Class is informal lecture and discussion. 

• Initial description of interests: 5% for a one-page statement handed in on time, the first day. 
• Midterm oral presentation of about 15 minutes illustrating progress to date; written statement (on diskette) of such to be handed in.Students will be encouraged to capture their work in some sort of electronic format for this presentation (such as PowerPoint---say, 20 slides). This presentation is worth 15% of the grade. 
• Midterm written project abstract, worth 10% of the grade. 
• Presentation of the final project (15 minutes for each, as above). Presentation worth 35% of the grade. 
• Website displaying final project. (Students who do not already know how to do this can get trained in this course, if they wish). Worth 35% of the grade. 

Class time will be spent in a combination of lecture on appropriate conceptual material and in a seminar/lab format dealing with student project concerns.  It will be instructive for all students to hear the concerns of individuals and to see what sorts of issues are problematic and how they are addressed. 

All software used is for the PC.

In using this website, note that material in table boxes is "official" material; all else is remnant or under construction material.

• Class list
• Overrides
• E-mail addresses
• Web pages

• Jordan Curve Theorem; implications for mapping. 
Page 1.  Page 2. Page 3.
• Four Color Theorem and one-point compactification (stereographic projection); implications for mapping. 
Blank, outline map.  Animated map.  Page 1.  Page 2. Page 3.  Page 4.  Page 5. Page 6.  Page 7.  Six color map

• National Geographic Magazine:  100 years on CD ROM
• National Geographic Society fold-out maps on CD ROM

• ifs space:  create an html subdirectory within Public directory
• enlarge ifs space:  at login type umce.itd.umich.edu and follow the links to your individual account and change the ifs amount to which you subscribe (you get 10 free MB--extra MB are charged at some rate per month--think it used to be 9 cents per MB per month).  After you have chosen an amount, hit the Perform Change link.  Then you should be all set.  You might want to wait to see how much more you need before getting more.
• Create a new page of your own.
• Insert images as .gif or .jpg format
• Insert links
• Put tables in
• Choose backgrounds
• ftp material to ifs space:  change name of file to index.html
• getting material from the Public directory of someone else

• Dot Density Maps:  use dot density maps to measure population or other densities in irregularly shaped regions superimposed on these maps.
• This map shows a single Census variable, non-white population, mapped by Block Group.  One dot represents one percent of the population  in a Block Group.

Within a Block Group, dots are scattered randomly.  Thus, position of dots within the block group is meaningless.  However, taking a smaller scale view, using Census Tracts, of the scatter at the Block Group scale, shows where there is clustering within a tract.
Scale change can reveal pattern!
• Formulas can be written in the GIS.  Thus, comparisons can be made using dots of different colors.  Here values satisfying the inquality
Non-white - white > 0
are green; values satisfying the inequality
Non-white - white < 0
are purple.
• Spatial Autocorrelation.  How are regions clustered in space?  Are similar ones next to each other or are dissimilar ones next to each other. 
• On this map, all non-white Block Groups whose adjacent Block Groups are also only non-white are light green.
• All non-white Block Groups adjacent to white Block Groups are darker green
• All white Block Groups adjacent to non-white Block Groups are darker purple.
• All white Block Groups adjacent to only white Block Groups are light purple.
• This map shows a pattern similar to the first one, but with adjacency counted through two stages.
• In both maps, when Block Group boundaries are removed, a continuous pattern emerges.
• Policy makers and municipal authorities may find maps such as these useful

     To pick out all block groups in which Nonwhite population dominates (layer test) 
             Query|Select by Layer :  choose the blockgroup layer 
             Query|Select by Value:  choose blockgroups; select subset (or replace selected), by expression, 
                     nonwhite-white>0 
             Edit|Copy to Layer, Selected features only, copy features to new layer--region, called "test" 

     To pick out all block groups in which White population dominates (layer test2) 
             Query|Select by Layer :  choose the blockgroup layer 
             Query|Select by Value:  choose blockgroups; select subset (or replace selected), by expression, 
                     nonwhite-white<0 
             Edit|Copy to Layer, Selected features only, copy features to new layer--region, called "test2" 

     To pick out all block groups that are touching a dissimilar blockgroup:  use test and test 2. 
             Query|Select by Location|Touching:  then, use test followed by test2 to select nonwhite block groups touching white block groups. 
             Query|Select by Location|Touching:  then, use test2 followed by test to select white block groups touching nonwhite block groups. 
             Note the lack of symmetry in adjacency once content is assigned to the blockgroups even though strict adjacency is symmetric (if A is adjacent to B then B is adjacent to A). 

Iteration of technique can produce a contouring of the region; also, weights can be introduced into the formulas. 

Graph-theoretic Background
The text/article shows one way to measure spatial autocorrelation using graph theory and adjacency matrices.

Research Seminar/Lab Material

Measurement of diffusion: a conceptual approach based on the work of Torsten Hagerstrand 

Diffusion is a process in which anything that moves, or that can be moved, is spread through a space, from a source, until it is distributed throughout that space. 

How can diffusion be measured? One way is to trace the positions of things that are being diffused at different times. Torsten Hagerstrand, a Swedish geographer at the University of Lund, used the following technique to trace the diffusion of an innovation. The ‘thing’ being diffused (communicated) is an idea; the agents of diffusion, or carriers of new information, are human beings; the space in which the idea is to be diffused is a region of the world. 

Hagerstrand traces the diffusion process by imitating it with numbers. Such imitation, leading to prediction or forecasting of the pattern of diffusion, is called a simulation of diffusion. To follow the mechanics of this strategy, it is necessary only to understand the concepts of ordering the non-negative integers and of partitioning these numbers into disjoint sets. 

INITIAL SET-UP. 

In Figure 1 contains a map of an hypothetical region of the world. After one year, a number of individuals accept a particular innovation; their spatial distribution is shown in Figure 1. 

MAP BASED ON EMPIRICAL EVIDENCE--REGION INTERIOR IS SHADED WHITE; CELLS WITH NUMERALS IN THEM INDICATE NUMBER OF ACCEPTORS IN LOCAL REGION. 
 

Figure 1. Distribution of original acceptors of an innovation--after 1 year--based on empirical evidence. After Hagerstrand, p. 380. 

In Figure 2, a map of the same region shows the pattern of acceptors after two years--again, based on actual evidence. Notice that the pattern at a later time shows both spatial expansion and infill. These two latter concepts are enduring ones that appear over and over again in spatial analysis---as well as in planning at municipal and other levels. 
 

Figure 2. Actual distribution of acceptors after two years. 

Might it have been possible to make an educated guess, from Figure 1 alone, as to how the news of the innovation would spread? Could Figure 2 have been generated/predicted from Figure 1? The steps below will use the grid in Figure 3 to assign random numbers to the grid in Figure 1, producing Figure 4 as a simulated distribution, as opposed to the actual distribution of Figure 2, of acceptors after two years. 
 

• Construct a "floating" grid (Figure 3) to be placed over the grid on the map of Figure 1, with grid cells scaled suitably so that they match. Center the floating grid on a square in Figure 1 in which there exists an adopter (say 2F)...this is the first cell, working left to right and top to bottom, which contains a numeral. The numbers in the floating grid, used with a set of four digit random numbers, will be used to determine likely location of new adopters. It is assumed that the adopter in F2 (or in any other cell) is more likely to communicate with someone nearby than with someone far away; velocity of diffusion is expressed in terms of probability of contact.
• This assumption regarding distance and probability of contact is reflected in the assignment of numerals within the grid--there are the most four digit numbers in the central cell, and the fewest in the corners. The floating grid partitions the set of four digit numbers {0000, 0001, 0002, ..., 9998, 9999} into 25 mutually disjoint subsets.

• Given a set (or sets) of four digit random numbers--as below. Center the floating grid on F2. Use the first set of random numbers below. The first number is 6248 and it lies in the center square of the overlay. So in the simulation, the previous Figure 1 acceptor in F2 finds another acceptor nearby in F2. Use the map in Figure 4 to record the simulated distribution (a red entry). In cell F2, enter a red +1 to represent the initial adopter.  Together with the original adopter, there are now two adopters in this cell.

• Pick up the floating grid and center it on G2. The second random number is 0925, located in the first cell northwest of the center cell of the overlay, cell F1. In Figure 4, a 1 in cell G2 to represent the original adopter, and a 1 in cell F1  represents the later (new) adopter.
• Continue this process, shifting the overlay so that it is centered on every cell with at least one early adopter (from Figure 1). Work from left to right, top to bottom. Use a different random number for each early adopter. If there are 3 early adopters in a given cell, use three different random numbers in entering the results in Figure 4. Just use the random numbers (supposedly already randomized) reading down the column.
• How does the simulation (Figures 4) compare to the actual distribution of adopters after two years (Figure 2)?

CONCEPTUAL BASE 

Construction of the floating grid--the so-called "Mean Information Field" (MIF)

Assumption: the frequency of social contact (migration) per square kilometer falls off (decays) rapidly with distance. 

Data from an empirical study: 

Units on axes: 

x-axis--distance in kilometers 
y-axis--number of migrating households per square kilometer. 

Definition: 

An area containing probabilities of receiving information from the central point of that region is called a mean information field, represented by Figure 3. 

To assign quantitites of four digit numbers to each cell in the MIF, it is necessary to use the curve derived from the empirical study (distance decay curve). 

It is used to 

• determine the size of the MIF
• assign particular probabilities.

The size of the MIF is 25 by 25 kilometers squared. Observation is that the typical household moves no more than 12.5 kilometers. This field is then split into cells 5 by 5 kilometers squared. 

Assignment of probabilities

From the graph of distance decay, a point 10 km from the center has a value of .167 associated with it. This is in households per square kilometer; there are 25 km squared in each cell; so the point value of the cell is 25*.0167=4.17. The center cell has a value of 110--an actual number of households. 

The total point value of all cells is 248.24--note the symmetry caused by assumptions about ease of movement in all directions outward from the center. 

Divide: 4.17/248.24=0.0168---so, assign 168 4 digit numbers to the cells that are 10 km from the center (two to the north, east, south, and west of center). 

Thus, the Mean Information Field is constructed. 

Some Basic Assumptions of the Simulation Method (Monte Carlo)

Assumptions to create an unbiased gaming table: 

• the surface is uniform in terms of population and transport
• all contacts are equally easy in any direction
• there are an equal number of potential acceptors in each cell

• There is a set of carriers at the start (as in Figure 1)
• information is transmitted at constant intervals
• when carrier meets a new person, acceptance is immediate
• the likelihood of a carrier and another meeting depends on the distance between them.

Hagerstrand, Torsten. Innovation Diffusion as a Spatial Process. Translated by Allan Pred. 
University of Chicago Press, 1967. 

Puu, Tonu. Mathematical Location and Land Use Theory.  Springer-Verlag, 1997. 

Possibilities for Application 
     Measure diffusion of neighborhood news over time on city parcel map (see Lawrence and Bar Nur). 
     Simulate animal movement:  large number of random numbers in a strip=stream; tightly bounded intervals = land (for frog, for example). 

Research Seminar/Lab Material

• Animated maps--show diffusion:  change in spatial properties over time.
• Mapping of large data sets of points:  geocoding.

Fractal base

• Space-filling curves, fractional dimensions, and self-similarity--motivational material.
• Fractal design for a marina: compact use of space; reprint address supplied in class.

• Outline  Slide 1. 
• Definition of a vertex of attachment (Tutte), involving relative attachment, as opposed to the concept of cutpoint, involving absolute location. Slide 2.
• Vertices of attachment, fixed subgraphs, network heart, and maximal fragmentation. Network fragments created downstream from a central delivery system.  Slide 3.
• Slide 4 Adding redundancy:  maximizing capability of extra links.  Bipartite graphs.  Slide 5.
• Slide 6
• Illustration of abstract material applied to a hypothetical real-world water plant.  Slide 7.
• Results.  The case of Detroit.  See also reprint of article by Arlinghaus and Nystuen (from Perspectives in Biology and Medicine). Slide 8.

• Material on a geometric model of Central Place Theory (Christaller/Losch).
• Classical central place theory.
• K=3 hierarchy, marketing principle
• calculation of distance between competing next nearest neighbors
• superimposed layers of hierarchy
• K=4 hierarchy, transportation principle
• calculation of distance between competing next nearest neighbors
• superimposed layers of hierarchy
• K=7 hierarchy, administrative principle
• calculation of distance between competing next nearest neighbors
• superimposed layers of hierarchy
• Fractal central place theory.
• K=3 hierarchy
• K=4 hierarchy
• K=7 hierarchy
• Illustration of exact fit of the two approaches, showing that the fractally-generated tiles fit together precisely to form the classical central place landscapes.
• K=3 hierarchy
• K=4 hierarchy
• K=7 hierarchy

Related links--click to go to article linked to the website of the 
Mathematica website:  www.wolfram.com
Institute of Mathematical Geography:
    Beyond the Fractal
    Fractal Geometry of Infinite Pixel Sequences: "Super-definition" Resolution?"
    Micro-cell Hex-nets

Related map--partially digitized Christaller map.
    Draft map

Research Seminar/Lab Material
Access to online archive
Consideration of needs for midterm presentation

PowerPoint
Data Show and Overhead
Laptop

• Use of the Diophantine equation K=x^2+xy+y^2 to generate classes of higher K values.
• Partition of higher values into mutually exclusive, exhaustive classes of K-values.
• Complete determination of fractal generator shape, that will generate a complete hierarchy, based solely on number-theoretic properties of K.
• Solution of sets of unsolved problems.
• Fundamental theorems.

• Sample of a higher K value to illustrate the difficulty in figuring out fractal generators to create the geometrically correct spatial hierarchy...Slide 17.
• Oblique axes used to separate K values into a number of different subsets:

• Procedure for working with K values on the y-axis; note, therefore that the square root of K is always an integer.

• Chart illustrating how to determine the number of generator sides and the fractal generator shape (in terms of "hex-steps") simply from the number-theoretic properties of K.  That is, the entire central place hierarchy (its geometry) can be generated by understanding the "genetic" code embodied in K.  Slide 20.
• Algebraic Table illustrating calculations in detail:  Slide 21, Slide 22.
• Geometric Table illustrating, a set of K values on the y-axis, the determination of generator shape and hierarchy type.  Slide 23.
• Geometric chart illustrating how to handle off-y-axis K-values (non-integral square roots).  Slide 24.
• Fractal generators solve the problem (Dacey) of twin K-values:  49 can be generated by the pair (0,7) and (5,3).  The procedure separates these geometrically and generates the correct spatial hierarchies for each of them.  Slide 25.
• Fractal dimension formula used to calculate space-filling.  Slide 26.
• Some implications:  from the urban to the electronic environment. Slide 27.
• Geometric suggestion of similar procedure for an environment of squares. Slide 28.

 

Research Seminar/Lab Material
Troubleshooting of projects.

• Graph-theoretic adjacency and quantifying the join-count measure of clustering.
• Given a set of polygons and two characteristics about them (say, B and W, as above).
• There are four different adjacency patterns that can arise between adjacent polygons:
BB, BW, WB, and WW.
Each of these patterns should occur, abstractly, 25% of the time.
• On your map, use the query tool to select out all the polygons that exhibit the BW adjacency pattern (as above on the website).
• Now, count the number of adjacency patterns that actually occur as BW:  thus, for example, polygon 1, which is B, is adjacent to 7 polygons that are W.  Then move to the next polygon and make a similar count.  Notice that this counts overlapping polygons and does not merge them as a single one.  This latter observation is critical--and, it makes for difficulty in doing the counting because the GIS table only puts in one copy of a polygon even though it might contribute more than once in the counting of the adjacencies.
• Now, count the number of actual adjacencies in the entire map--call this number A.  Look at BW/A.  Similarly, look at WB/A and WW/A and BB/A.  If BW/A+WB/A=50% then the distribution is random.  If it is >50% then the dissimilar regions are clustered and autocorrelation is said to be negative.  If it is <50% then the similar regions are clustered and autocorrelation is said to be positive.
• Often it is obvious from looking at a map whether or not there is positive or negative spatial autocorrelation.  But, if you need quantitative back up for such observations, this is one way to get it.  There are others, as well.  Indeed, you can even consider designing your own index and crafting theorems about that index (as we have done in a variety of situations above).
• One way to estimate the number of adjacencies is to consider the "jig saw puzzle" model--corner pieces are adjacent to three other pieces; edge pieces are adjacent to five other pieces; and, interior pieces are adjacent to 8 other pieces.  This works well when the polygons are close to a grid pattern.  Notice that adjacency is considered to be across edges and vertices (unlike the Four Color Theorem in which adjacency is only across edges).  The reason for including the point-level adjacency is that the GISs use that when calculating "touching" polygons.
An hypothetical situation:  given the following distribution of characteristics.  The entries B and W may be calculated by simple majority, by weighting the population in each cell according to, for example, economic indicators or others.  Count the adjacency pattern--number of WW, BW, WB, and BB adjacencies.  Each is expected 25% of the time, abstractly.

W	W	W	W	W	W	W
W	W	B	B	W	W	W
W	W	B	B	B	B	W
W	W	B	B	B	B	W

Number the cells from upper left to lower right, reading from left to right.  Adjacency is measured across both lines and points.
 

cell	WW	BW	WB	BB	jig saw sum
1	3				3
2	4		1		5
3	3		2		5
4	3		2		5
5	4		1		5
6	5				5
7	3				3
8	5				5
9	3		2		5
10		5		3	8
11		4		4	8
12	4		4		8
13	6		2		8
14	4		1		5
15	5				5
16	5		3		8
17		3		5	8
18		1		7	8
19		2		6	8
20		5		3	8
21	3		2		5
22	3				3
23	3		2		5
24		2		3	5
25				5	5
26				5	5
27		2		3	5
28	1		2		3
actual sum	67	24	24	44	159

WW:  67/159 * 100 = 42.1%
WB:   24/159 * 100 = 15.1%
BW:   24/159 * 100 = 15.1%
BB:    44/159 * 100 = 27.7%

WW+BB=69.8%; BW+WB=30.2%, similar regions are clustered.  One might go on and on, but this is just a simple measure...divide the smaller by the larger--expect 1, get 0.43, here.

• Converting files from one GIS to another.

• Website work on an individual basis
• Stella, very brief view.
• Compound interest--money flows.

1000*(1+.06/12)^12=1061.68

Compounded daily, you have at the end of a year:

1000*(1+.06/365)^365=1061.83

Flows other than money might exhibit a "compounding" effect.  Generally, if P is the principal and i is the interest rate per period,

at the end of period 1, interest earned is P*i and total amount is P+P*i=P(1+i)

at the end of period 2, interest earned is the principal times the interest rate, but now the principal is the total amount (the compounding effect) from the end of period 1, or P(1+i)i.  The total amount is the principal plus the interest or
P(1+i)+P(1+i)i=P(1+i)(1+i)=P(1+i)^2.

• Books brought to class by Tonu Puu, Rhonda Ryznar, and handbook.

• Coordinates
• Latitude and longitude, simple geometry; p01, p02, p03, p04
• Geosystems latitude and longitude flyer
• Diagram, showing measuring scheme for latitude and longitude 
• How long is a degree of a parallel/meridian...graphic and calculation, p01, p02, p03 
• Map Projections
• Map Projections -- Overview with links to various projections. 
• Geosystems Handouts on Map Projections 

• A few references (samples brought to class; check in graduate library for circulating copies).
• John P. Snyder, Flattening the Earth:  Two Thousand Years of Map Projections, Chicago:  The University of Chicago Press, 1993.
• Arthur H. Robinson, Elements of Cartography, 2nd Edition shows classical constructions (there are also more recent Robinson et al. books), New York:  John Wiley, 1962.
• D'Arcy Wentworth Thompson, On Growth and Form.  Cambridge.  Various editions.

    With map projections, the challenge is to choose a projection whose favorable qualities match the aspects of the map that you wish to emphasize.  The same basic strategy applies when mapping in real-world situations and particularly in harsh environments.

MAPPING AND REAL-WORLD APPLICATIONS: 
DIGITAL MAPPING IN HARSH ENVIRONMENTS

• Adjacency matrix
• Powering of matrices
• Application from British History:  drawing space and time together
• Transportation application
(chapter in forthcoming book by Arlinghaus, S., Arlinghaus, W., and Harary, F.)

Per Hage and Frank Harary, Sturctural Models in Anthropology, 1983, Cambridge University Press. 

Per Hage and Frank Harary, Island Networks, Communication, Kinship, and Classification Structures in Ceaniia, 1996, Cambridge University Press. 

Per Hage and Frank Harary, Exchange in Oceania, A Graph Theoretic Analysis, 1991, Oxford University Press.

• Heads-up digitizing on the screen as opposed to use of a digitizer; for many purposes heads-up is just as good as using a digitizer (a digitizer allows selection of an arbitrary number of control points; heads-up does not).  A digitizer also is useful for dealing with maps of physically large dimension.  Primitive heads-up "digitizing"...useful only in harsh environments (putting a transparency on the screen).

• Inserting a raster image behind the vector map and then digitizing it is a powerful way to digitize easily.  For Atlas GIS there is a plug-in, called Raster Underlay, that works well.  For ArcView 3.0 and higher there is a script (can be found on schlossb; thanks to Marc!).

• Sample extensions of various sorts, including Projector! and Datum converter loaded up.  Cannot convert from NAD27 to NAD83 using these extensions.

• Linking spreadsheets of data you find to existing GIS databases:  Cleaning the data base to fit with a GIS base map:  make sure to leave the cursor in Column A1.  Remove all formats from the existing database.  Check for multiple entries using the same ID code.  Uniqueness of ID code is critical.

• Written communications.  Samples of journal articles, abstracts, magazines; scholarly and other documents.
• Other connections:  music clips, video clips, and so forth can be found on the web.
• Web strategy

 
            1.  If you link to another site, inform that site by e-mail and offer to disconnect if your site becomes a high volume site...at their request.
            2.  Grabbing an image
                    a.  request permission to use and behave according to answer to request;
                    b.  give cutoff date;
                    c.  put in own directory to avoid bandwidth drain
                    d.  cite source
            3.  Do not use an image if there are disclaimers on it;
            4.  U.S. Government materials; these do not generally require written requests for printed materials...so, apparently, use them...cite source and so forth and send a request if it indicates that one should be sent.
            5.  Link to search engines...do not inform them.  Cite them:  "here's what Yahoo says about Leelanau...", copy link and paste...pass along your search skills...can increase traffic on your site.
           6.  I will assemble all of it and make CDs for each.  Check out security issues and let me know.
            7.  Citation of software and electronic sources:  get citation from metadata--give links to websites.
            8.  Student UM accounts can be kept past graduation (contact Alumni Association)
            9.  Outside US copyright and related laws/ethics vary.
           10. Citations...be clear as to what kind of material is being cited--maybe one category for hardcopy, one for software.
           11. Concepts...tie to spatial concepts...broad connections send broad message: centrality, hierarchy, scale, density, transformation, distance, orientation, geodesic, minimization, connection, adjacency to name a few.
• Trouble-shooting, for your reference.
• To capture the screen:  Alt + PrintScrn--useful with Powerpoint (among others)--lets you place a large PowerPoint image into a table on a web site, for example.
• To multiply matrices in Excel (Office 97):  Select region into which matrix product is to fall, Function Wizard (Math/Trig), MMULT, fill out appropriately, and then CTRL+SHIFT+ENTER

• City of Ann Arbor maps.
• CSF web site.
• Sample CSF project:  digital mapping in a harsh environment--turning evidence from the field into GIS maps as part of a broader management information system.  The CSF approach is through Assessment, Analysis, and Action, with opportunity for Feedback. 
• Child Info--current CSF work.
• Sculpture of David Barr:  spatial relations and art.
• Splitting regions:  real-world interpretation, ArcticArc.
• One thing leads to another...a reference about the art connections.
• Another such reference--see Monograph 1 on this site.

• Agricultural Land Use Model, von Thunen (based loosely (partially) on a description in Kolars and Nystuen)
• Simplifying assumptions:
• There is a single, isolated market at the center of the region
• The region is a homogeneous plain
• Labor costs are homogeneous
• Transportation costs are homogeneous
• The system is in economic equilibrium
• The market price of a single commodity is fixed.
• Fundamental Concept
Rent (as opposed to "rental"):  net return associated with a unit of land, rather than with a unit of commodity.
• Conjecture
On any given parcel of land, the activity yielding the highest rent will dominate all others.
In what follows, the conjecture will lead to certain consequences; whether or not the conjecture can then be confronted with real-world evidence and proven as a "theorem," has been, and remains, a matter of discussion.
 
• Equations
• Net return, r, on a unit of commodity, given
p is the market price
c represents production costs
d is distance to market
t is transport rate
(hence, dt is total transport cost)
is:
r = p - c - dt
• Rent (net return), R, on a unit of land is given by r times the yield of the land (over a fixed time frame, such as a year).  If Y is the yield,
R = rY = Y(p - c) - Ydt.
Rewriting the equation, setting
P = Y(p - c), profit on amount of crop produced per acre (market margin)
T = Yt,
the equation becomes:
R = P - Td
• Implications of equations
• Whether the return is measured for unit of commodity or for unit of land, the measure of distance is invariant.
• In rewriting the rent equation, it becomes clear that the equation is of the linear slope-intercept form y = mx + b.  That is, rent is expressed as a linear function of distance R = f(d), in which the slope of the line representing the equation is -T and the intercept of the line with the vertical axis is P.  The units along the horizontal axis are "distance" and those along the vertical axis are "rent".
• Implications of the conjecture
• Different crops give rise to different equations, based on the idea that a given parcel will produce that which generates the greatest net return (and the variability of transport rates for different types of crops).
• Given an agricultural system with three crops:
R = P1 - T1 d
R = P2 - T2 d
R = P3 - T3 d
When the crop that yields the greatest rent is produced on all parcels of land, the resulting land use pattern is one of concentric circles centered on the market.
• Directions for extension
• Test model against reality: "prove" the conjecture.  Here, GIS can provide easy testing of large amounts of data.
• Vary the simplifying assumptions:  when, for example, is swamp reclamation feasible?
• Create non-linear split-domain Thunen functions.  Here, concepts of the calculus become important.

• Industrial Location (in the manner of Weber)
• Issue:  Cities are foci of agricultural consumption (marketplaces) and therefore have a strong influence on rural land use patterns.  Consider the parallel situation for industry.  Cities are marketplaces for industrial output.  What influence does that fact have on the location of facilities for resource processing and on consequent industrial location patterns?
• Simplifying Assumptions:
• Assume a uniform plane
• Assume a single urban market at a point
• Assume resources are point locations
• Assume equal transport costs per unit of weight
• Underlying Concept:  The point of lowest total transport cost will serve as an optimal location site for an industry.  This total is a sum of cost distances from resources and market.
• Mechanical model, Weber's weight analysis:  the accompanying figure shows a mechanical model of weights on strings.  Weights may represent cost in transport; the knot settles at the point minimizing total transport costs among the locations (holes) on the uniform (circular) plane region.
• Analysis:  Given one market, M, and two resource supply sites, R1 and R2
• Contour the plane surrounding each point according to transport costs from that point.  Given the simplifying assumptions, the pattern will be a set of concentric circles surrounding each point, evenly spaced, of increasing value as one moves outward from the resource site or marketplace.
• Model with equal transport costs (after Haggett)
• isotims:  contours of equal transport costs surrounding single points
• isodapanes:  contours of equal aggregate transport costs among a set of locations.
• Model with unequal transport costs (modify the simplifying assumptions) (after Haggett)
• isotims:  contours of equal transport costs surrounding single points
• isodapanes:  contours of equal aggregate transport costs among a set of locations.
In both cases, an optimal location (according to the Underlying Concept) is found within the lowest contour on the resulting topographic map of isodapanes.
• Challenge to use GIS to consider this sort of analysis.  Modifying constraints on the fly, and seeing immediate spatial consequences, offers richness in the analysis previously impossible.
• References:
von Thunen, J. H.  von Thunen's Isolated State, trans. Carla M. Wartenberg, edited with an Introduction by Peter Hall, London: Pergamon Press, 1966.

Weber, Alfred.  Alfred Weber's Theory of the Location of Industries, trans. C. J. Friedrich.  Chicago:  The University of Chicago Press, 1928.

Textbooks (out of print; contact instructor)
Kolars and Nystuen, Human Geography; Haggett, Peter, Geography:  A Modern Synthesis; Abler, Adams, and Gould, Spatial Organization.

Tying up loose ends 
• Konigsberg bridge puzzle (1636, Leonhard Euler)--there is no circuit (same starting and ending point) that crosses each bridge exactly once.  Theorem that covers this states such a traverse is possible if and only if each node in the associated graph is of even degree (counts number of "ins" and "outs" to and from each node).
• Question about pinching together a cylinder in order to add the north pole to a cylindrical projection--if done in Euclidean 3-space, then the cylinder is truncated and more is lost than gained; if done with a point at infinity, we move into the non-Euclidean realm and one then needs to consider matters such as uniqueness of that point (will the cylinder then become a torus?).
• Troubleshooting of individual projects.