Information and research about the geometry invented by Herman Minkowski
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Taxicab Geometry > History / Applications

The metric (distance formula) underlying what has become known as taxicab geometry was first proposed as a means of creating a non-Euclidean geometry by Herman Minkowski (1864-1909) early in the 20th century. (Minkowski was an early teacher of Albert Einstein.) The metric was one of a whole family of metrics Minkowski proposed to easily create non-Euclidean geometries.

In 1952, Karl Menger created a geometry exhibit at the Museum of Science and Industry in Chicago. For visitors to the exhibit, Menger also created a booklet entitled You Will Like Geometry. It was in this booklet that the term "taxicab geometry" was first used. It has remained associated with the geometry ever since.

By 1975, work to develop a full geometry based on the taxicab metric had still not been done. At this time Eugene Krause commented, "It would seem the time has come to do so." Krause's book published in 1975, Taxicab Geometry: An Adventure in Non-Euclidean Geometry, is still the standard introduction to the geometry.

Modern research in taxicab geometry first began appearing sporadically in the early 1980s. It would not be until about 1997 that continuous, earnest research would start in taxicab geometry. This research began with near simultaneous, independent work on taxicab angles and trigonometry by Kevin Thompson at Oregon State University and also Rüstem Kaya in Turkey. Thompson's research was conducted in graduate school in 1996 with Tevian Dray and published in 2000 with Kaya's research being published in 1997. From 1997 to 2010, Rüstem Kaya has been the most productive researcher in the geometry.

It would seem that Krause's observation is being carried out in earnest today.

Real-world Applications of Taxicab Geometry

Taxicab geometry finds itself useful in a number of real-world sitations.
• In chess, the distance between squares on the chessboard for rooks and bishops is measured in taxicab distance.
• An extended version of taxicab geometry is used in fire-spread simulation with square-cell, grid-based maps.

References
[1] Caballero, David. Taxicab Geometry: some problems and solutions for square grid-based fire spread simulation, V International Conference on Forest Fire Research, D. X. Viegas (ed.), 2006.
[2] Golland, Louise. Karl Menger and Taxicab Geometry, Mathematics Magazine, Vol. 63, No. 5 (Dec 1990), pp. 326-327.
[3] Krause, Eugene F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry, 1975.
[4] Menger, Karl. You Will Like Geometry, Guidebook of the Illinois Institute of Technology Geometry Exhibit, Museum of Science and Industry, Chicago, Illinois, 1952.
[5] Minkowski, Herman. Gesammelte Abhandlungen, Chelsea Publishing Co., New York, 1967.
[6] Reynolds, Barbara E. Taxicab Geometry, The Pi Mu Epsilon Journal, Worcester, MA. Vol. 7, No. 2 (Spring 1980), pp. 77-88.

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