In existographies, William W. Warntz (1922-1988) (CR:18) was an American economist-geographer, a genius hiatus effect product, noted for his extension of American sociologist Henry Carey’s demographic gravitation theory, and for his later work with American sociophysicist John Q. Stewart on physics-based models of population potentials.

Population potential | Maps
Warntz, in his collaboration with Stewart early on, emphasized the importance of recognizing general geographical patterns through the spatial analysis of large sets of geo-coded data—thinking, supposedly, that the patterns could be represented and explained by models and theories of physics, particularly the Lagrangian potential model of Newtonian mechanics. [13]

In 1964, Warntz built a 2x3.25 feet, tree-dimensional so-called “potential population” map of the US using 3,100 nails, the exact height and location of each calculated using an IBM 7090 computer. [10]

he following are a few images of Warntz’s three-dimension potential population model or “potential of population”, as Warntz seems to have called it, indicating the economic importance of all the cities, towns, and rural areas in the US, which was done for a special exhibit of the American Geographical Society at the New York State Fair. The varying heights of each nail represent the varying demographic “potential” of population groupings based on “formulas analogous to those in field quantity physics”, according to one article summary, having something to do with movement of people, mail, and economic goods into and out of each location: [11]

In 1960, supposedly, Warntz made the following plaster-style social gravitation potential (Ѻ) countered map:

The equation Warntz used to make these peaks seems to have been a variant of the following:

which measures the influence of a given population at one location (j) on another location (i), where vi is the population potential at location i, Pj is the size of the population at j, and Dij is the distance between locations i and j.

Each of these “population potentials”, shown as peaks on the map, as summarized by Donald Janelle, indicate an index of sociological intensity, according to which the population at any point in space contributes to the population potential of all other points, is a measure of influence at a distance. [11]

Warntz found, accordingly, that the strength of the interaction I or interaction force Fi, i.e. exchange force, as measured by number of telephone calls, traffic flows, etc., is proportional to the product of the populations of the respective populations P1 and P2 and inversely proportional to their distance of separation squared:

Overview
In 1947, American sociophysicist John Q. Stewart published his “Empirical Mathematical rules Concerning the Distribution and Equilibrium of Population”, in the Geographical Review, wherein he introduced geographers to his idea about isomorphic (equal form or structure) relations between social behavior and the laws of physics, stating, among other things, that: [4]

“There is no longer an excuse to ignore the fact that human beings, on average and at least in certain circumstances, obey mathematical rules resembling in a general way some of the primitive ‘laws’ of physics.”

Stewart pointed out isomorphic relationship between empirical studies, which indicated that the movement of persons between two urban centers was proportional to the product of their populations and inversely proportional to the square of the distance between them, and Newton’s law of gravitation. This postulate, in quantitative geography, soon came to be known as the gravity model, as detailed below by Norwegian geographer Arild Holt-Jensen, from his 2009 Geography: History and Concepts: [3]

 The Carey-Stewart-Warntz "gravity model" of the so-called "social physics school", one of the four schools of quantitative geography, according to human geography historian Ron Johnston. [7]
Stewart, in his 1947 paper, also published maps of population potentials. [5]

Warntz, in 1984, remarked that he was introduced to Stewart’s ideas via his 1945 book Coasts, Waves and Weather; in his own words: [6]

“[Stewart’s book] was prepared primarily to explain to marine and air navigators the physical environment … [but] Stewart could not resist the temptation to include an exotic chapter describing potential of population and its sociological importance.”

Warntz, sometime thereafter, began working with Stewart, resulting in at least two papers in 1959 and 1964, in which they were said, according to Holt-Jensen, to have borrowed models from physics in their studies of population potentials. [1]

Warntz suggesting that the mathematics of a population potential is the same as that which describes a gravitational field, a magnetic potential field and an electrostatic potential field. [2]

Education
Warntz completed his BS, AM, in economics, and PhD, with a dissertation on “The Geography of Prices” in 1955 in economics, all at the University of Pennsylvania. Warntz then taught at the University of Pennsylvania, and was a research associate at the Department of Astrophysical Sciences at Princeton University and at the American Geographical Society. In 1968, Warntz then became professor and director of theoretical geography and regional planning in the graduate school of design, Harvard. In 1971, he became professor of geography and departmental chair at the University of Western Ontario. [9]

Quotes
The following are related quotes:

“Every educated person should carry about in his mind’s eye an instantaneously available globe. The globe should be in life colors and rotating slowly in the sunlight. On it the mind’s eye should see at least the continental outlines, major political divisions, vegetation and climatic belts, primal atmospheric and oceanic circulation, the earth’s outstanding cities and their economic flows, and ultimately the people themselves and the quality of life.”
— William Warntz (1968), on a proposal to initiated a Department of Geography at Harvard [12]

Mateo Gil

References
1. (a) Warntz, William. (1959). Towards a Geography of Price: A Study in Geo-Econometrics. University of Pennsylvania Press.
(b) Warntz, William. (1964). “A New Map of the Surface of Population Potentials for the United States, 1960” (abs), Geographical Review, 54(2):170-84.
(c) Holt-Jensen, Arild. (2009). Geography: History and Concepts (social physics + John Q. Stewart, pg. 87). Sage.
2. (a) James, P.E. (1972). All Possible Worlds: A History of Geographical Ideas (pg. 517). Odyssey Press.
(b) Holt-Jensen, Arild. (2009). Geography: History and Concepts (social physics + John Q. Stewart, pg. 87). Sage.
3. Holt-Jensen, Arild. (2009). Geography: History and Concepts (social physics + John Q. Stewart, pg. 87; gravity model, pg. 88). Sage.
4. (a) Stewart, John Q. (1947). “Empirical Mathematical rules Concerning the Distribution and Equilibrium of Population” (pg. 485), Geographical Review, 37: 461-85.
(b) Holt-Jensen, Arild. (2009). Geography: History and Concepts (social physics + John Q. Stewart, pg. 87). Sage.
5. Warntz, William. (1964). “A New Map of the Surface of Population Potentials for the United States, 1960” (abs), Geographical Review, 54(2):170-84.
6. (a) Warntz, William. (1984). “Trajectories and Coordinates”, in: Recollections of a Revolution (editors: M. Millinge, D. Gregory and R. Martin) (pgs. 134-52). MacMillan.
(b) Johnston, Ron J. (1997). Geography and Geographers: Anglo-American Human Geography Since 1945 (pgs. 62-73). London: Arnold.
7. (a) Johnston, Ron J. (1997). Geography and Geographers: Anglo-American Human Geography Since 1945 (pgs. 62-73). London: Arnold.
(b) Holt-Jensen, Arild. (2009). Geography: History and Concepts (social physics + John Q. Stewart, pg. 87; gravity model, pg. 88). Sage.
8. Barnes, Trevor J. and Wilson, Matthew W. (2014). “Big Data, Social Physics, and Spatial Analysis: the Early Years” (pre, pdf) (abs) (pdf), Big Data & Society, Apr-Jun:1-14.
9. (a) Warren, M.E. (2003). “Guide to the William Warntz Papers [c. 1955-1985]” (Ѻ), Carl A. Kroch Library, Cornell University.
(b) Janelle, Don. (2012). “William Warntz and the Legacy of Spatial Thinking at Harvard” (ppt: pdf), Presentation to the GIS Colloquium, Center for Geographic Analysis, Harvard University, Sep 5.
10. (a) Anon. (1964). “Article”, The Princeton Packet, Feb 12.
(b) Janelle, Don G. (1997). “In Memoriam: William Warntz, 1922-1988”, Annals of the Association of American Geographers, 87:723-31.
(c) Barnes, Trevor J. and Farish, Matthew. (2006). “Between Regions: Science, Militarism, and American Geography from World War to Cold War” (pdf) (nails, pg. 819), Annals of the Association of American Geographers, 96(4):807-26.
11. (a) Anon. (c.1964). “Geographer Uses 3,100 Nails to Construct Map”, Periodical.
(b) Janelle, Don. (2012). “William Warntz and the Legacy of Spatial Thinking at Harvard” (ppt: pdf), Presentation to the GIS Colloquium, Center for Geographic Analysis, Harvard University, Sep 5.
12. Janelle, Don. (2012). “William Warntz and the Legacy of Spatial Thinking at Harvard” (ppt: pdf), Presentation to the GIS Colloquium, Center for Geographic Analysis, Harvard University, Sep 5.
13. (a) Rich, David C. (1980). Potential Models in Human Geography CATMOG 26, Study Group in Quantitative Methods, Institute of British Geographers, University of East Anglia, Norwich: Geo Abstracts.
(b) Barnes, Trevor J. and Wilson, Matthew W. (2014). “Big Data, Social Physics, and Spatial Analysis: the Early Years” (pdf) (abs) (pdf), in: Big Data & Society, Jun.