Sandra Lach Arlinghaus
The University of Michigan
In 1911, Thiessen and Alter  wrote on the analysis of rainfall using polygons surrounding rain gauges. Given a scatter of rain gauges, represented abstractly as dots, partition the underlying plane into polygons containing the dots in such a way that all points within any given polygon are closer to the rain gauge dot within that polygon than they are to any other gauge-dot. The geometric construction usually associated with performing this partition of the plane into a mutually exclusive, yet exhaustive, set of polygons is performed by joining the gauge-dots with line segments, finding the perpendicular bisectors of those segments, and extracting a set of polygons with sides formed by perpendicular bisectors. It is this latter set of polygons that has come to be referred to as "Thiessen polygons" (and earlier names such as Dirichlet region or Voronoi polygon, see Coxeter ). The construction using bisectors is tedious and difficult to execute with precision when performed by hand. Kopec (1963)  noted that an equivalent construction results when circles of radius the distance between adjacent points are used. Indeed, that construction is but one case of a general construction of Euclid. Like Kopec, Rhynsburger (1973)  also sought easier ways to construct Thiessen polygons: Kopec through knowledgeable use of geometry and Rhynsburger through the development of computer algorithms. The world of the Geographical Information System (GIS) software affords an opportunity to combine both.
Traditionally one might have used a drawing compass and a straightedge to construct a perpendicular bisector between two points. It is an easy matter to do so, however, using a GIS, as suggested above. If there are more than two points, the matter can become quickly tedious. Again, the GIS offers a quick and accurate way to calculate positions (Figure 2).
To actually position the lines of partition, or Thiessen polygon edges, in the GIS, use the "split polygon" feature available in ArcView or other GIS software, creating a sort of bubble foam (Figure 3 shows one split created in this manner). Numerous websites offer suggestions for use of Thiessen polygons ranging from rainfall regions, to hydrological modelling, to road centerline location (and others) [12, 13, 14].
Again, the GIS is helpful: ArcView (Spatial Analyst extension) offers a single tool that quickly calculates Thiessen polygons. Use "Assign Proximity" to create zones around each point. Within each zone, all points are nearer to the distribution point in that zone than they are to any other point in the distribution. In Figure 4, the relationship between perpendicular bisector, buffer (construction of Euclid), proximity zone/Thiessen polygon becomes clear.
Whether one considers rail networks within sausage-like
linear buffers, counts population in buffered bus routes, or selects minority
groups from within a circular buffer intersecting census tracts, the buffer
has long served, and continues to serve, as a basis for making decisions
from maps. Buffers have a rich history in geographical analysis.
Jefferson [10, 2]
rolled a circle along lines on a map representing railroad tracks to create
line-buffers representing proximity to train service and suggested consequent
implications for population patterns in various regions of the world.
Julian Perkal and John Nystuen saw buffers in parallel with delta-epsilon
arguments employed in the calculus to speak of infinitesimal quantities
(reprint of Perkal, "An Attempt at Objective Generalization," Michigan
Interuniversity Community of Mathematical Geographers, [16,
Jefferson's mapping effort in 1928 was extraordinary; today, buffers of
points, lines, or regions are trivial to execute in the environment of
Geographic Information Systems software. To paraphrase Faulkner (1949),
'good ideas will not merely endure, they will prevail' .
In a recent invited lecture (2001) to The University of Michigan Lecture Series in GIS Education, Arthur Getis noted  that he had used circular buffers around point observations and that he used a sequence of nested buffers to successively fill space to eventually include all individuals in the underlying point distribution gathered from field evidence. Thus, viewed abstractly, each set of buffers serves as a base map, with the sequence successively filling more space and including more individual observations in the analysis. A different view might see the Thiessen map as the base map (Figure 5). When it is calculated at the outset, it can serve as a standard against which to test more specialized views at varying buffer radii, on a continuing basis, as the research within buffers evolves. The Thiessen base map serves, therefore, as an "absolute" base map against which to view the "relative" base maps of varying local radii (and other configurations): it is a limit of a sequence of measures based on buffers that increasingly fill more space (but still leave gaps). Getis noted  that he and Ord had recently completed an article involving issues of global and local spatial statistical measures . What is suggested here is the appropriate use of a geometric foundation: a use for a Thiessen, space-filling, base against which to test the results of sequences of successive measures in buffers that may not fill the underlying universe of discourse.
References and selected related readings.
1. Arlinghaus, Sandra L. 1985. "Fractals take a central place," Geografiska Annaler, 67B, (1985), pp. 83-88. Journal of the Stockholm School of Economics.
3. Bogue, Donald J. 1950. The structure of the metropolitan community: a study of dominance and subdominance. Ann Arbor, MI.
4. Coxeter, H. S. M. 1961. Introduction to Geometry. New York: John Wiley & Sons, pp. 53-54.
5. Dacey, Michael F. 1965. The geometry of central place theory. Geografiska Annaler, B, 47, 111-124.
6. Edit Tools 3.1. http://www.ian-ko.com/howto.htm
7. Faulkner, William. 1949. Nobel Prize Acceptance Speech. http://www.nobel.se/literature/laureates/1949/faulkner-acceptance.html
8. Haggett, P.; Cliff, A. D.; and, Frey, A. 1977. Locational Analysis in Human Geography 2: Locational Methods. Second Edition. New York: John Wiley & Sons. (Section 13.5).
9. Hargrove, William W., Winterfield, Richard F. and Levine, Daniel
A. Dynamic segmentation and Thiessen polygons: a solution to the
river mile problem.
10. Jefferson, Mark. 1928. The Civilizing Rails. Economic Geography, 1928, 4, 217-231.
11. Kopec, R. J. 1963. An alternative method for the construction of Thiessen polygons. Professional Geographer, 15, (5), 24-26.
12. Ladak, Alnoor and Martinez, Roberto B. Automated Derivation of High Accuracy Road Centrelines Thiessen Polygons Technique. http://www.esri.com/library/userconf/proc96/TO400/PAP370/P370.HTM
13. Murray, Alan and Gottsegen, Jonathan. The Influence of Data Aggregation on the Stability of Location Model Solutions http://www.ncgia.ucsb.edu/~jgotts/murray/murray.html
14. Natural Neighbor Interpolation. http://www.ems-i.com/smshelp/Scatter_Module/Scatter_Scattermenu/Natural_Neighbor.htm
15. Nystuen, J. and Brown, D. 2001. Coordinators, UM GIS Lecture Series. Lecture by Arthur Getis: "Spatial Analytic Approaches to the Study of the Transmission of Disease: Recent Results on Dengue Fever in Iquitos, Peru".
16. Nystuen, J., 1966. "Effects of boundary shape and the concept of local convexity;" Michigan Interuniversity Community of Mathematical Geographers (unpublished). Reprinted, Ann Arbor: Institute of Mathematical Geography.
17. Okabe, A.; Boots, B.; and, Sugihara, K. 1992. Spatial Tesselations: Concepts and Applications of Voronoi Diagrams. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley & Sons.
18. Ord, J. Keith and Getis, Arthur. 2001. "Testing for Local Spatial Autocorrelation in the Presence of Global Autocorrelation." Journal of Regional Science, Vol.. 41, No. 3, pp. 411-432.
19. Perkal, Julian, 1966. "An attempt at objective generalization;" Michigan Interuniversity Community of Mathematical Geographers (unpublished). Reprinted, Ann Arbor: Institute of Mathematical Geography.
20. Rhynsburger, Dirk, 1973. Analytic delineation of Thiessen polygons. Geographical Analysis, 5, 133-144.
21. Thiessen, A. H., and Alter, J. C.. 1911. Climatological Data for July, 1911: District No. 10, Great Basin. Monthly Weather Review, July:1082-1089.
22. Thiessen Polygons. http://www.geog.ubc.ca/courses/klink/g472/class97/eichel/theis.html
Solstice: An Electronic Journal of Geography
Volume XII, Number 2.
Ann Arbor: Institute of Mathematical Geography.