An electronic journal of geography and mathematics

Earth:  with 23.5 degrees north latitude as the central parallel.

Volume X
Number 1
June, 1999

Volume X, Number 1, 1999
Front matter: Summer, 1999.  Editorial Board, Advice to Authors, Mission Statement.
Dedication  To John D. Nystuen, on the occasion of his retirement from The University of Michigan
John D. Nystuen, Metropolitan Mining: Institutional and Scale Effects on the Salt Mines of Detroit
Sandra L. Arlinghaus and William C. Arlinghaus. Animaps III:  Color Straws, Color Voxels, and Color Ramps.
Richard Wallace.  Book Review:  Andre I. Khuri, Thomas Mathew, and Bimal K. Sinha, Statistical Tests for Mixed Linear Models, John Wiley & Sons, 1998, 352 pp., $69.95 (cloth).
Seema Desai Iyer.  Book Review:  Castells, Manuel (1996). The Rise of the Network Society (The Information Age: Economy, Society and Culture, Volume 1).
Malden, MA: Blackwell Publishers, Inc. (556 pages, bibliography 51 pages, index 23 pages).

SUMMER, 1999

Founding Editor-in-Chief:
     Sandra Lach Arlinghaus, University of Michigan;
        Institute of Mathematical Geography (independent)

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.

Book Review Editors:
     Richard Wallace, University of Michigan.
     Kameshwari Pothukuchi, Wayne State University

Web Design:
     Sandra L. Arlinghaus
     (with early input from William E. Arlinghaus).


Electronic address:

     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

     Back issues of Solstice are available on the WebSite of the Institute
of Mathematical Geography, and at various sites
that can be found by searching under "Solstice" on the World Wide Web.
Thanks to Bruce Long (Arizona State University, Department of Mathematics)
for taking an early initiative in archiving Solstice using GOPHER.

     The electronic files are issued yearly as copyrighted hardcopy in the
Monograph Series of the Institute of Mathematical Geography.  This
material will appear in a Volume in that series, ISBN to be announced.
To order hardcopy, and to obtain current price lists, write to the
Editor-in-Chief of Solstice at 1964 Boulder Drive, Ann Arbor, MI 48104, or
call 734-975-0246.
     Suggested form for citation:  cite the hardcopy.  To cite the
electronic copy, note the exact time of transmission from Ann Arbor, and
cite all the transmission matter as facts of publication.  Any copy that
does not superimpose precisely upon the original as transmitted from Ann
Arbor should be presumed to be an altered, bogus copy of Solstice.  The
oriental rug, with errors, serves as the model for creating this weaving of words and graphics.

It is with the greatest of pleasure that this issue of Solstice is dedicated,
on the occasion of his retirement from The University of Michigan,
to the career of
John D. Nystuen,
pioneer in spatial analysis and in modern mathematical geography.
John Nystuen outside IMaGe in 1991.

Metropolitan Mining: 
Institutional and Scale Effects on the Salt Mines of Detroit

John D. Nystuen 
The University of Michigan 
Revised June 1999; based on earlier work as noted at the end. 


Mining, as with most industrial activities, is constrained by logistics, which involves technological matters of transportation, material conversion and energy costs. Convention and law also influence the activity. These are institutional matters involving mineral rights and access to resources. Both logistical and institutional configurations exist in a space/time context and in metropolitan areas, where geographic space is a complex mosaic of private and public property, the limits to an industrial activity are nicely illustrated in the example of the salt mines of Detroit. 


Metropolitan mining refers to industries that extract minerals or other materials from locations within highly urbanized regions. The example of a metropolitan mine in Detroit reveals some interesting interplay between technological and institutional constraints. Industrial activities operate in a space/time envelope partially confined by logistics, the physical task of moving people, things and energy through space and time. Physical movements depend on the speed and cost of transportation and communication--technological matters employed to overcome the cost of space. Social/economic behavior is also constrained by conventions; bound by abstract, imaginary barriers that establish entitlement and prescribe behavior--institutional matters that have temporal components associated with expectation and risk. Social behavior often has geographical manifestations. We know how to behave on account of where we are: on sacred ground, on private property, or in a public forum. Private property, although a very abstract and multi-dimensional concept, often includes strong geographical connotation. In this paper we make reference to one kind of private property, mineral rights used to gain access to the salt deposits under Detroit. 

Every geographical process whether social or physical also has an intrinsic or operational scale associating relative size of the elements contained in the process. Two components are involved: (a) the diameter of the system which may be defined as the longest extent or distance between interacting parts and (b) the unit size or smallest extent of an elementary element. An elementary element is a part which is treated as a single unit in the system and which cannot be subdivided. This requires some elaboration. Everything and every action take some minimum time and space. We allocate time and space among activities according to their needs. We jostle about and shoulder one another aside to arrange our affairs spatially and temporally---that is geographically. If by chance or otherwise some elements cannot maintain their minimum unit size, they cease to exist, in Hägerstrand's words; everything has a minimum extent and duration, a kind of kernel or minimum unit of existence (Hagerstrand, 1970). This is true of abstract, institutional content of our environment as well as in the logistical matters we face. The interplay between space and time and scale are nicely illustrated in the Detroit Salt Mines of the International Salt Company. 

The Geography of Salt Mining

The Detroit salt mine was started 1906 and finally closed operations in 1985 after millions of tons of salt had been removed. The work created extensive man-made caverns under the city that remain today. The Detroit mine has a rather complex shape that is intriguing to geographers and that calls for some explanation (Figure 1). 

In mining the first issue is the matter of the location of the natural resource. As it happens salt deposits underlay much of the Michigan Basin and extend all across the Midwest into New York State. Anywhere in this region "straight down" carries one back in time. Nearly one quarter of a mile under Detroit we are brought back 390 million years into the Paleozoic Era to a Silurian Sea in which deep salt deposits were made in a series of layers now covered with shale, limestone and sandstone overburden. The Detroit salt mine worked a 30 foot thick seam of rock salt at 1135' below the surface, one of several layers of salt (Figure 2). The top 90 to 100 feet from the surface is unconsolidated glacial drift full of water under high pressure and permeated with hydrogen sulfide. This proved to be a difficult mix of material through which to drive a mineshaft. 

The mineshaft was started in 1906 eleven years after the salt was discovered under the city. The mineshaft proved very difficult to dig and eight men lost their lives in the effort. By 1914 after bankruptcy and acquisition by a rival salt company the mine began production and shortly reached a production of about 10,000 tons per year. A second larger shaft 16 feet in diameter was sunk in 1922. Despite this width the largest opening is only 6 foot by 6-foot square as room for ventilation, the salt skips (the containers to lift the salt), power lines, elevators for men and equipment must all fit in the shafts. Both shafts were used. Large diesel trucks, front loaders, drilling rigs, conveyor machines, milling machine and machine shops to maintain them are all underground brought down the narrow shafts in pieces and even cut into pieces by acetylene torch and reassembled underground. 

At a regional scale the location of the mine may be taken as market oriented. Because the resource is spatially ubiquitous, that is, available anywhere in the Midwest, proximity to the highest market potential dictates choice of location, hence the metropolitan location. There are only four rock salt mines in the northeast quarter of the country. Each is in a metropolitan area with an upstate New York site as an exception. The cost of sinking the shafts appears too great for widespread use. For some decades there has been an alternative to open shaft mines. Brine wells are more common. In such installations the salt is removed by pumping hot water into the salt bed and withdrawing brine. Large Midwestern industrial users such as chemical companies can sink their own brine wells and are no longer customers for the rock salt mines. The overwhelming proportion of the rock salt is used to clear road of ice during winter months. That market is seasonal and varies with the severity of the winters. As it is always more efficient for an industrial operation, including mining, to have steady production, the older, closed parts of the salt mine are used for storage of processed salt. Storage is a time transfer process. The mine is very dry which means that the stored salt does not deteriorate over time. 

The salt sells for about $18 to $20 per ton f.o.b. the mine. (f.o.b. means free on board--the customer pays for hauling it away). The mining company sells either f.o.b. or delivers and adjusts the price accordingly. They lease or contract for trucks from hauling companies when they offer to deliver. Transportation costs vary by size of truck, 25 cents per ton-mile for trucks with 10 to 15 ton capacity down to a minimum of 12 cents per ton-mile for truck/trailer rigs with 55 to 60 tons capacity. Great Lake carriers are much less, perhaps 3 cents per ton-mile. At these prices it doesn't take much distance to double the price of the salt: 167 miles by truck, 667 miles by water (Figure 3). The diagram gives some sense of when customer, principally municipalities, county and state road maintenance departments, will forego salt and turn to sand and plows to some other alternative to clearing streets in snow emergencies. 

In 1960, the International Salt Company opened a new rock salt mine in Cleveland. It is located exactly on the shore of Lake Erie. This location has both logistical and institutional advantages. They are able to ship salt in bulk by lake carrier at much reduced costs per ton-mile than overland shipments. Cleveland is able to ship salt past Detroit to lake ports in the upper Great Lakes at costs lower than it could be delivered from the Detroit mine. The Detroit mine has shipped by water in the past and is not much more than one-half mile from the turning basin of the River Rouge where they have loading facilities. This half-mile gap must, however, be bridged by trucking and loading costs that exceed the cost of the sixty-eight nautical mile shipment from Cleveland. The Cleveland mine also, by mining out under Lake Erie, leases mineral rights from a single owner, the State of Ohio. This permits a more efficient mine layout. In 1987 the Detroit mine ceased to operate, put out of business in part by its awkward shape. The Detroit Metropolitan area now gets its salt from Cleveland and Windsor, Ontario, where in the later location the mine is of optimal shape and extends out under the Detroit River on the Canadian side. In Canada, mineral rights laws were more favorable to the salt companies. 

Site Conditions. Pure salt crystals make a very hard rock and hard rock mining techniques must be employed in the mine. The active mine face is undercut ten feet, (the undercut is called a kerf), powder holes drilled and the rock salt blasted free. Very large trucks, primary crushers, conveyor belts and milling machines all underground, are used to create the finished rock salt graded by several sizes. Up to twenty-five percent of annual production can be stored in the empty rooms of the mine (200,000 to 250,000 tons) as annual productions of up to one million tons are mined in normal years. The active face of the mine is 23 to 25 feet high and 50 to 60 feet wide. Eight hundred to nine hundred tons of salt are freed in each shot. Salt weighs about one ton per cubic yard in place. The seam being worked yields about 40,000 tons per acre of recoverable salt. These dimensions are important when we turn to considering technological and institutional scale effects in the mining operation. Long rooms, fifty feet wide and twenty-five feet high are formed by the mining operations. Huge salt pillars sixty by eighty feet on a side are left in place to hold up the roof. The salt is strong enough that no shoring is necessary. No cave-in has ever occurred. This type of hard rock mine is called "pillar and room" (Figure 4). Sixty-two percent of the salt is recovered using this method. The mine has been operating since 1914. Something like 1700 acres have been excavated. Figure 1 shows a plane view of the mine as it exists today. These boundaries are not exact because the mine managers were reluctant to release a map of the mine to me for reasons that will become obvious as I continue this story. Notice the irregular shape of the mine and that it extends essentially only westward from the mineshaft. This does not make sense technologically. Logistics are a big part of the mining costs and logistically the best shape would be compact, nearly circular with the mineshaft at the center. Two transport technologies are employed underground, trucks costing at best perhaps 20 cents per ton-mile and conveyor belt at perhaps eight cents per ton-mile. The spatial problem is to minimize the sum of these two costs from the active mine face to the shaft. The radius of a circle of 1700 acres is 0.92 miles. The active mine face is currently over four miles away: 434% farther than the ideal. If half the distance were by truck and the rest by conveyor belt this distance would cost 35.3 cents per ton more than the ideal. For 800,000 tons per year that is an additional $280,000. So why does the mine have this shape? The answer is institutional and relates back to that concept of the kernel of existence. 

This permits a more efficient mine layout. In 1985 the Detroit mine ceased to operate, put out of business by its bad shape. The Detroit Metropolitan area now gets its salt from Cleveland and Windsor, Ontario, where in the later location the mine is of optimal shape and extends out under the Detroit River on the Canadian side. In Canada, mineral rights laws were more favorable for the salt companies. 

Mineral Rights and Transaction costs. In Michigan every landowner having free title to his or her land owns the mineral rights for all minerals beneath it. The salt company will offer around $2000 per acre for mineral right or about $0.05 per ton. This is highly variable depending upon the size and strategic position of the property under consideration. Under some circumstances the company might be willing to buy the land outright only in order to obtain the mineral rights. In other circumstances mineral rights would be worth very little. The size and location of the property is the key to its value. This can be understood through analysis of institutional factors. There is a minimum transaction cost associated with each mineral rights transaction. First negotiations must be made, and upon agreement, the transfer or leasing of mineral rights must be assigned in each property deed and recorded at the county court house records office. If two lawyers are involved, one for each side, there exists a minimum institutional friction for each transaction that in a rock bottom estimate would total more than one thousand dollars at ninety to a hundred dollars per hour per lawyer for a day or day and a half of work. It could be much more. If the legal fees were $1200, this translates into all the value of the mineral rights for a lot just under a two-thirds of an acre in size. Most city lots in high-density residential blocks are 1/8 to 1/6 of an acre in size. The mineral rights for a lot 1/6-acre in size are worth perhaps $333. Would you like to sell your mineral rights to the salt company? Never mind that your lawyer would probably get most of this payment. 

There is another problem. Time as well as space is involved. For security in continuity of operations the company is interested in procuring mineral rights ten years or more in advance of actual use. That means that they are not willing to pay more than the present worth for the mineral rights they will use in ten years. For a lot that has about $400 worth of mineral rights--a lot 91 feet on the side or just under 1/5 acre, the present worth at 5.5% interest rate is $234 -- perhaps under two hours of a lawyers time. Under these conditions, the salt company preferred to take options on the mineral rights and promise to pay royalties whenever they actually mined under your property. Under this arrangement they were willing to offer to pay $2000 per acre in ten years or so, and to make cash payments as the salt is mined. The seller needed to evaluate this option based on the present worth of that future payment using the same interest formula. For both parties the transaction costs (i.e., the lawyers' fees) had to be paid up front. There were also some accounting expenses associated with this procedure and again it did not pay to deal with small landowners--the mining company was not interested in anything under an acre, in fact, deals involving several acres at a time are clearly preferable. Therefore residential land use marks the limit to the mining activities. This is abundantly clear from looking at the map. The shape of the mine is understandable when considering institutional constraints in addition to technological ones. Although the mine management did not discuss the matter with me, some simple calculations are sufficient to give a sense of the minimum property size a metropolitan mining company would be willing to consider for acquiring property rights. The present value of $2000 for each acre of mineral rights to be used in ten years is $1171 at 5.5% interest. If legal and closing costs were to be kept at, say 5% of total mineral rights costs, then supposing an efficient law firm could handle the matter in one day, $800 worth of legal fees in current money would require a $16,000 transaction to be attractive to the mine operators, ($800 = .05P, P= $16000). The figure $16,000 divided by $1170.86 (present value per acre) yields 13.7 acres at 5.5% interest and 22.8 acres at 11% interest. The salt company would not be interested in any place under twelve or so acres with more normal interest rates and nothing under twenty-two acres given high interest rates characteristic of the 1980's unless some special strategic location existed that might affect mine operations. Both a unit space and a time duration, twelve acres and ten years, can be seen to affect the overall dimensions and actual shape of the mine. 

There are more subtleties. Space and time combine to create velocity. The velocity at which things happen affects geographic patterns as well. Twenty acres are mined in a typical year amounting to 800,000 tons of salt. The tonnage must move from the active face of the mine to the shaft. Some time ago the company negotiated a purchase of mineral rights from a group of small lot owners in which all had to agree to the sale of their mineral rights or no deal was to be made. The plan was to cut off about 3000 feet of underground travel route to reduce underground transportation costs. Diesel trucks with 22 ton load capacities are used underground along with conveyor belts. I estimate the diesels may cost 20 cents/ton-mile (I did not have exact figures from the salt company). A saving of 3000/5280 of a mile at 20 cents/ton-mile would be 11.4 cents per ton. A similar saving if the conveyor system were extended through the bypass would be 4.5 cents per ton. At 800,000 tons per year the savings in truck operating costs would be $91 thousand per year. An amount of $36 thousand would be saved if the conveyor system were shortened by this much. The bypass opened up approximately one-quarter square mile (160 acres), which if mined at about 20 acres per year would mean eight years of operation. What is the present worth of a stream of income (savings) of $91 thousand per year for eight years? The interest formula for an annuity or stream of savings for this period yields a present value of $576,000. 

The bypass involved extending mining operations down a residential street where property owners on both sides owned the mineral rights to the center of the street. A corridor 200 feet wide and 1900 feet long was sought. The by-pass corridor is shown in Figure 1 located on the north side of the central part of the mine. It makes the shape of the mine more complex topologically by creating a hole in the shape. The corridor contains 8.17 acres. Using the truck technology, the by-pass was worth $70 thousand per acre in savings. The strategic location was thus worth thirty-five times the usual mineral rights payments. To realize this fact one must account for the effects of space and time simultaneously, that is, by considering the velocity of activities. Gross sales at twenty dollars a ton and 800,000 tons per year amount to sixteen million dollars. Savings of $91,000 by better spatial arrangement within the mine amounts to one half percent of the gross per year. I have no idea what profit margins for a mine of this sort amount to, but I suspect five percent of gross might be generous. Perhaps the corridor was worth ten percent of profits per year. It pays to pay attention to geography. The prospects for the mine are good insofar as acquisition of property rights are concerned. The mine abuts parcels that exceed ten acres in size at several points on its perimeters. These in turn open up to territory several times the area that has been mined up to the present. The access was greatly improved once they acquired mineral rights under a railroad right-of-way and more recently under the Interstate Highway in the City of Allen Park. These linear forms create many links to large parcels under various industrial properties in these communities. I conclude that the mining company had opportunities for acquiring mineral rights sufficient to carry them well into the next century. The outside dimension and shape of the mine can thus be seen to be a function of an elementary element that would be no smaller than ten acres and which, in turn, depended upon institutional factors interacting with logistical considerations. Certain strategic locations might be exceptions. 

Metropolitan salt mining may seem to be a rather special topic but I detect a generalization here that sheds light on how spatial and temporal parameters can be used in understanding other urban patterns. I have thought it odd that high-density town house developments have sprung up at the edge of metropolitan regions and unfortunate also because of the increase in travel effort this pattern creates. I suspect that changes in construction costs and in the working of the financial market have increased the size of minimum viable developments to the extent that suitable large properties can only be found on the edge of the metropolitan areas. The urban region is a mosaic, made up of discrete elements and not a continuous surface as is implied in certain urban models. When a system is made up of discrete units the minimum viable unit space for an activity affects the larger dimensions of the activity and should enter into calculations used to explain the general patterns. 

In spatial terms alone the key variable is density of the activity measured in dollars per unit area. In temporal terms the key variable is the annual return on initial investments and transaction costs measured as an intensity or dollar amount per unit of time. In simultaneous space and time the measure is dollars per unit area per unit time. Where/when there is a moving front as in the case of metropolitan mining or subdivision expansion, the key variable is a velocity or rate of advance. In discrete space/time, one must take into account an appropriate estimate of the minimum extent and duration of all elementary elements (kernels) in the system and their interactions. 


Rodolfo J. Aguilar (1973) Systems Analysis and Design in Engineering, Architecture, Construction and Planning, Englewood Cliff, N. Jersey: Prentice Hall, Inc. 

Hagerstrand, Torsten (1970) "What about people in Regional Science?" Papers, Regional Science Association, v. 24, pages 7-21. 

International Salt Company (1971) Salt City Beneath Detroit (brochure). 

I extend thanks to Mr. Jim McDonald, the Manager of these mines, for the information he shared with me about their operations.

"Metropolitan Mining: Institutional and Scale Constraints on the Salt
Mines of Detroit", paper presented to the  Northeast Regional Science
Conference, Hunter College, New  York City, May 7, 1983.

  A version of the paper was also presented by the author as part of a
lecture entitled: "Place, Location, Time and Timing: Form and Texture in Space/Time" given as the Reginald G. Golledge Invited Lecture, Department of Geography, University of California, Santa Barbara, April 8, 1999.

Sandra L. Arlinghaus
The University of Michigan
William C. Arlinghaus
Lawrence Technological University


    Background is important not only in color visualization but also in fostering a deep understanding of a variety of abstract concepts. One place to begin any background study of color is with the four-color problem (now, "theorem;" Appel and Haken, 1976). For centuries, mathematicians have concerned themselves with how many colors are necessary and sufficient to color complicated maps of many regions. (Two regions are said to be adjacent, and therefore require different colors, if and only if they share a common edge; a common vertex, alone, is not enough to force a new color.) The answer depends on the topological structure of the surface onto which the map is projected. When the map is on the surface of a torus (doughnut) seven colors are always enough. Surprisingly, perhaps, the result was known on the torus well in advance of the result for the plane (then again, the plane is unbounded and the torus is not). The same number of colors that work for the plane will also work for the sphere (viewing the plane as the sphere with one point removed). However, it was not until the last half of the twentieth century, aided by the capability of contemporary computing equipment to examine large numbers of cases, that the age-old "four color problem" became the "four color theorem." Appel and Haken (1976) showed that four colors are always enough to color any map in the plane (hence the University of Illinois postage meter stamp of "four colors suffice" announcing this giant result).

    The world of creating paper maps and publishing them has traditionally been one that is black and white: color processing is expensive and often has been prohibitive. Nonetheless, cartographers, photographers, and others have developed a number of strategies for considering color, independent of how many colors suffice to color a map in the plane. Indeed, Arthur Robinson noted (Robinson, 1960, p. 228),

"Color is without a doubt the most complex single medium with which the cartographer works. The complications arise from a number of circumstances, the major one being that even yet we do not know precisely what color is.  The complexity is due to the fact that, so far as the use of color is concerned, it exists only in the eye of the observer." Like the mathematician, the cartographer, too, has significant unsolved problems associated with the concept of color.

    Thus, color choice and use is typically tailored to "standard" reactions, by a typical observer, to color. The effect of color on an observer is often captured using the following terms as primitive terms: hue, saturation, and luminosity.

In the more contemporary environment of the desktop computer, users of various software packages in common use are exposed to the hue-saturation-luminosity set of primitive terms on a regular basis. In addition, they see the RGB (Red-Green-Blue) description also using three primitive terms and the printer's (photocopier's) environment of separations into layers based on CMYK (Cyan-Magenta-Yellow-Black). A color wheel (Figure 1) can help the user to design strategies for color change: to decrease magenta, for example, subtract magenta, or add cyan and yellow (opposite from magenta).


    One obvious way to look at color, given two sets of primitives each with three elements, is as an ordered triple in Euclidean three-space. Indeed, that is how color maps are set up in contemporary software such as Netscape, Microsoft Office, and so forth. Hue is measured across a horizontal x-axis (Figure 2) and saturation is measured along a vertical y-axis (Figure 3). The result is a square or rectangle with vertical strips of color corresponding in order to the pattern on the color wheel. A third axis of luminosity (a gray scale) is often seen as a strip to the right of this square (Figure 4). It serves to match the selected color against light/dark values.

Figure 2. Animated color map: shows change in resulting hue as one moves across the x-axis.
Figure 3. Animated color map: shows change in resulting saturation as one moves along the y-axis.
Figure 4. Animated color map: shows change in resulting luminosity as one moves along the z-axis.
These animated color maps fix two dimensions and allow a third one to vary.  That variation shows up in the small rectangle to the lower left of the color map and also in "straw" to the right of the plane region.   In all three cases, hue is the variable mapped on the horizontal axis, saturation is the variable mapped on the vertical axis, and luminosity is the variable mapped in the straw to the right.  Thus, in Figure 2, luminosity is fixed at 120 as indicated by the small arrow to the right of the straw.  Saturation is fixed at 180 along the left side of the rectangle.  Only hue is allowed to vary, as shown in the progression of the crosshair movement.  The small rectangle to the lower left of the color map changes in color to show the hue of the current position of the crosshair.   Thus, to see a hue-straw, one would need to take all 256 colors available in the flashing rectangle and stack them up in order of progression.  Similarly, one can allow saturation to vary and keep hue and luminosity fixed (Figure 3).  When luminosity is once again fixed at 120, and hue at 180, a structurally identical situation occurs (to that above).  To see a saturation-straw, one would need to take all 256 colors available in the flashing rectangle and stack them up in order of progression.  The final case, in Figure 4, keeps hue and saturation fixed and allows luminosity to vary.  Thus, one imagines a point in the base hue/luminosity plane fixed at (180, 120) and variable height shown in the luminosity straw reflecting changes in the single color-point as one alters luminosity.  In this latter case, the obvious straw that appears is in fact the actual luminosity straw sought.  In two cases, there is no evident straw of color and in the third there is; visualization is not impossible but it is made difficult.
     An alternate way to visualize all of this is to think of a cube (in 3-space) of 256 units on a side.  Label the x-axis as hue, the y-axis as saturation, and the z-axis as luminosity.  Then, draw a plane parallel to the base plane (bottom of the cube) at height 120.    Fix lines at 180 within that plane:  one with hue=180 and one with saturation=180.  These two lines trace the paths of the crosshairs, respectively, in Figures 2 and 3.  What the cube approach also shows clearly is that there are really a set of voxels (volume pixels) making up the cube:  there are 256 straws available for each of the three variables.  Since 256=2^8, there are therefore 2^8 * 2^8 * 2^8 = 2^24 = 16,777,216 voxels within the color cube (note the reliance on discrete mathematics and discrete structuring of a normally continuous object).
     The notion of looking only at voxel subsets within a single plane parallel to a face of the cube is limiting within this large, but finite, set of possibilities.  In choosing sequences of color there may well be reason to follow a diagonal, to tip a plane, or to find various other ways of selecting subsets of color, as a smoothed color ramp, from this vast array.  It is to these possibilities that we now turn.

      The problem of finding color ramps linking one color to another can be captured simply as follows.  To find a ramp joining two colors, A and B, first represent each of A and B as an ordered triple in color voxel space.  Then, the problem becomes one of find a path from A to B.  Because one is limited to integer-only arithmetic, divisibility of distances often will not be precise; thus, one is thrown from the continuous realm of the Euclidean metric into considering the non-Euclidean realm of the Manhattan metric (of square pixel/cubic voxel space).  Algorithms for finding shortest paths between two arbitrary points using integer-only arithmetic will therefore apply to colors mapped in color space as well as to physical locations mapped on city grids.  To see how these ideas might play out with colors, we consider an example that will lead to an animated color ramp.

Find a path through color voxel space from (80, 100, 120), shown below as a medium green

to (200, 160, 60), shown below as a fairly deep purple.

One set of points through which to pass, spaced evenly (not always possible), is given in the table below.  The left-hand column shows values of hue, the middle column values of saturation, and the right-hand column values of luminosity.

80 100 120
90 105 115
100 110 110
110 115 105
120 120 100
130 125 95
140 130 90
150 135 85
180 140 80
170 145 75
180 150 70
190 155 65
200 160 60

Figure 5 shows an animation using the path outlined in the table above.  The crosshairs show the movement along the path while the flashing color in the rectangle below the color map shows the associated color ramp.  Clearly, the choice of path is not unique:  geodesics are not unique in Manhattan space.  From this analysis, we see that the following theorem will hold.

The determination of color ramps joining two colors is abstractly equivalent to finding paths in Manhattan space between two arbitrary points (where geodesics are not unique).

    One might wonder what would happen when other color characterization schemes are considered.  We suspect that a similar analysis will follow.  For, in a related, but not identical, manner the RGB scheme may also be represented as describing color using 3-space. In that scheme, the gray scale comes out as a 45 degree diagonal. Computer scientists offer a color code containing six alpha-numeric characters, appearing in pairs of hexadecimal code that also serve as a 3-space. Generally, though, the various schemes offer only visual slices through this three-dimensional color space along axes or in other "expected" ways. Different vantage points offer different perspectives, however. Pantone color formula guide books offer one physical set of straws by which to probe 3D color space. The theorem above offers a comprehensive mathematical set.


Appel, K., and Haken, W. A proof of the 4-color theorem. Discrete Mathematics, 16, 1976, no. 2 (and related references).

Arthur H. Robinson, Elements of Cartography, 2nd Edition, 1960. New York: Wiley.

Sandra L. Arlinghaus, William C. Arlinghaus, John D. Nystuen.  The Hedetniemi Matrix Sum:  An Algorithm for Shortest Path and Shortest Distance, Geographical Analysis, Vol. 22, #4, Oct. 1990, pp. 351-360.

Sandra L. Arlinghaus, William C. Arlinghaus, John D. Nystuen.  The Hedetniemi Matrix Sum:  A Real-world Application, Solstice, Vol. I, No. 2, 1990.

Sandra L. Arlinghaus, William C. Arlinghaus, John D. Nystuen.  Discrete Mathematics and Counting Derangements in Blind Wine Tastings, Solstice, Vol. VI, #1, 1993.

Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary, John D. Nystuen.  Los Angeles 1994--A Spatial Scientific Study, Solstice, Vol. V, #1, 1994.


Andre I. Khuri, Thomas Mathew, and Bimal K. Sinha, Statistical Tests for Mixed Linear Models, John Wiley & Sons, 1998, 352 pp., $69.95 (cloth).

The decomposition of variance components is an essential part of data analysis for researchers employing mixed models i.e., those containing both fixed and random effects. In recent years, analysts have made significant breakthroughs regarding statistical tests for such models. Statistical Tests for Mixed Linear Models, written by André Khuri and his co-authors, presents a comprehensive, mathematical overview of these methods and extends past work to include hypothesis testing.

Traditional, analysis of variance (ANOVA) models are well developed for fixed effects models, which are those in which the researcher has complete control over assignment of factors and factor levels. For models with random effects (as often exists in observational studies, where for example, subject educational level varies but is not under the control of the researcher), too, ANOVA models have long existed. Models with both types of effects, however, present some special challenges, and Statistical Tests for Mixed Linear Models lays out appropriate solutions. 

Covering both balanced (those with equal numbers of observations in all subclasses) and unbalanced models (those with at least one subclass with a different number of observations compared to the others), Statistical Tests for Mixed Linear Models presents derivations of both exact and optimal tests for variance component models, as well as guidance on using such tests for hypothesis testing. While little attention is paid to conducting such tests with commonly available statistical software (e.g., SPSSTM or SASTM) in many instances such software cannot directly perform the tests described -- the authors usually provide sufficient information to allow users (especially advanced users) to complete the tests on their own, generally aided by specific output given in standard ANOVA tables. In several places, conceptual algorithms are given to allow the reader to conduct tests not offered in standard software.

Designed primarily as a course textbook, Statistical Tests for Mixed Linear Models includes student exercises at the end of each chapter, an appendix that gives the solutions to selected problems, and an ample bibliography. Beyond formal use in the classroom, the book also may serve as a reference guide for researchers beyond their student years who wish to know more about exact or optimal tests for mixed linear models. Interested readers, however, should be aware that this is not an introductory text on experimental design or ANOVA. To make best use of Statistical Tests for Mixed Linear Models, readers should be well versed in both. For a good overview of experimental research design, see, for example, Montgomery (1991). A classic work on ANOVA is Scheffe (1959), and many more fine texts have come since.

For those interested specifically in the optimal tests presented in Statistical Tests for Mixed Linear Models, the authors recommend previous familiarity with the concept of optimal tests and the methods for deriving such tests (such as Lehmann, 1986). Readers would do well to heed this advice; indeed, Khuri and his co-authors would have greatly aided their readers had they included an introduction to optimal tests in Statistical Tests for Mixed Linear Models.

Viewed as either a textbook or a reference guide, Statistical Tests for Mixed Linear Models suffers from one major drawback for researchers who primarily use statistics (as opposed to statisticians who advance statistical methods)--too few applications of developed procedures to real data. No doubt, the almost purely mathematical exposition is not a drawback for statisticians or mathematicians, but it can be frustrating for those who want to learn how best to apply advanced methods to actual data. Working the sample problems may alleviate some of this concern.

Those already comfortable with mixed models will find much of use in Statistical Tests for Mixed Linear Models. The tests described therein will enable researchers to make stronger and more certain inferences from their data. Finally, teachers of advanced courses in experimental data analysis will have collected in one place many of the most recent advances in the field.


Lehmann, E.L. 1986. Testing Statistical Hypotheses, Second Edition. New York: Wiley.

Montgomery, D.C. 1991. Design and Analysis of Experiments. New York: Wiley.

Scheffe, H. 1959. The Analysis of Variance. New York: Wiley.

Reviewed by
Richard Wallace
University of Michigan


Castells, Manuel (1996). The Rise of the Network Society (The Information Age: Economy, Society and Culture, Volume 1). Malden, MA: Blackwell Publishers, Inc. (556 pages, bibliography 51 pages, index 23 pages).

Manuel Castells has helped to alter the direction of social research with such works as The Urban Question (1977) and The City and the Grassroots (1983). In his latest book, The Rise of the Network Society (part one of a three-part series), Castells attempts both to synthesize decades of intellectual thought (his own and others) and to generate a conceptul structure to embody the myriad societal changes occurring worldwide. The book provides a thought-provoking description of the collective human experience during the current Information Age. While the book attempts to cover many aspects of the economy, society, and culture, the most novel aspects of the book, which Castells refers to as "the architecture and geometry" of the network society, should be of particular interest to geographers and mathematicians, alike.

Castells begins the book with a description of how the Information Technology (IT) Revolution is distinct from the Industrial Revolution. The distinguishing characteristic of the new IT paradigm that particularly affects social and economic transformations is its "networking logic". As opposed to the linear or serial set of relationships during the Industrial Revolution, epitomized by Fordist mass production, new information technologies are facilitating more complex interactions that are organized by networks. Clearly, network structures are not new, but Castells argues that new information technologies, such as the Internet, allow such structural types to pervade social and economic processes.

Castells describes how the fundamental aspects of networks allow for changes that are leading to a variety of transformations, such as decentralization within firms, telecommuting of workers, interactions in the virtual community and economic globalization. Networks can expand without limits by simply integrating new nodes that share the same means of communication with other nodes. Networks are much more flexible and malleable, because there is no overarching organizational or institutional shape.

Building upon his previous research in political economy and urban sociology, Castells views the current transformations in urban form around the world as the manifestation of the interconnections and linkages between cities. The "space of flows," which pertains to flows of capital, flows of information, flows of technology, etc., intertwines the fates of nodes in the network, but does not predetermine them. Winners and losers in the global urban network are difficult to predict and are continuously emerging from the space of flows. Perhaps an example of Castells's view is the economic uncertainty that ensued after the East Asian crisis in 1997. The path of the pandemic affected the Pacific Basin, but did not travel to the US (yet), as widely feared.

Potential future research in planning based on Castells' framework should center upon the policy implications of the new IT paradigm. When are local planning initiatives held hostage by the global forces in the space of flows? Are different networks destined to remain infinitely apart due to incompatible means of communication? Are there policy remedies for the segmentation of society based on those who are networked and those who are not? Geographers and mathematicians will recognize the applicability of a graph theoretic approach to decipher complex networks, which may be appropriate for a planning context. Such methods can, for example, identify critical linkages that would cripple a section of the network if severed.

Castells interjects several heady topics throughout the book such as the logic of capital accumulation, the relationship between society and postmodern architecture, and the social arrhythmia of the natural lifecycle. The book is written like a very long essay, since Castells does not provide rival explanations for many of the issues included in the book. There are also some unclear aspects of his framework. For example, why do some networks have nodes that dominate flows, as in the urban network, whereas others do not, as in the Internet? Although Castells specifically states that "this is not a book about books", the amount of detail compiled from a wide variety of sources tends to detract from the originality of his thoughts. For the reader who already knows about the rise of Silicon Valley and the Latin American debt crisis, Castells; synopses are redundant. For the reader new to topics related to the high-technology economy and globalization, however, this book provides a comprehensive survey of the literature.

The Rise of the Network Society is a book to read neither quickly nor only once. If time is a limiting concern, however, the final three chapters, which include a provocative discussion about space and time in the network society, incorporate the crux of Castells's vision of society at the turn of the 21st century.

Reviewed by

Seema Desai Iyer
University of Michigan