From sarhaus@umich.eduTue Jun 20 20:37:59 1995
Date: Tue, 20 Jun 1995 20:21:06 -0400 (EDT)
From: Sandra Arlinghaus 
Subject: Solstice, Volume VI, No. 1, 1995.

>From sarhaus@umich.eduTue Jun 20 11:37:37 1995
Date: Tue, 20 Jun 1995 11:17:31 -0400 (EDT)
From: Sandra Arlinghaus 
Subject: Solstice, Volume VI, No. 1, 1995


                              SUMMER, 1995

                           VOLUME VI, NUMBER 1
                           ANN ARBOR, MICHIGAN

Founding Editor-in-Chief: 
     Sandra Lach Arlinghaus, University of Michigan

Editorial Advisory Board:
     Michael F. Goodchild, University of California, Santa Barbara
     Daniel A. Griffith, Syracuse University
     Jonathan D. Mayer, University of Washington (also School of Medicine)
     John D. Nystuen, University of Michigan
     William C. Arlinghaus, Lawrence Technological University
     Neal Brand, University of North Texas
     Kenneth H. Rosen, A. T. & T. Bell Laboratories
  Engineering Applications.
     William D. Drake, University of Michigan
     Frederick L. Goodman, University of Michigan
     Robert F. Austin, Austin Communications Education Services.

Technical Editor:  
     Richard Wallace, University of Michigan.

     The purpose of Solstice is to promote interaction between geography 
and mathematics.  Articles in which elements of one discipline are used 
to shed light on the other are particularly sought.  Also welcome are 
original contributions that are purely geographical or purely 
mathematical.  These may be prefaced (by editor or author) with 
commentary suggesting directions that might lead toward the desired 
interactions.  Individuals wishing to submit articles or other material 
should contact an editor, or send e-mail directly to
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     Figures and graphs are transmitted in a file separate from the text 
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5.  INDEX TO VOLUMES I (1990) TO V (1994).
     The current issue marks the beginning of the second half of the 
first decade of Solstice.  Thanks to all who have participated in this 
project--editors, authors, and readers, alike!
     Over the course of the past five years, Solstice has garnered 
attention from the media.  If you know of additional citations, please 
share them with us for our electronic scrapbook!  Thank you.

Science (AAAS) "Online Journals," [Joseph Palca] 29 November 1991, Vol. 

Science News "Math for all seasons" Ivars Peterson, Jan. 25, 1992, Vol. 
141, No. 4.

Newsletter of the Association of American Geographers, June, 1992.

American Mathematical Monthly, September, 1992.

Harvard Technology Window, 1993.

Graduating Engineering Magazine, 1993.

Earth Surface Processes and Landforms, 18(9), 1993, p. 874.

On Internet, 1994.

Papers in Regional Science:  The Journal of the Regional Science 
Association.  "Wide Area Computer Networks and Scholarly Communication in 
Regional Science."  Gunther Maier and Andreas Wildberger.
     With this issue, we work to make Solstice available to a wider 
readership.  For the first five years, all articles were typeset using 
TeX, the typesetting program of Donald Knuth and the American 
Mathematical Society.  Our goal is to continue to provide text that is 
available to a wide variety of readers; thus, we do transmit directly so 
that those without Gopher access can read Solstice.  Surely one great 
advantage of e-mail is the ease with which it can deliver information to 
points remote from its source.  We also wish to push the text delivery in 
the directions of current technology, as well.
     Richard Wallace has kindly agreed to serve as Technical Editor of 
Solstice, working in conjunction with the Editor-in-Chief, to continue to 
develop innovative presentations that take advantage of current 
technology.  With this issue, we transmit a separate packet of figures to 
accompany the single text file.  In the future, we hope to have World 
Wide Web access to Solstice, in addtion to the direct delivery via e-mail 
and continuing archiving on a Gopher.  When the mathematics used requires 
it, we intend to offer that notation within the direct e-mail 
transmission (as we have in the past).
                            POLICY SCENARIOS
                             Richard Wallace
                       The University of Michigan
           Urban, Technological, and Environmental Planning

[Except where noted, figures for this article are being sent in a 
separate transmission with instructions enclosed]

     Driven largely by rapidly increasing atmospheric levels of greenhouse
gases, such as carbon dioxide, methane, nitrous oxide, and
chlorofluorocarbons (CFCs), the global climate appears to many observers
(e.g., Meadows, Meadows, and Randers 1992) to be in a period of change. 
Computer models suggest that throughout most of the temperate zones of the
world this change will take the form of rising temperatures.  Motor
vehicles, which are powered by fossil-fuel burning engines that emit
carbon dioxide and nitrous oxide, contribute significantly to the
production of potentially climate altering greenhouse gases.  Simply put,
carbon dioxide "is an inevitable byproduct of fossil fuel consumption and
it streams out of tail pipes in direct proportion to the quantity of fuel
burned" (Wilkinson 1993).  On average, for every kilogram of standard
motor vehicle fuel (gasoline) consumed, three kilograms of carbon dioxide
are released into the atmosphere (Faiz 1993).  Furthermore, motor vehicles
emit other greenhouse gases, too, such as CFCs that leak from
airconditioners.  Thus, while improved fuel economy can have a mitigating
effect on total greenhouse emissions, in general the greater the number of
motor vehicles, and the more miles that they are driven, the larger their
contribution to global climate change.  Therefore, understanding trends in
the number of vehicles registered and examining policy options to curb
growth in this number can play a significant role in combating global
climate change. 
     Worldwide, transport energy accounts for about 20 percent of total
emissions of greenhouse gases (Lashof and Tirpak 1990), but this figure
varies from nation to nation.  In the OECD nations as a whole, transport
contributes between 26 and 31 percent of all greenhouse emissions produced
there, while in the U.S. transport accounts for 38 percent of domestic
greenhouse emissions (International Energy Agency 1993).  In the Less
Developed Countries (LDCs) and in Eastern Europe, largely due to lower
rates of car ownership and use, the transport sector currently accounts
for a smaller contribution to greenhouse emissions (Faiz 1993), but these
nations represent a burgeoning new market for vehicles.  As a result,
greenhouse emissions from the LDCs are expected to rise in the near
     By contributing significantly to greenhouse gas emissions and,
therefore, to global climate change, the motor vehicle sector plays a role
in the general class of phenomena known as population-environment
dynamics.  This dynamic, as described by Drake (1993), is characterized by
transitions from one stable state to another.  While the second stable
state may be a more or less desirable state than the original condition,
and it is often worse (e.g., the transition from forest to desert that is
taking place in some regions), Drake argues, and offers supporting data,
that the transition phase itself is a period of vulnerability,
characterized by the potential for extremely negative outcomes. 
Transitions, however, also offer opportunities, and positive outcomes can
occur, too, especially if appropriate policies are pursued to manage the
transition.  Thus, while human society and environmental systems both may
survive under a new climatic regime, the transition period may prove to be
the most dangerous period of all. 
     The field of population-environment dynamics thus leads us to see not
only that transport affects the global environment, but also that
transport emissions of greenhouse gases are not purely a function of the
number of vehicles, miles driven, fuel use, and emissions technology. 
Worldwide, the ratio of people to motor vehicles varies considerably from
nation to nation.  While we might expect this ratio to be highly
correlated with GNP per capita, a simple linear regression fails to detect
a significant relationship between these two variables.  Approaching this
relationship geographically (see Figure 1), however, reveals a clear
relationship--wealthier nations in general have a lower ratio of vehicles
per person.  Population growth, too, can have an effect on the number of
vehicles.  By examining trends and relationships in and between these
technological and social factors, this paper seeks to investigate the
efficacy of different policy options on reducing the quantity of
greenhouse emissions from the transport sector.  This analysis will be
performed for six nations that typify the range of transport, economic,
and population dynamics across the globe.  By examining nations that
differ in these key respects, the analysis will illustrate different
dimensions of the dynamic and provide policy guidance tailored to the
specific circumstances of each nation and beyond to the world community. 
The six nations examined here, and the patterns that they represent, are
listed in Table 1. 
                                Table 1.
          United States
          Wealthy, high use of autos and energy
          Wealthy, more emphasis on public transit and
          more energy efficient
          Former Eastern Block, dirty cars
          Less developed, low car ownership, booming
          Latin American, Industrializing    
          South Korea
          Asian Newly Industrialized Country (NIC)
     While no rigorous attempt will be made to justify this
categorization, a few observations backed by data collected by the World
Resources Institute (WRD) and the American Automobile Manufacturers
Association (AAMA) provides it some legitimacy.  First, vehicle technology
varies substantially between the most highly industrialized nations and
the rest of the world.  The typical vehicle manufactured in Eastern Europe
and the LDCs is only about half as fuel efficient as typical
OECD-manufactured vehicles (Faiz 1993).  Second, an examination of trends
in the number of registered vehicles in Hungary, India, Mexico, and South
Korea (see Figure 2) reveals a clear distinction between them, with the
two industrializing nations (South Korea and Mexico) showing an especially
steep growth curve, India showing a slower rate of growth, and Hungary
showing little growth at all.  Based on this evidence, these six nations
do appear to represent distinct patterns. 

Figure 2, of Motor Vehicle Registrations in the six
nations considered, is described well in the text--
it appears in hard copy only.

Patterns of Vehicle Use
     Across the six nations listed in Table 1, reliance on motor vehicles
for transportation varies greatly.  In 1992, for example, India and South
Korea each had roughly the same number of motor vehicles registered, but
India's population was nearly twenty times that of South Korea.  On the
other hand, of these six nations, India, which had by far the highest
ratio of people to vehicles in 1992, experienced the largest absolute drop
in this figure over the last twenty years (see Figure 3, which is drawn to
log scale so that all six nations may be viewed on one graph).  Thus,
while the U.S. and Japan currently are the largest contributors to
greenhouse emissions from the transportation sector, population and
consumption trends suggest that other nations will account for an
ever-increasing percentage of transportation's contribution to greenhouse
gas emissions in the future. 

Figure 3 appears in hard copy only; its description
is straightforward in the text.  Specific information
may be read from appropriate figures following it.

     Figure 3 shows that all six nations, with the exception of the U.S.,
still are experiencing a decline in the ratio of people to vehicles.  What
is driving this trend, however, varies from nation to nation.  In some
cases, rising consumption, as measured by GNP growth, appears to be the
driving factor, while in others increasing urbanization as measured by
urban population, appears to be the culprit.  Epitomizing these two
dynamics are Japan, South Korea, and Hungary.  In Japan (see Figure 4),
the increase in vehicle registrations appears to have been driven largely
by increases in population and urbanization in the early post-war years,
with more recent gains appearing to be more associated with increased GNP. 
By comparison, South Korea (see Figure 5), displays a relationship between
increased vehicle registrations and a rising GNP, with population
appearing to have little effect.  Finally, Hungary (see Figure 6) displays
a combination of the two effects, with both urban population growth
(despite a steady total population) and GNP growth associated with
increased vehicle registrations during the study period. 

Policy Options
     As described by Meadows, Meadows, and Randers (1992), environmental
impacts can be viewed as a function of population, consumption patterns,
and the state of technology.  These variables also appear within the
policy options available to reduce the contribution of the transportation
sector to greenhouse gas emissions.  Among these are: (1) increased fuel
efficiency, (2) reliance on alternative fuels, (3) reliance on public
transportation and other travel behavior approaches, (4) consumption
limits, and (5) population-control policies.  While these approaches range
from technological fixes to changes in societal norms, even the
technology-based approaches demand a corresponding societal component. 
Increasing fuel efficiency, for example, requires the political will to
raise minimum standards, and increasing use of public transportation
requires alteration of travel behavior.  If we are to explore the likely
effectiveness of each of these policies, we must first understand how and
what each contributes to the reduction in emissions of greenhouse gases
and, where possible, gauge how this dynamic might play out in the near
future--not an easy undertaking. 
     Given a rather large body of research and literature in the field,
gauging future technical abilities is perhaps the simplest forecasting
task.  DeLuchi (1993) has estimated the reduction in greenhouse gas
emissions from a variety of alternative fuel sources.  His findings
indicate a broad range of outcomes depending on the source of the
alternative power.  Electric vehicles powered by coal-burning plants, for
example, can be expected to lead to an increase in the amount of
greenhouse emissions compared to a standard gasoline- or diesel-burning
vehicle.  Emissions reductions, however, would be realized from a variety
of alternative fuels, including solar-powered electric, compressed natural
gas, methanol (from wood), and ethanol (from several sources).  The
International Energy Agency (IEA; 1993) produced similar findings, also
adding liquid hydrogen to the list of alternative fuels that would reduce
greenhouse emissions.  Using middle estimates from the latter source,
anywhere from a 25 to 50 percent reduction in emissions appears feasible. 
Recent technological breakthroughs in the manufacture of photovoltaic
panels, which suggest the possibility of producing solar energy at very
competitive market rates by next year, promise to make this figure even
higher (Myerson 1994).
     The IEA also studied potential improvements to fuel economy and found
that by 2006 a 10 to 20 percent improvement is feasible for OECD nations. 
This estimate is conservative and is based only on marginal improvements
to currently employed vehicle technologies and materials.  Other
researchers, however, have cast aside such industry-bound restraints and
discussed what could be done even today using technical inputs from beyond
the traditional steel-centered perspective of the global auto industry. 
Typifying this approach, Lovins and Lovins (1994) tout the revolutionary
potential of a new car design known as an ultralight hybrid.  These
light-weight vehicles, manufactured from high-tech composite materials
such as carbon fiber, and powered by a combination of liquid fuel and a
battery-powered flywheel that captures and stores energy now lost during
braking, offer a tenfold improvement in fuel economy over today's typical
car.  These vehicles also offer equal or better safety, mostly because,
pound for pound, these ultralight materials possess far superior crash
resistance than does steel.  As the Lovins's recognize, bringing such
vehicles to market will require a major restructuring of the auto
industry, but large auto companies that resist making these changes may
soon be leapfrogged by other manufacturers, such as former defense
contractors, more experienced with these new materials. 
     Public transportation provides another alternative to reliance on
motor vehicles.  Although increased urbanization seems to be a factor
associated with increased vehicle registrations, public transportation
works best in urbanized areas.  Which effect seems to dominate may be a
consequence of urban density.  As Newman and Kenworthy (1991) showed in
their study of 32 of the world's major cities, per capita consumption of
gasoline can vary considerably, even across affluent nations.  They found
that, generally, in high-density cities such as Tokyo, per capita gasoline
consumption is far less (about 1/6) than in relatively low-density U.S.
cities.  Much of this difference can be explained by examining the rate of
transit use between these cities.  The average resident of Tokyo takes 472
transit trips per year, while the typical New Yorker takes 58 and a
Detroiter only 17; vehicle ownership rates may be similar in Japan and the
U.S., but vehicle use is lower in high density areas.  Given the
relationship between fuel consumption and greenhouse emissions, less
vehicle use results in less greenhouse emissions. 
     Consumption limits represent perhaps the most difficult policy option
to address.  While draconian laws certainly hold out the promise of being
effective in reducing reliance on motor vehicles, such approaches carry
high social costs and appear incompatible with dominant social and
political standards in much of the world.  Fortunately, policies exist
that achieve the goal of reduced consumption without serious intrusions on
individual liberties.  On the regulatory side, for example, implementation
and enforcement of vehicle occupancy regulations can reduce the
preponderance of singly-occupied vehicles.  On the market-based side, high
fuel taxes provide economic incentives for people to choose non-vehicular
travel modes, as does the elimination of free parking in employment
centers (Wachs 1981). 
     Finally, in those nations experiencing rapid population growth,
policies aimed at slowing or reducing population growth may have some
promise in reducing their transportation sector's contribution to
greenhouse emissions.  On the other hand, it may be that controlling
population will increase wealth in these nations, thereby increasing
demand for motor vehicles.  Either way, India, with its large population
of would-be motorists, has an enormous potential to contribute to
greenhouse emissions, and the ratio of people to motor vehicles in India
has been falling. 

Tailoring Policy to Population-Environment Conditions
     Given differences in the forces influencing increased motor vehicle
registrations, and therefore increased greenhouse emissions, the range of
potentially effective policy options available to each nation also is
likely to be different.  Each of these policy options thus is best suited
to a particular set of political, technological, economic, and societal
conditions.  A policy that promises to be effective in one country,
therefore, may be inappropriate for another.  Population-control measures,
for example, appear to be an unlikely candidate to reduce greenhouse
emissions from the transportation sector in nations such as Hungary that
are experiencing little or no population growth.  The task at hand,
therefore, is to match policies to nations in the most effective manner. 
Doing so requires forecasting growth in the motor vehicle population,
particularly the fossil-fuel powered motor-vehicle population, within the
societal context of each nation. 
     To start this analysis, let us begin with the number one emitter of
greenhouse gases via motor vehicles--the United States.  Because the U.S.
has so many more motor vehicles than any other nation, reducing emissions
in the U.S. alone would signal progress toward worldwide reduction.  As
can be seen in Figure 7, the U.S. is undergoing a steep rise in GNP, along
with a moderate rise in both total and urban population.  The rate of
increase in the number of registered automobiles, however, appears to be
declining.  The already enormous number of vehicles registered in the U.S.
points to the need of reducing either the per-capita use of each vehicle
or the amount of emissions per vehicle or both.  This state of high wealth
and an abundance of motor vehicles suggests pursuing one or more of the
technological approaches listed above--California's mandate that two
percent of vehicles sold be zero-emissions vehicles by 1998 (and 10
percent by 2005) is a good example--along with some modifications to
travel behavior, such as increasing the importance of public
transportation.  Based on these policy pursuits, future vehicle
registrations for the U.S. can be projected. 
     To begin forecasting, we first need to find a good fit to the current
trend in vehicle registrations.  From Figure 7, the growth in the number
of motor vehicles in the U.S. appears to be S-shaped, suggesting a
logistic (or other similar) function.  Assuming a maximum value of
225,000,000 cars by 2025, an S-shaped curve can be fit successfully to the
data.  Furthermore, following California's lead, we can mandate that one
percent of all vehicles registered be zero emissions by 1998.  Next, we
can go a step further and assume that the zero-emissions portion of the
vehicle fleet will increase by one percent each year thereafter.  As shown
in Figure 8, this scenario implies a gradual decrease in the number of
fossil-fuel powered vehicles, with this number falling below current
levels by 2005 and continuing down to 160 million (more than 30 million
below the current number) by 2025.  While this decrease is modest,
combining it with increased fuel economy in remaining fossil-fuel-powered
vehicles, and at worst, no increase in vehicle miles traveled per vehicle,
could multiply the result by a factor of ten or so.  In effect, this means
the equivalent of about 16 million of today's motor vehicles--a figure not
seen in the U.S. since the mid-1920s. 
     For Japan, the other economic powerhouse in the analysis,
technological solutions also are appealing.  As can be seen in Figure 4,
however, growth in motor vehicle registrations in Japan continues at what
appears to be a linear rate (with r^2=0.99 when beginning with the year
1960, this trend is fit better by a straight line than either an
exponential or logistic curve), driven by a growing GNP.  If this linear
trend continues, Japan will have about 125 million registered motor
vehicles by 2025 (see Figure 9).  Assuming a similar pattern of switching
to zero-emissions vehicles as was forecast for the U.S., we find that the
number of fossil fuel vehicles does begin to assume an S-shaped curve. 
Like the U.S., Japan would appear to have the technical and economic means
to also pursue fuel-economy improvements, too.  Again, another tenfold
increase in effectiveness is possible, meaning the equivalent of a little
more than 9 million of today's cars by 2025.  As discussed earlier, Japan
also displays less reliance on the cars that are registered.  If this
pattern also continues, then the effectiveness of Japanese efforts to
reduce greenhouse emissions can be further enhanced. 
     In Hungary, as we saw in Figure 3, the population growth rate is
close to zero.  Nonetheless, the urban population continues to grow, as
does the number of registered vehicles.  Currently, however, Hungary has
the fewest registered vehicles of all six nations included in this study,
but many of these cars are relatively high polluting and inefficient
vehicles based on the engine technology typical of Eastern European
nations (Michelberger 1991).  To a large extent, then, Hungary's
contribution to greenhouse emissions is related to poor vehicle
technology.  Thus the ongoing replacement of its fleet of older, outdated
vehicles with newer, Western-style vehicles offers a sure path to fewer
overall emissions.  This dynamic, however, competes with growth in the
size of the vehicle fleet.  This growth is fit well by both a linear and a
logistic curve (with a maximum value of 5 million), with these two not
diverging by much until around 2010 (see Figure 10).  This result suggests
that Hungarian transportation policy stands able to influence future
growth in vehicles: Hungary can choose to maintain and improve alternate
modes (especially public transit) or accept the consequences of a
vehicle-centered society.  Either way, however, Hungary will continue to
have the smallest vehicle fleet in this study and can be expected to make
only a small contribution to greenhouse emissions. 
     India, which has yet to see its population growth rate begin
declining, has been projected to soon overtake China as the world's most
populous nation (see Figure 11).  Relative to this enormous population,
however, India has a very low number of registered vehicles.  In the near
term, the rate of growth in the vehicle population, however, is
exponential, regardless of whether an exponential or logistic function
(with a maximum value of either 140 or 200 million) is fit to the actual
data (see Figure 12).  Thus India's growth rate in this area may not be
boundless, but current trends suggest that India eventually will meet or
surpass the number of motor vehicles currently registered in the U.S. 
     Given this growth in vehicle registrations, India can be expected to
experience increased negative consequences of fossil-fuel based travel,
including worsening congestion and pollution, and to dramatically increase
its contribution to greenhouse emissions from the transportation sector. 
The level at which this growth is bounded, along with which technologies
and policies India applies to mitigate these problems (e.g., alternative
fuels, emissions controls, demand management), will determine the severity
of future outcomes and India's contribution to greenhouse emissions. 
Unfortunately, however, India does not appear to have the economic
strength to address this problem with technological approaches, save what
may accrue naturally from advancing vehicle technology.  Developing
infrastructures for alternative fuels, for example, would appear unlikely
in the near future.  These leaves India with only one option from the
above list of policies for addressing greenhouse gas emissions from
transportation--travel behavior measures.  In this regard, India already
relies predominantly on non-motorized forms of personal transit in rural
areas and on high vehicle-occupancy rates in urban areas.  Restricting
access to vehicle purchases--perhaps unpopular with the growing middle
class--and financial disincentives, such as a high fuel tax, for vehicle
travel both are options worth considering. 
     Excluding the U.S. and Japan, Mexico has far more registered vehicles
than the other nations in this study and, as Figure 1 shows, trails only
OECD nations and Australia in having a low people to vehicle ratio.  As in
India, in Mexico both the population and the motor-vehicle fleet are
growing quickly, but Mexico possesses a much stronger economy than does
India.  As can be seen in Figure 13, Mexico's GNP has shown a strong
increase--despite a brief setback in the early 1980s--over the last 25
years; meanwhile, both its total and urban population have grown steadily. 
During the period of economic decline in the early 1980s, motor vehicle
registrations also stagnated, but did not decline.  This suggests a strong
and tight temporal relationship in Mexico between GNP and vehicle
registrations (r^2=0.87 for a simple linear model since 1970 for a
relationship that would seem to demand a time lag).  Therefore, if
Mexico's GNP continues to rise--perhaps as a result of NAFTA--then vehicle
registrations also should soar, barring a policy intervention. 
     In many ways, the situation in South Korea closely resembles that in
Mexico: GNP, vehicle registrations, population, and urban population all
are rising--most of these rapidly.  In South Korea, however, the economy
did not decline in the early 1980s and the rate of population growth began
declining shortly after 1950, while Mexico is just now beginning to see
its rate of population growth decline.  As a result, motor vehicle
registrations have grown exponentially in South Korea with no break since
1970.  Clearly, however, this growth cannot be unbounded, because it were
it would lead to nearly one billion registered vehicles by 2025--about 200
cars per person!  Therefore, we must assume a logistic growth rate,
bounded at about 50 million, or one per person (see Figure 14). 
     This curve fit indicates a nearly tenfold increase in South Korea's
motor vehicle fleet, which can be expected to increase South Korea's
contribution to greenhouse gas emissions.  Given South Korea's strong
economic position and world class domestic vehicle production facilities,
however, South Korea, like the U.S. and Japan, stands in a good position
to pursue technology as one policy approach for mitigating the effects of
a growing vehicle fleet.  Furthermore, given that most of South Korea's
population is urban, the maintenance of public transportation and the
restriction of parking in city areas offer parallel options aimed at the
behavioral side of the equation. 

     Overall, all nations in this study displayed a propensity toward more
motor vehicle registrations and, hence, increased greenhouse emissions. 
Furthermore, all save Hungary, either already are or soon will be large
contributors on a global scale.  Analysis of trend data, however, also
shows that all five nations faced with this problem have available policy
options that will allow them to mitigate the problem, perhaps even leading
to a net decline in greenhouse emissions from the motor vehicle sector. 
As expected, the likely solutions that emerged from the data vary from
nation to nation. 
     In general, the nations with high GNPs, domestic auto industries, and
low rates of population growth seem best poised to pursue technologically
based remedies, including transitioning to alternative fuels and mandating
improved fuel economy.  Such nations also appear able to benefit from
changes in travel behavior.  While numerous approaches to reducing the
reliance on vehicular travel have been proposed, including following the
European lead of internalizing some of the social costs of driving into
car and fuel prices, none have been greeted enthusiastically in the U.S. 
In Japan and South Korea, where such policies do exist, car ownership is
associated with status, prestige, and other difficult-to-overcome
attributes.  Two related possibilities that have some promise in all three
nations are road pricing and congestion pricing.  These two policies are
both aimed at capturing some of the social costs of driving by introducing
per-trip tolls based on mileage, time of travel, or both, thereby
increasing the marginal cost of a vehicle trip (Gomez-Ibanez 1992). 
Congestion pricing already has succeeded in reducing traffic volume in
places such as Singapore, and California soon will unveil a congestion
pricing scheme in the Anaheim-to-Riverside corridor. 
     In Hungary and the rest of Eastern Europe, typified by little or no
population growth and outdated vehicle technology, technological advances
again appear to be the best path to reduced greenhouse emissions, but the
relevant technologies are different than what was prescribed for the U.S.
and Japan.  Rather than expecting a transition to ultralights and
super-efficient cars of tomorrow, Hungary and other Eastern European
nations can be served well simply by allowing market forces to engender a
transition from 1970s to 1990s technology, while at the same time taking
steps to protect against the development of an autocentric culture.  In
this regard, following the lead of their Western-European neighbors and
internalizing the social costs of driving should provide an ameliorative
     For third world nations, characterized here by India, technological
solutions appear to be out of reach due to economic factors.  Although the
transportation sector in such nations currently makes a relatively small
contribution to global greenhouse emissions, large populations and the
hope of future economic development leave these nations poised to
dramatically increase their vehicle use and greenhouse emissions.  In
these nations, a policy emphasis on travel behavior patterns appears to be
most appropriate.  If pursued, such a policy may allow such nations to
avoid developing an emissions problem in the first place. 
     Finally, for nations such as Mexico caught squarely in the middle of
larger and more unstable population and economic transitions, the
motor-vehicle future is far more difficult to predict.  Compared to South
Korea--a developing nation nearing the end of its development transition
and on a par economically with the developed world--Mexico has yet to
enter an exponential growth phase in motor vehicle registrations (a simple
linear model better fits vehicle registrations than does an exponential
curve).  Probably, Mexico and similar nations have the possibility of
pursuing all policies available--bringing the latest technology into their
vehicle fleet when possible, promoting public transit, while at the some
time bringing population growth under control--to avoid entering this
vehicle growth spurt.  Therefore, these nations have the opportunity to
approach or match the OECD nations in economic development without falling
prey to at least one of the ills of life in the industrialized world--over
reliance on motor vehicles. 
     Thus, the transition period presents nations with two paths, and the
selection of one or the other is to a large extent within the control of
policy.  One path--chosen explicitly or implicitly by South Korea--closely
follows the Western world and leads to over-reliance on motor vehicles and
increased environmental degradation on a global scale.  The other path,
less traveled and partially unexplored, appears to lead to economic
development that is more friendly to the global environment.  As described
by chaos theory (see, e.g., Prigogine and Stengers 1984), periods of
uncertainty may quickly give rise to irreversible outcomes.  Therefore,
the selection of a path to follow is crucial, and evidence from the
industrialized world suggests that choosing the motor-vehicle dominated
path is costly to the global environment and may dictate an undesirable
future from which there may be no escape. 

     The ultimate effects of greenhouse gases on the global climate remain
uncertain.  Even more uncertain is how these global changes will play out
at local and regional scales.  Some still argue that global climate change
is either not occurring or not dangerous even if it is occurring.  Others,
of course, predict dire consequences if global climate change is not
halted (Brown, et al. 1994).  Following Drake, however, we must not
neglect the possibility that the period during which the climate is
changing, and not the resultant state of the future global climate, is the
period most deserving of close scrutiny and remedial action.  That period
is now.  By following some or all of the policy recommendations discussed
in this paper, we may yet be able to slow the pace of change, which should
serve to mitigate the negative consequences of a transitional period. 
Furthermore, by not delaying our policy response, we allow scientific
research on global climate change to find more answers before we
needlessly condemn future generations to a changed global climate of our


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                             JOHN D. NYSTUEN
                 College of Architecture and Urban Planning
                         The University of Michigan

                           SANDRA L. ARLINGHAUS
                 School of Natural Resources and Environment
                         The University of Michigan

                           WILLIAM C. ARLINGHAUS
                Department of Mathematics and Computer Science
                      Lawrence Technological University

     The statistician Fisher explained the mathematical basis for the
field of "Design of Experiments" in an elegant essay couched in the
context of the mathematics of a Lady tasting tea (Fisher, in Newman 1956;
Fisher 1971).  In Fisher's text, the problem is to analyze completely the
likelihood that the Lady can determine whether milk was added to the tea
or tea added to milk.  Problems associated with the tasting of wines have
a number of obvious similarities to Fisher's tea-tasting scenario.  We
offer an analysis of this related problem, set in the context of Nystuen's
wine tasting club.  To begin, a brief background of the rules of that club
seems in order; indeed, it is often the case that the application is
forced to fit the mathematics in order to illustrate the abstract.  Here,
it is the real-world context that guides the mathematics selected. 

Wine Tasting Strategy
   The Grand Crew wine club of Ann Arbor has been blind-tasting wines
monthly for years.  In a blind tasting, several wines are offered with
their identity hidden.  Not only are labels covered, but the entire bottle
is covered as well because the shape and color of the bottle provides some
clues as to the identity of the wine.  The wines are labeled 1 through n
in the order presented.  Six to eight wines are tasted at a sitting. 
Members sip the wines and score each on a scale from 1 to 20, using a
scoring method suggested by the American Wine Association.  The wines are
judged on the basis of quality and individual taster preference.  The
evening's host is in charge of choosing and presenting the wines.  Usually
wines of a single variety but from different vineyards, wineries, prices,
or distributors are tasted.  Two sheets of paper are provided to each
taster.  One is a blank table with a row for each wine numbered 1 through
n in the order presented.  The columns on this sheet provide space for
comments, the individual's numerical ratings of the wines, average ratings
of the group, and the range in scores for each wine.  One column is
reserved for the member's guess as to the identity of the wine.  The
second sheet contains information about each wine to be used to match the
wines tasted.  On this sheet the wines are labeled a, b, c, and so forth,
along with information on age, winery, negotiant, and price.  The tasters
try to match the identity of the wine with their individual rating on
sheet 1. 
    The wines are listed in unknown order on the second sheet.  The
tasters make their decisions by matching the letter identification with
the numerical order of presentation.  On rare occasions one or more
members correctly identifies every wine.  More often two or more wines are
mislabeled, and quite often the identities seem hopelessly scrambled. 
Guessing at random would seem just as effective.  The question then
arises, "what are the chances of getting one, two, more, or all correct by
chance alone?"  Discrete mathematics and the algebra of derangements
provides the answer to this question. 
    Probabilities are a matter of counting.  In what proportion does a
particular combination of correct and incorrect identifications occur
purely at random out of all possible combinations?  The denominator in
this proportion is a count of all possible arrangements and the numerator
is a count of all possible ways a particular event occurs, such as one
right, all the rest wrong.  The denominator is easily determined.  If one
has five things any of the five might be chosen first; there remain four
things any of which might be chosen next.  The process continues until the
last stage in which only one can be chosen.  Thus, there are 5*4*3*2*1=120
ways to arrange five bottles of wine--the customary notation for this
product is 5! (read five factorial).  This notation extends to arbitrarily
large positive integers in the obvious way; 0! is defined to be 1.  The
factorial of a number grows rapidly with an increase in the size of the
number; thus, 7!=5040 while 8!=40,320.  
     The numerator of the proportion sought is not found as easily. 
Consider the case of a blind tasting of three bottles of wine.  Suppose
the first one is correctly identified; the remaining two outcomes must be
both right or both wrong.  It is not possible to identify two wines
correctly and the third one incorrectly.  Table 1 illustrates all possible
patterns of identification for three bottles of wine, a, b, and c, with
bottle "a" presented first, bottle "b" presented second, and bottle "c"
presented third.  As this table indicates, there is only one arrangement
in which all are correct, two arrangements with none correct and three
arrangements with two correct.  There are, of course, no arrangements with
exactly one correct. 

Table 1.  All possible arrangements of three items, a, b, and c.
          Number of matches and non-matches to the arrangement abc.
        Matches         Non-matches
 abc       3                 0
 acb       1                 2
 bac       1                 2
 bca       0                 3
 cab       0                 3
 cba       1                 2

     When all possible outcomes, shown in Table 1, are enumerated, it is
an easy matter to calculate the probability of each type of event-- to
obtain the probability, divide each outcome from Table 1 by 3!, the number
of total possible arrangements.  Table 2 shows the probability of each
outcome:  P(0) denotes none right, P(1) denotes exactly one right, and so
forth.  The sum of all probabilities adds to 1.00, as it should. 

Table 2.  Probability of a correct labeling.

 P(0) = 2/6 = 0.33
 P(1) = 3/6 = 0.50
 P(2) = 0/6 = 0.00
 P(3) = 1/6 = 0.17

    A total enumeration approach to finding the probabilities is
satisfactory for introductory purposes and for very small samples. Even
for six, seven, or eight wines at a single tasting it is, however, not
satisfactory; Table 1 would expand to 720, 5040, or 40,320 columns for
each of those cases.  Clearly more clever and mathematically elegant ways
of counting, rather than brute force listings, are required.  In this
latter regard, one is reminded of the story of Gauss who, as a young
child, astounded his German schoolteacher with an instant result for what
the teacher had planned as a tedious exercise.  The teacher, in order to
keep his students busy, told them to add all the numbers from 1 to 100. 
Gauss immediately wrote the answer on his slate.  He had apparently
discovered for himself that the sum, S, of the first n positive integers
is given by the recursive relationship S=(n(n+1)/2).  Thus, all he had to
do was multiply 50 by 101 to obtain the answer: an elegant solution to an
otherwise tedious problem.  It was the more mature Gauss and later Laplace
that would do pioneering work in the Theory of Errors of Observation which
in turn would serve as a significant part of the base for applications of
mathematics and statistics (in Design of Experiments) in the Scientific

    For our problem, we need a way to count the number of times a taster
can get all the wines right, one wine right and all the others wrong, two
wines right and all the others wrong, and so forth.  To convert the
tedious, brute force task of listing permutations and combinations for
this problem, to a more tractable situation, we employ the concept of
"derangement," that will eliminate, notationally, combinations that we do
not wish to consider. 
    A "derangement" is a permutation of objects that leaves no object in
its original position (Rosen 1986; Michaels and Rosen 1991).  The
permutation badec is a derangement of abcde because no letter is left in
its original position.  However, baedc is not a derangement of abcde
because this permutation leaves d fixed.  Thus, the number of times a wine
taster gets all the wrong answers in tasting n bottles is the number of
derangements of n numbers, D(n), divided by n!:  D(n)/n!. The value of
D(n) is calculated as a product of n! and a series of terms of alternating
plus and minus signs: 


Readers wishing more detail concerning this formula might refer to Rosen
(1988); for the present, we continue to consider the use of derangements. 
    In order to see how derangements can be enumerated visually, we
construct the following tree of possibilities for arrangements of 5
letters which do not match the natural order of abcde.  On the first
level, the natural choice is a--so choose some other letter instead.  The
second level would be b in the natural order so choose all others,
instead, and continue the process until all possibilities have been
exhausted.  Following each path through the tree will give all possible
derangements beginning with the letter b--there are 11 such routes.  Thus,
there are 11*4 derangements. 

 Tree of derangements for 5 bottles.

 Level 1: b
          |             \               \               \
 Level 2: a              c               d               e
          |   \          |   \    \      |   \           |   \ 
 Level 3: d    e         a    d    e     a    e          a    d
          | \  |         |    | \  |     | \  | \        |    | \
 Level 4: c  e c         e    a  e a     c  e a  c       c    a  c 
             | |         |       | |        | |  |       |    |  |
 Level 5:    c d         d       a d        c c  a       d    c  a

Indeed, when there are five wines, D(n)=5!(1/2!-1/3!+1/4!-1/5!)

     What is of particular significance is that derangements focus only on
wrong guesses:  because a non-wrong guess is a correct guess, it is
possible to focus only on one world.  The Law of the Excluded Middle, in
which any statement is "true" or "false"--with no middle partial truth
admitted, is the basis for this and for most mathematical assessments of
real-world situations.  It is therefore important to use the tools
appropriately, on segments of the real-world situation in which one can
discern "black" from "white." 

Derangements and Probability in Random Guesses
    In the case of the five wine example, the number of ways of choosing
(for example) three correctly out of five is the combination of five
things taken three at a time: C(5,3)=5!/2!3!=10.  Exhausting all possible
combinations reflects an expected connection with the binomial
theorem--these values are the coefficients of (x+y)^5. 
 The total number of right/wrong combinations is therefore 2^5 or 32. 
Notice, though, that the pattern within each grouping is disregarded; to
discover the finer pattern, of how right/wrong guesses are arranged we
need permutations.  To limit the number of permutations necessary to
consider, we investigate the derangements. 
    If we can count derangements, we can now address the question of how
many times a taster, guessing randomly, gets exactly one wine correct. 
The answer is simply the number of ways one bottle can be chosen from n
bottles times the number of derangements of the other (n-1) bottles of
wine.  When this value is divided by n!, the probability P(1) of guessing
exactly one wine correctly is the result. That probability is: 


This idea generalizes in a natural manner so that the probability of
choosing exactly k wines correctly is given as: 


Table 3 displays all the probabilities for outcomes in blind tastings in
which random choices are made in situations for which from 2 to 8 wines
are offered by the evening's host.  Notice that there is less than a one
percent chance of guessing all wines correctly by chance alone whenever
the host offers five or more wines in the evening's selection.  Evidently,
some knowledge of wines is displayed by a taster who accomplishes this
feat with any regularity.  On the other hand, one could expect, by chance
alone, to guess none of the wines correctly about 37 percent of the time,
independent of the number of wines offered for tasting. The same situation
holds for guessing exactly one wine correctly. 
    The reason that this is so, as readers familiar with infinite series
will note, is that the alternating series contained in the parenthetical
expression in the formula for counting derangements is precisely 1/e,
where e is the base of natural logarithms (a transcendental number of
value approximately 2.71828).  That is, e^x = 1+x/1!+(x^2)/2!+(x^3)/3!+...
so that when x=-1, then e^(-1), or 1/e, is precisely the parenthetical
expression in the formula for D(n). The larger the value of n, the closer
the approximation to 1/e=0.3678797.  In a blind tasting with an infinite
number of bottles of wine, random choices will result in approximately a
0.368 probability that all will be in error! 

 Table 3.  Probability of correctly matching K wines from tasting
 a total of n wines

     2        3        4        5        6        7        8

 0   .5000    .3333    .3750    .3667    .3681    .3679    .3679
 1   .0000    .5000    .3333    .3750    .3667    .3681    .3679
 2            .0000    .2500    .1667    .1875    .1833    .1840 
 3            .1667    .0000    .0833    .0555    .0625    .0611
 4                     .0417    .0000    .0208    .0139    .0156
 5                              .0083    .0000    .0042    .0028
 6                                       .0014    .0000    .0007
 7                                                .0002    .0000
 8                                                         .0000       

     Table 3 suggests some rules of thumb about how well a taster has
done.  In a normal-sized tasting of six, seven, or eight wines,
identifying at least five of them correctly occurs less than 1% by chance
alone.  Identifying four correctly happens by chance about 2 percent (or
less) of the time.  However, identifying three correctly occurs by chance
from near five to six percent of the time:  in every 16 to 18 tastings. 
Usually there are ten to twelve tasters at a sitting in this one club. 
None to one member at a sitting rates to guess three wines correctly by
chance alone; the group usually does substantially better than this,
suggesting some expertise in identifying the wines. 

The Principle of Inclusion and Exclusion:  The Basis for Counting.
    The expression for counting derangements, as a product of n! and and a
truncated series for 1/e, has some interesting properties, most notably
perhaps the alternating plus and minus signs preceding terms of the
series.  This alternation occurs because the principle of inclusion and
exclusion has been used as the basis for the counting. 
     Readers versed in elementary set theory, Boolean algebra, or symbolic
logic, are familiar with the idea of including the intersection, and then
subtracting it out, in order to count the number of elements in
intersecting sets.  This idea, in this context, was clearly familiar to
Augustus DeMorgan in the late nineteenth century.  Indeed, in a wider
context, it dates back to the time of Eratosthenes of Alexandria and his
sieve for determining which numbers are prime:  those that are multiples
of numbers early in the ordering of positive integers are excluded.  Only
those numbers not excluded have divisors of only themselves and 1, and so
are exactly the set included as prime numbers. 
     The following example illustrates how inclusion and exclusion is used
in counting derangements; the reader interested in the general proof is
referred to Rosen (1988).  It is easy to visualize cases when n is small
using Venn diagrams--thus, the linkage between inclusion/exclusion, set
theory, and derangements becomes clear. 
   Consider for example, a tasting of two wines.  Let a be the event that
the first wine is correctly identified; let b be the event that the second
wine is correctly identified.  Draw a rectangle on a sheet of paper and
within the rectangle draw two intersecting circles, a and b--a familiar
Venn diagram.  The content of the rectangle is the universe of discourse. 
The content of circle a is the set of all events that the first wine is
correctly identified (either alone or with another), denoted N(a).  The
content of circle b is the set of all events that the second wine is
correctly identified, denoted N(b).  The intersection of the two circles
has content ab, the set of all events in which both the first wine and the
second wine are correctly identified, denoted N(ab). The set of all
derangements is the content of that area of the rectangle outside the two
circles.  The content of the two circles is the sum of the content of the
first circle plus the sum of the content of the second circle:  N(a)+N(b). 
This sum however includes N(ab) in the first term and also N(ab) in the
second term; thus, N(ab) must be excluded from the sum to get an accurate
count of the content of the union of the two circles--hence inclusion and
exclusion.  The accurate count of one or more wines correct is thus given
as N(a)+N(b)-N(ab).  The case for three circles is more complicated to
visualize but can be enumerated carefully as a set of three two-circle
problems.  With values greater than 3, visualization in this manner
becomes impossible and one must rely on extension of the notation and
visualization in the world of language rather than in the world of
pictures--both subsets of "the world of mathematics."  Indeed, geographers
interested in spatial statistics should be familiar with this issue in
using the statistical forms to capture what becomes increasingly too
complex to map. 

    These classical ideas, whether cast in the number theoretic
context of prime numbers, in the discrete mathematics context of 
inclusion and exclusion,  or in the set theoretic context of 
intersections, served once again, when cast in the context of 
derangements and the counting of incorrectness, to permit a clever 
solution to a complicated, uncontrived, real-world problem.
What this sort of analysis offers is a challenge to look at the world
in different ways:  from the use of classical theoretical material in
new real world situations, to the development of new theoretical material
which can foster further theoretical exploration and application. 

 Fisher, R. A.  The Design of Experiments.  Eighth edition, reprinted, 
   New York, Hafner and Co., 1971.  First edition, Edinburgh, London, Oliver 
   and Boyd, 1935.

 Fisher, R. A.  "The Mathematics of a Lady Tasting Tea" in
   Newman, J.R., ed.  The World of Mathematics, Simon and Shuster,
   New York, 1956 (pp. 1512-1521).

 Michaels, John G. and Rosen, Kenneth H. (eds.)  Applications of
   Discrete Mathematics, New York, McGraw-Hill, 1991.

 Polya, G.; Tarjan, R. E.; and, Woods, D.R.  Notes on Introductory
   Combinatorics, Boston, Birkhauser, 1983.

 Rosen, Kenneth R.  Discrete Mathematics and Its Applications.  
   First edition, New York, Random House, 1988.

                    INDEX TO VOLUMES I, 1990, TO V, 1994.

Volume V, No. 2, Winter, 1994.

  Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary: The Paris 
Metro:  Is its Graph Planar?
     Planar graphs; The Paris Metro; Planarity and the Metro; 
Significance of lack of planarity.

  Sandra Lach Arlinghaus:  Interruption!
     Classical interruption in mapping; Abstract variants on interruption 
and mapping; The utility of considering various mapping surfaces--GIS; 
Future directions.

  Michael F. Dacey:  Imperfections in the Uniform Plane.  Forewords by 
John D. Nystuen.
     Original (1964) Nystuen Foreword; Current (1994) Nystuen Foreword; 
The Christaller spatial model; A model of the imperfect plane; The 
disturbance effect; Uniform random disturbance; Definition of the basic 
model; Point to point order distances; Locus to point order distances; 
Summary description of pattern; Comparison of map pattern; Theoretical 
order distances; Analysis of the pattern of urban places in Iowa; Almost 
periodic disturbance model; Lattice parameters; Disturbance variables; 
Scale variables; Comparison of M(2) and Iowa; Evaluation; Tables.

  Sandra L. Arlinghaus:  Construction Zone:  The Brakenridge-MacLaurin 

  William D. Drake:  Population Environment Dynamics:  Course and 

Volume V, No. 1, Summer, 1994.

  Virginia Ainslie and Jack Licate:  Getting Infrastructure Built.
     Cleveland infrastructure team shares secrets of sucess; What 
difference has the partnership approach made;  How process affects 
products--moving projects faster means getting more public investment; 
How can local communities translate these successes to their own settings?

  Frank E. Barmore:  Center Here; Center There; Center, Center Everywhere.
     Abstract; Introduction; Definition of geographic center; Geographic 
center of a curved surface; Geographic center of Wisconsin; Geographic 
center of the conterminous U.S.; Geographic center of the U.S.; Summary 
and recommendations; Appendix A:  Calculation of Wisconsin's geographic 
center; Appendix B:  Calculation of the geographical center of the 
conterminous U.S.; References.

  Barton R. Burkhalter:  Equal-Area Venn Diagrams of Two Circles:  Their 
Use with Real-World Data
     General problem; Definition of the two-circle problem; Analytic 
strategy; Derivation of B% and AB% as a function of r(B) and d(AB).

  Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary, John D. Nystuen.
Los Angeles, 1994 -- A Spatial Scientific Study.
     Los Angeles, 1994; Policy implications; References; Tables and 
complicated figures.

Volume IV, No. 2, Winter, 1993.

  William D. Drake, S. Pak, I. Tarwotjo, Muhilal, J. Gorstein, R. Tilden.
Villages in Transition:  Elevated Risk of Micronutrient Deficiency.
     Abstract; Moving from traditional to modern village life:  risks 
during transtion; Testing for elevated risks in transition villages; 
Testing for risk overlap within the health sector; Conclusions and policy 

Volume IV, No. 1, Summer, 1993.

  Sandra L. Arlinghaus and Richard H. Zander:  Electronic Journals:  
Observations Based on Actual Trials, 1987-Present.
     Abstract; Content issues; Production issues; Archival issues; References

  John D. Nystuen:  Wilderness As Place.
     Visual paradoxes; Wilderness defined; Conflict or synthesis; 
Wilderness as place; Suggested readings; Sources; Visual illusion authors.

  Frank E. Barmore:  The Earth Isn't Flat.  And It Isn't Round Either:  
Some Significant and Little Known Effects of the Earth's Ellipsoidal Shape.
     Abstract; Introduction; The Qibla problem; The geographic center; 
The center of population; Appendix; References.

  Sandra L. Arlinghaus:  Micro-cell Hex-nets?
     Introduction; Lattices: Microcell hex-nets; References

  Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary:
Sum Graphs and Geographic Information.  
     Abstract; Sum graphs; Sum graph unification:  construction; 
Cartographic application of sum graph unification; Sum graph 
unification:  theory; Logarithmic sum graphs; Reversed sum graphs; 
Augmented reversed logarithmic sum graphs; Cartographic application of 
ARL sum graphs; Summary.

Volume III, No. 2, Winter, 1992.

  Frank Harary:  What Are Mathematical Models and What Should They Be?
     What are they?  Two worlds:  abstract and empirical; Two worlds:  
two levels; Two levels:  derivation and selection; Research schema; 
Sketches of discovery; What should they be?

  Frank E. Barmore:  Where Are We?  Comments on the Concept of Center of 
     Introduction; Preliminary remarks; Census Bureau center of 
population formulae; Census Bureau center of population description; 
Agreement between description and formulae; Proposed definition of the 
center of population; Summary; Appendix A; Appendix B; References.

  Sandra L. Arlinghaus and John D. Nystuen:  The Pelt of the Earth:  An 
Essay on Reactive Diffusion.
     Pattern formation:  global views; Pattern formation:  local views; 
References cited; Literature of apparent related interest.

Volume III, No. 1, Summer, 1992.

  Harry L. Stern:  Computing Areas of Regions with Discretely Defined 
     Introduction; General formulation; The plane; The sphere; Numerical 
examples and remarks; Appendix--Fortran program.

Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg:  The 
Quadratic World of Kinematic Waves.

Volume II, No. 2, Winter, 1991.

  Saunders Mac Lane:  Proof, Truth, and Confusion, The Nora and Edward 
Ryerson Lecture at The University of Chicago in 1982.
     The fit of ideas; Truth and proof; Ideas and theorems; Sets and 
functions; Confusion via surveys; Cost-benefit and regression; 
Projection, extrapolation, and risk; Fuzzy sets and fuzzy thoughts; 
Compromise is confusing.

  Robert F. Austin:  Digital Maps and Data Bases:  Aesthetics versus 
     Introduction; Basic issues; Map production; Digital maps; 
Computerized data bases; User community.

Volume II, No. 1, Summer, 1991.

  Sandra L. Arlinghaus, David Barr, John D. Nystuen:
The Spatial Shadow:  Light and Dark -- Whole and Part.
     This account of some of the projects of sculptor David Barr attempts 
to place them in a formal   systematic, spatial setting based on the 
postulates of the science of space of William Kingdon Clifford (reprinted 
in Solstice, Vol. I, No. 1.).

  Sandra L. Arlinghaus:  Construction Zone--The Logistic Curve.  
Educational feature--Lectures on Spatial Theory.

Volume I, No. 2, Winter, 1990.

  John D. Nystuen:  A City of Strangers:  Spatial Aspects of Alienation 
in the Detroit Metropolitan Region.
     This paper examines the urban shift from "people space" to "machine 
space" (see R. Horvath, Geographical Review, April, 1974) in the Detroit 
metropolitan regions of 1974.  As with Clifford's Postulates, reprinted 
in the last issue of Solstice, note the timely quality of many of the 

  Sandra Lach Arlinghaus:  Scale and Dimension:  Their Logical Harmony. 
     Linkage between scale and dimension is made using the Fallacy of 
Division and the Fallacy of Composition in a fractal setting.

  Sandra Lach Arlinghaus:  Parallels Between Parallels.
     The earth's sun introduces a symmetry in the perception of its 
trajectory in the sky that naturally partitions the earth's surface into 
zones of affine and hyperbolic geometry.  The affine zones, with single 
geometric parallels, are located north and south of the geographic 
parallels.  The hyperbolic zone, with multiple geometric parallels, is 
located between the geographic tropical parallels.  Evidence of this 
geometric partition is suggested in the geographic environment--in the 
design of houses and of gameboards.

  Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen:  The 
Hedetniemi Matrix Sum:  A Real-world Application.
     In a recent paper, we presented an algorithm for finding the 
shortest distance between any two nodes in a network of n nodes when 
given only distances between adjacent nodes (Arlinghaus, Arlinghaus, 
Nystuen, Geographical Analysis, 1990).  In that previous research, we 
applied the algorithm to the generalized road network graph surrounding 
San Francisco Bay.  Here, we examine consequent changes in matrix entries 
when the underlying adjacency pattern of the road network was altered by 
the 1989 earthquake that closed the San Francisco--Oakland Bay Bridge.

  Sandra Lach Arlinghaus:  Fractal Geometry of Infinite Pixel Sequences:  
"Super-definition" Resolution?
     Comparison of space-filling qualities of square and hexagonal pixels.

  Sandra Lach Arlinghaus:  Construction Zone--Feigenbaum's number; a 
triangular coordinatiztion of the Euclidean plane; A three-axis 
coordinatization of the plane.

Volume I, No. 1, Summer, 1990.

  William Kingdon Clifford:  Postulates of the Science of Space.
     This reprint of a portion of Clifford's lectures to the Royal 
Institution in the 1870s suggests many geographic topics of concern in 
the last half of the twentieth century.  Look for connections to boundary 
issues, to scale problems, to self-similarity and fractals, and to 
non-Euclidean geometries (from those based on denial of Euclid's parallel 
postulate to those based on a sort of mechanical `polishing').  What else 
did, or might, this classic essay foreshadow?

  Sandra Lach Arlinghaus:  Beyond the Fractal.
     The fractal notion of self-similarity is useful for characterizing 
change in scale; the reason fractals are effective in the geometry of 
central place theory is because that geometry is hierarchical in nature.  
Thus, a natural place to look for other connections of this sort is to 
other geographical concepts that are also hierarchical.  Within this 
fractal context, this article examines the case of spatial diffusion.
     When the idea of diffusion is extended to see "adopters" of an 
innovation as "attractors" of new adopters, a Julia set is introduced as 
a possible axis against which to measure one class of geographic 
phenomena.  Beyond the fractal context, fractal concepts, such as 
"compression" and "space-filling" are considered in a broader 
graph-theoretic setting.

  William C. Arlinghaus:  Groups, Graphs, and God.

  Sandra L. Arlinghaus:  Theorem Museum--Desargues's Two Triangle Theorem 
from projective geometry.  Construction Zone--centrally symmetric hexagons.