of marine systems.
Elsevier Oceanography Series. Elsevier
Scientific Publishing Co., New York.
Odum, H. T. 1976. Energy quality and carrying
capacity of the earth. Tropical Ecology
Odum, H. T. 1976a. Macroscopic minimodels of
man and nature. Pages 249-80 in B. Patten
(ed.), Systems analysis and simulation in
ecology, vol. 4. Academic Press, New York.
Odum, H. T. 1977. Energy, value, and money.
Pages 174-96 in C. S. Hall and J. W. Day
(eds.), Ecosystem modeling in theory and
practice: An introduction with case histor-
ies. John Wiley and Sons, New York.
Odum, H. T. 1978a. Energy analysis, energy qual-
ity and environment. Pages 55-87 in W.
Gilliland (ed.), Energy analysis: Anew pub-
lic policy tool. AAAS Selected Symposium 9.
Odum, H. T. 1978b. Net energy from the sun.
Pages 196-211 in S. Lyons (ed.), Energy for
a livable future comes from the sun: A hand-
book for the solar decade. Friends of the
Earth, San Francisco.
Odum, H. T. 1981. Systems. John Wiley and
Sons, New York. Forthcoming.
Odum, H. T., and M. T. Brown (eds.). 1976.
Carrying capacity of man and nature in south
Florida. Final contract report to the U.S.
Department of Interior. National Park Ser-
vice, Washington, D.C.
Odum, H. T., M. Brown, and R. Costanza. 1976.
Developing a steady state for man and land:
Energy procedures for regional planning.
Pages 343-61 in Science for better environ-
ment. Proceedngs of the International Con-
gress on the Human Environment (HESCO),
Kyoto. Asahi Evening News, Tokyo.
Odum, H. T., J. E. Cantlon, and L. S. Kornicker.
1960. An organizational hierarchy postulate
for the interpretation of species-individual
distribution, species entropy, ecosystem
evolution, and the meaning of a species-var-
iety index. Ecology 41(2):395-99.
Odum, H. T., and E. C. Odum. 1976. Energy basis
for man and nature. McGraw-Hill, New York.
Orians, G. H. 1975. Diversity, stability, and
maturity in natural ecosystems. Pages
139-50 in U. H. VanDubben and R. H. Lowe-
McConnell (eds.), Unifying concepts in
ecology. The University Press, Belfast,
Pugh, A. L., III. 1970. Dynamo users manual.
MIT Press, Cambridge.
Purver, D. 1975. A programming model of central
place theory. Journal of Regional Science
Simon, H. 1962. The architecture of complexity.
Proceedings of the American Philosophical
Simon, H. 1973. The organization of complex
systems. Pages 1-28 in H. Patte (ed.),
Hierarchy theory. George Braziller, New
Stewart, J. W. 1947. Empirical mathematical
rules concerning the distribution and equil-
ibrium of population. Geographical Review
Thompson, S. 1967. Organizations in action.
McGraw Hill, New York.
Timbergen, J. 1968. The hierarchy model of the
size distribution of centers. Papers,
Regional Science Association 20:65-8.
United Nations, Department of International Econ-
omic and Social Affairs. 1978. 1977 sta-
tistical yearbook (29th issue). United
Nations, New York.
Van Boventer, B. 1969. Walter Christaller's
central places and peripheral areas: The
central place theory in retrospect. Journal
of Regional Science 9:117-24.
Von Bertalanffy, L. 1933. Modern theories of
development: An introduction to theoretical
biology. (Transl. by J. H. Woodger.) Oxford
University Press, London.
Von Bertalanffy, L. 1968. General system
theory. George Braziller, New York.
Weber, M. 1946. Essays in sociology. Oxford
University Press, New York.
Weber, M. 1947. The theory of social and
economic organization. Free Press, Glencoe,
Weiss, P. A. 1971. Hierarchically organized
systems in theory and practice. Hafner Com-
pany, New York.
Whyte, L. L. 1949. The unitary principles in
physics and biology. Henry Holt and Com-
pany, New York.
Whyte, L. L. 1969. Structural hierarchies: A
challenging class of physical and biological
problems. Pages 3-16 in L. L. Whyte, A. G.
Wilson, and D. Wilson Teds.), Hierarchical
structures. American Elsevier, New York.
Wilson, A. C. 1969. Hierarchical structures in
the cosmos. In L. L. Whyte, A. C. Wilson,
and D. WilsonTeds.), Hierarchical struc-
tures. American Elsevier, New York.
Woodger, J. H. 1929. Biological principles.
Cambridge University Press, London.
Woodger, J. H. 1937. The axiomatic method in
biology. Cambridge University Press,
Zipf, G. K. 1941. National unity and disunity.
Principia Press, Bloomington, Indiana.