For most of us, the model of the Earth as a sphere is sufficient for our needs.  Geodetic scientists and others do important research to develop models that are more accurate than the simple, but clearly imperfect, spherical model of the Earth's surface.  When the spherical model is used as the domain in a transformation carrying the Earth's surface to the plane, to create flat maps of the Earth's surface, further distortion is necessarily the result. These derived maps reflect both the inaccuracy of the base model and of its transformation to a different surface.  It is these imperfect maps, nonetheless, that often serve as foundation material for planning and other decision making processes that circumscribe our everyday lives.  The nature of the base models which subtly envelop our lives is critical to understand:   specific applications often fall from broad conceptual models.  To illustrate the thought processes developed here, we begin with a detailed analysis of a familiar surface:  the sphere.  Thus, the reader has that model in mind as a benchmark against which to visualize less familiar surfaces and processes.

THE SPHERE
To navigate a surface, be that navigation in the real or virtual world, one needs some sort of chart and method for finding paths from here to there.  On the spherical model of the Earth, one conventional system involves the use of parallels and meridians.  To understand the nature of this system consider a plane and a sphere in three dimensional space.  There are two broad logical relationships between them:  either the plane does not intersect the sphere or the plane does intersect the sphere.  This relationship serves as a mathematical partition of the space:  it separates the space into mutually exclusive, yet exhaustive, classes.  Within the second class, one might further specify the partition:  either the plane is tangent to the sphere or it is not tangent to the sphere.  In the latter instance the plane cuts the sphere, as a cutting plane, in a circular cross-section, much as a knife might cut into an orange.  With the "orange" idea in mind, the class of circular cross sections can then be further partitioned into circles containing the center of the sphere and circles not containing the center of the sphere.  A diagram illustrating the hierarchical arrangement of this partition appears in Figure 1.
 

Figure 1.  Hierarchical arrangement of one partition of the relationship of a plane and a sphere.

When the sphere in the above paragraph is specified to be a representation of the Earth's surface, then two other points, in addition to the center of the sphere, become involved in the system:  the north and south poles.  These points lie at opposite ends of a diameter of the sphere:  they are antipodal (opposite feet).  Circles containing the center of the Earth are conventionally called great circles:  they are the largest circles possible and are geodesics on this surface.  A geodesic is a line along which shortest paths are measured and these differ, both in shape and in number, according to the surface traversed.  In the Euclidean plane, geodesics are unique; on the sphere they are not (antipodal points, for example, have two distinct shortest paths joining them along any single great circle).  All other circles, those not containing the center of the Earth, are called small circles. There is an infinite number of great circles and an infinite number of small circles.  There is, however, exactly one great circle that bisects the distance between the poles:  that great circle is called the equator.   Its role is like that of the horizontal axis in the Cartesian coordinate system.  A set of small circles, whose cutting planes are parallel to the cutting plane of the equator, become  grid lines on either side of the horizontal axis.  This set is called a set of parallels.  Notice, however, that it is the cutting planes that generate the small circles, and not the small circles themselves, that are parallel.  The parallels specify location to the north and south of the equator but not east to west.  One way to do so is to introduce a set of lines that cut across the horizontal set in an orthogonal fashion: a set of great circles that intersect each other at the poles functions in this manner.  The halves of these great circles, from pole to pole, are called meridians (half circles reflecting the earth-sun relationship that on the equinoxes, and only on the equinoxes, any full meridian spends half the day in light and half in night).  One meridian is typically singled out for unique characterization as the "vertical axis" meridian; historical convention followed recently in many parts of the world is to use the meridian passing through the Royal Observatory in Greenwich, England.  This unique meridian is the Prime Meridian.   This arrangement of a grid on a sphere, often called a graticule, is shown in Figure 2.  The location of a point P on the sphere may be specified in terms of parallels and meridians.  However, different observers may create different graticules.  Thus, location is relative to graticule choice as is shown Figures 2a and 2b:  in Figure 2a, P is located at about one and a half parallels to the north of the Equator and almost three meridians to the west of the Prime Meridian while in Figure 2b, P is about eight parallels to the north of the Equator and about 14 meridians to the west of the Prime Meridian. Both are correct; the navigation to P depends on how the system was set up.  This relative location strategy does not permit replication of results.
 

Figure 2a*.

Figure 2b*.

What is needed to produce a natural replication of results is the introduction of some standard system of measure:  to convert relative to absolute location.  Most people agree to a measure of circles that splits them into some number of equally sized wedge-shaped pieces, measured from the center to the perimeter.  Some use degrees, some use radians, but basically the idea is the same:  transformation from one system to another is easy.  In either of the graticules in Figure 2, the point P lies 42 degrees north of the equator (measure of latitude) and 71 degrees west of the prime meridian (measure of longitude).  Conventional circular measure standardizes location on the sphere.

On flat maps, however, circular measurements can seem odd.  When one understands clearly, however, the base model from which the flat map was derived and the need for circular measure as a standardizing device to ensure replication of results, then the presence of circular measure notation on a flat, rectangular map does make sense.  Specific use of circular measure notation for locational precision falls from the broad spherical model of the Earth; the structure of the geometry is clear independent of sphere radius.
 

ZIPF'S RANK-SIZE RULE
The view of the Earth as a sphere, and its geometry that has served as a conventional abstract model, has endured for centuries.  Beyond the natural world derived from the Earth's planetary relation to the sun, one might wonder what other surfaces could arise as interesting models and what their associated geometries might be.  One set of systems of interest to both authors (albeit in different ways) is the set of systems of cities.  Batty has investigated this set from the standpoint of "power laws" and particularly in relation to the basic power law of Zipf.  A set of lectures given by Batty at The University of Michigan, in 2003, stimulated a response from Arlinghaus to investigate the geometry of the Zipf configuration from an unconventional approach.

A set of cities may be rank-ordered according to population (or any number of other variables).  When it is, and the largest city is assigned the rank of 1, the next largest the rank of 2, and so forth, it follows that there is an inverse relationship between rank and population size.  Zipf characterized this relationship as the "rank-size rule."  The pattern inferred from plotting actual data corresponds to the equation xy=K which represents an hyperbola in the plane.  West, Brown, Enquist, and Savage characterize this rule as:

The empirically observed regularity is that settlements of rank r in the descending (size) array of settlements have a size equal to 1/r of the size of the largest settlement in the system. In other words, when the population size of towns is plotted against their frequency on a logarithmic scale, we see an approximately straight line with an exponent of -1. This relationship is known as the 'rank-size rule'.

Figure 3a* and Figure 3b*.  City Size Distributions:  The Rank-Size Rule.  If we examine the size distribution of cities, we find they are not normally distributed by lognormally distributed, and can be approximated by a power law.
 

Figure 4*.  Original Zipf plot using incorporated places in the U.S.A. from 1790 to 1930.
 

Figure 6a*.

Figure 6b*.

THE HYPERBOLOID OF TWO SHEETS
Part of the rationale for plotting on log paper and converting to straight lines is that lines are easier than curves to handle.  Such conversion, however, implies a functional relationship that may or may not be there.  Why convert curves as models, whose equations are known, to what they in fact are not?  Log paper masks the true shape of the curves.  Instead, Arlinghaus wished to look for a surface from which to derive these curves, as a set, in order to understand, rather than to mask, fundamental geometric structure.

The branches of the hyperbola in the 1st and 3rd quadrants may be rotated about the line y=x to generate a surface of revolution composed of two dish or bowl-shaped objects facing outward from the origin; follow this link and scroll down to find an image containing an hyperboloid of two sheets that the user can manipulate ("sheets" is the 3-dimensional equivalent of the 2-dimensional "branches").

To ease visualization of the pattern of intersecting planes with the hyperboloid sheets, consider the shapes in an upright position.  In analogy with the sphere, one sees, looking at the animations in Figure 7, that the parallels are circular sections sliced by the cutting plane in Figure 7a and that the meridians are hyperbola of varying curvature (Figure 7b) reminiscent of the pattern in the Zipf plot and its hyperbolic representation (Figure 5).
 

Figure 7a*.  Click here to link to movie version in which the reader can control frames and stop the animation at any point desired.

Figure 7b*.  Click here to link to movie version in which the reader can control frames and stop the animation at any point desired.

The basic geometric configuration, in terms of underlying structure is similar to that of the sphere although of course the shapes of the hyperboloid of two sheets and the sphere appear very different from each other.  Once one understands the geometry of the unit sphere, one understands the geometry of any sphere; the same is true for the hyperboloid of two sheets.  The center of the solid, in each case, is the origin of the three dimensional coordinate system.  Each solid has two poles:  on the sphere, these are opposite ends of a diameter of the surface.  On the hyperboloid, these are opposite ends of the axis of bilateral symmetry of the surface, piercing each bowl at its base.  The "great hyperbolae" on the hyperboloid pass through a pole; they serve as geodesics on the surface.  The "small hyperbolae" do not pass through a pole.  Indeed, a hierarchy for the hyperboloid of two sheets matches the one for the sphere shown in Figure 1.  It is shown below in Figure 8.

Figure 8.  Hierarchical arrangement of one partition of the relationship of a plane and an hyperboloid of two sheets.

MAPPING THE HYPERBOLOID OF TWO SHEETS
Cartographers have a permanent job because it is not possible to map precisely a sphere on a plane (stereographic projection is the best one can do by the one-point compactification theorem of topology).  Nonetheless, flat maps are easily portable and have other nice characteristics.  Thus, if the hyperboloid of two sheets is the one that drops out the ideal models of hyperbolae for the Zipf plot, then how might one map this surface back to the plane?  Harrison, on a cover of Scientific American in November of 1975 suggested one possibility (Figure 9) in his rendering of Raisz's earlier orthoapsidal projection of the orthographic projection to the plane of a globe on an hyperboloid of two sheets.  Harrison's "globe" is depicted on part of an hyperboloid of two sheets.
 

Figure 9*.  Harrison's hyperboloidal globe, 1975.

Beyond that particular projection, however, one might ask about general mapping strategy.  Once again, a look to the past finds answers.  When the hyperboloid of two sheets is orthographically mapped to the plane (through the origin) the visualization is as follows.  First, imagine that the plane is extended to include points at infinity so that the concept of parallel lines is removed.  The Euclidean space is converted to a non-Euclidean one.  Now, orthographic projection from the origin sends the upper sheet to the interior of a disk whose boundary is composed of the points at infinity:  what had been unbounded become compressed inside a single disk.   Points on the boundary of the disk are places where the hyperbolae touch the asymptotes (at infinity).  In this model, a great hyperbola, a geodesic passing through a pole (bottom of the bowl), maps to a diameter of the disk; small hyperbolae map to circular arcs with arc endpoints on the disk boundary (images in this link suggest how some of these mappings occur).  This systematic mapping of the hyperboloid of two sheets carries the viewer into the non-Euclidean world of hyperbolic geometry. The disk is the "Poincaré" disk.  The Poincaré disk boundary served as the limit circle of M.C. Escher's "Circle Limit" series [Peterson, Science News]; the interiors reflect the characteristics described above of hyperbolic geometry modeled within the limit circle.  This hyperbolic geometry model served as the base for Lobachevsky's geometry which has often been realized in relation to space-time problems in a variety of disciplines.

ZIPF'S HYPERBOLOID?
If Lobachevsky's hyperboloidal geometry is a natural base from which to drop Zipf curves, as Zipf's Hyperboloid, then how might that geometry be useful in understanding the emergence, timing, and decay of these systems?  Figure 10 shows mappings of Zipf curves in a compact form in the Poincaré disk.  Because the hyperbolae that pass a pole on the hyperboloid of revolution of two sheets map to diameters of the Poincaré Disk model of the hyperbolic plane, one has the freedom in the Zipf model to choose any Zipf hyperbola as the one about which to perform the revolution.  Thus, one might tie such choice to cultural or other considerations.  Suppose one wished, for example, to consider pre- and post-World War II patterns.  Then, 1950 might be selected as a date about which to create the surface of revolution.  Thus, in Figure 10a, imagine that the red diameter represents a U.S.A. rank-size database from 1950, extrapolated according to some procedure to infinity in either direction.  Then, the red circles, each with 10 percent fill, on either side, represent the corresponding extrapolated U.S.A. rank-size curves on either side of 1950; those farther from the diameter are farther from 1950.  Further, in Figure 10b, imagine that the green diameter represents a European rank-size (extrapolated) database from 1950.  The green circles on either side of the diameter, each with 10 percent fill, represent the corresponding European rank-size (extrapolated) databases on either side of 1950.  These figures have both been drawn with no crossings of rank-size curves.  Were there crossings, the corresponding circles would have perimeters that intersect, affecting fill concentration (which might be used to suggest urban or other concentration).  They have also been drawn assuming data on both sides of the date selected for revolution; naturally, if no data were available on one side, then no fill would appear on that side.
 
 

Figure 10a.  Poincaré Disk with one set of rank-size curves.

Figure 10b.  Poincaré Disk with another set of rank-size curves.

When the two Disk layers are superimposed, as in Figure 11, one advantage to the compact Poincaré Disk model becomes clear:  sets of rank-size curves from different parts of the world can be clearly visualized, simultaneously.  It is perhaps an interesting question to consider what overlapping regions might represent:  perhaps an opportunity for a Thiessen-style of transformation leading to rank-size art?  Indeed, the simple pattern in Figure 11 suggests the style of deeper pattern that can arise in association with complex issues:  it is the method of pattern creation so beautifully depicted in the art of Escher's Circle Limit series (inside Poincaré Disks).

Naturally, extrapolated rank-size curves need not be used; if instead, actual data only is used, the circles do not extend to the edge of the limit circle (points at infinity) of the Poincaré Disk.  We show the extrapolated curves here to indicate the power of the method.  Not only may it be used for the simultaneous display of complex datasets from disparate locations but also it handles finite as well as infinite processes.
 

a Figure 11.  Poincaré Disk model of the hyperbolic plane with two sets of rank-size curves, from disparate locations, embedded.

Conjecture 1:  Raisz's 1943 orthoapsidal projection of the hyperboloid to the plane is intellectually part of the broader 19th century Poincaré-Beltrami model of the hyperbolic plane.

Conjecture 2:  The non-Euclidean Lobachevskian geometry of the hyperbolic plane is a natural geometry in which to view Zipf rank-size processes.

Conjecture 3:  The same or other non-Euclidean geometries are natural geometries in which to view power-law processes.

Indeed, an Atlas of non-Euclidean maps based on appropriate surfaces would then become useful for Zipf and power-law processes.  Beyond that, examination of the uses of the non-Euclidean Lobachevskian geometry in space-time problems from the past should be enlightening.  Further, how might the capability to characterize broad systems through a single model enhance our understanding of the multiple connection patterns between and within urban systems?  Recent applications of the non-Euclidean Lobachevskian geometry in computer scientific analysis of internet connections patterns would seem important to consider.  There are many exciting avenues to probe in this clearly multidisciplinary approach; thus, we offer a glimpse of our thinking about these complex issues in order to entice others, from a variety of disciplines and with a varied set of experiences, to join us in these thought processes.

FUTURE DIRECTIONS

• Create actual Poincaré Disk models from real-world data
• Work on Conjectures.
• Work out detail of hyperboloidal model
• Is one hyperboloid sufficient to model a system?
• Are multiple hyperboloids helpful and if so what is their meaning (perhaps in terms of complex curve crossings or other complexity)?
• Recast the model characterization in terms of level curves, vertical profiles, and conventional topographic considerations.  Take advantage of the substantial science already built-up in the field of terrestrial mapping.
• Connection to Diophantine equations:  the equation xy=K admits only composite solutions for K when all of x, y, and K are to be integers.
• View of the hyperboloidal geometry as one part of broader classifications of
• Transformations of a cartographic nature [Tobler]:  here, only orthographic was mentioned, in the spirit of Raisz's orthoapsidal projection.
• Transformations of a synthetic geometric nature, particularly in association with fractals and their characterization of central place geometry [Arlinghaus, 1985; Arlinghaus and Arlinghaus, 1989].
• Assessment of properties of Lobachevskian geometry and its implications for thinking about systems of cities.
• Assessment of use of applications of Lobachevskian geometry in disparate academic fields [Rhodes and Semon, for example] and implications for thinking about systems of cities.
• Evaluation of existing use of hyperboloid of two sheets for analysis of complex internet connections [Munzner], particularly as they mirror complex urban connections.
• Evaluation of existing space-time use of the geometry and implications for thinking about the rise and fall of systems of cities [Batty].
• Virtual reality of figures to allow readers interesting visual experiences.

• Arlinghaus, Sandra L. 1985.  "Fractals take a central place," Geografiska Annaler, 67B, pp. 83-88. Journal of the Stockholm School of Economics.
• Arlinghaus, Sandra L. and Arlinghaus, William C.  1989.  "The fractal theory of central place hierarchies: a Diophantine analysis of fractal generators for arbitrary Loschian numbers," Geographical Analysis: an International Journal of Theoretical Geography. Ohio State University Press. Vol. 21, No. 2, April, 1989; pp. 103-121.
• Batty, Michael.  29 May, 2003.  CUPUM '03:  Sendai, Japan.  The Emergence of Cities:  Complexity and Urban Dynamics.  Powerpoint display.  Contact the author: m.batty@ucl.ac.uk
• Calvet, Ramon Gonzalez.  Treatise of Plane Geometry through Geometric Algebra.  Found at:  http://www.lomont.org/Math/GeometricAlgebra/ GA%20Treatise%20A%20-%20Calvet.pdf
• Castner, Henry W.  2000.  Manipulative Map Projections:  Linking Projection Concepts to the Barbara Petchenik Competition.  Found at:  http://lazarus.elte.hu/ccc/10years/ea/castner2.htm
• Coxeter, H. S. M.  1961, 1965 (fourth printing).  Introduction to Geometry.  New York:  John Wiley and Sons.
• Coxeter, H. S. M.  1961 (fourth edition). Non-Euclidean Geometry.  Toronto:  University of Toronto Press.
• Escher, M. C.  Circle Limit Series; see Peterson reference below.
• Furuti, Carlos.  1997.  Other Interesting Projections.  Found at:  http://www.progonos.com/furuti/MapProj/Normal/ProjOth/projOth.html
• Harrison, Richard E.  Scientific American, cover, November, 1975, "Novel Map Projections."
• Hyperbolic Non-Euclidean World: Found at: http://www1.kcn.ne.jp/~iittoo/us40_mode.htm.
• Mathworld, MathWorld.Wolfram.com, Hyperboloid.  Found at:  http://mathworld.wolfram.com/Hyperboloid.html
• Munzner, Tamara.  2000.  Interactive Visualization of Large Graphs and Networks.  Ph.D. Dissertation, Computer Science, Stanford University. Found at:  http://graphics.stanford.edu/papers/munzner_thesis/allbw.pdf   Html version found at:  http://graphics.stanford.edu/papers/munzner_thesis/; Related articles appear in  general server space.
• Putz, John F.  Animated Demonstrations for Multivariable Calculus.  Found at: http://archives.math.utk.edu/ICTCM/EP-10/C9/html/paper.html
• Peterson, Ivars.  2001.  Visions of Infinity.  Science News Online  Found at:  http://www.sciencenews.org/articles/20001223/bob8.asp
• Raisz, Erwin J.  1943.  Orthoapsidal world maps.  Geographical Review 33(1):  132-134.
• Rhodes, John A. and Semon, Mark D.  2004. Relativistic velocity space, Wigner rotation and Thomas precession.  American Association of Physics Teachers, Am. J. Phys. 72 (7), July.  Found at: http://www.bates.edu/~msemon/RhodesSemonFinal.pdf
• Snyder, John P.  1993.  Flattening the Earth:  Two Thousand Years of Map Projections.  Chicago:  University of Chicago Press.
• Thompson, D'Arcy W.  1917 (original publication date with others later).  On Growth and Form.  Cambridge University Press.
• Tobler, Waldo R.  1962.  A classification of map projections.  Annals of the Association of American Geographers 52(2):  167-175.
• Waterman, Steve.  Polyhedron website.  Found at:  http://watermanpolyhedron.com/
• West, Geoffrey; Brown, Jim; Enquist, Brian; and Savage, Van.  Network Dynamics:  Scaling and Distribution Networks.  Santa Fe Institute.  Found at:  http://www.santafe.edu/research/scalingNetworks.php
• Zipf, G.K. 1949. Human Behavior and The Principles of Least Effort, Addison Wesley, Cambridge, MA.

• Figures 2a and 2b.  Use of part of a globe found at Geosystems.com in the 1990s that is no longer available.  Permission was granted earlier for use of this globe elsewhere.  Here, a modified form is used as a backdrop to figures created in ArcView GIS (ESRI, TM).
• Figures 3a and 3b.  Batty, Michael, Sendai 2003 Powerpoint presentation.
• Figure 4.  Zipf, G. K.  Based on the 1949 reference as found in Batty, above, and modified in Adobe Photoshop (TM).
• Figures 6a and 6b.  Batty, Michael, Sendai 2003 Powerpoint presentation.
• Figures 7a and 7b.  Animations are modified from base animations of John F. Putz, Professor of Mathematics and Computer Science, Alma College, Alma Michigan 48801, putz@alma.edu.  
• Figure 9.  Modified form of a scanned image found on eBay.