Taxicab Geometry MAEN 504 George Carbone August 18, 2002
Table Of Contents • Why Taxicab Geometry? • History of Taxicab Geometry • The General Equation for Taxicab Distance • Taxicab Circles • An Application - Taxicab Treasure Hunt • Taxicab Ellipse • More Applications • Resources and Bibliography
Why Taxicab Geometry? The best way to understand “Why Taxicab Geometry” is through a practical application or problem... John wants to walk from his home to the library. Based on the diagram at the left where each square represents one square block, what is the shortest distance he must walk? Analysis The library is 4 blocks north and 3 blocks west of John's house. Euclidean geometry would suggest that the shortest distance is along the dotted line. Moreover, since we have a right triangle with legs equaling 3 and 4, then based on the Pythagorean Theorem, the hypotenuse or dotted line would equal 5 blocks. However, unless John is a bird, he can't follow the dotted path. Instead he must walk along the various streets or blocks. Therefore, what is the least number of blocks he must walk? One can see that least number of blocks is 7. Specifically, he would need to walk some combination of 4 blocks north and some combination of 3 blocks west. Taxicab Geometry provides us with a Non-Euclidean framework for analyzing problems based on blocks - much like the grid of an urban street map - hence the name Taxicab Geometry.
History Of Taxicab Geometry Hermann Minkowski (1864-1909) introduced taxicab Geometry over 100 years ago. Minkowski was a German mathematician and professor who studied at the Universities of Berlin and Konigsberg. He taught at several universities in Bonn, Konigsberg and Zurich. Interestingly, Albert Einstein was among his students in Zurich and Minkowski's Non-Euclidean work on a four-dimensional space-time continuum provided a framework for Einstein's later work on the Theory of Relativity. Euclidean Geometry measures distance as the crow flies. Minkowski recognized that this was not necessarily the best model for many real world situations, particularly for problems involving cities where distances are determined along blocks and not as the crow flies. Another valuable aspect of Taxicab Geometry is its simplicity as a non-Euclidean Geometry. It is more easily understood than many other non-Euclidean geometries. In fact, given its grid or Cartesian based orientation, it can be taught with the aid of graph paper as early as in the middle school years. Since its introduction in the late 1800s, Taxicab Geometry has undergone periods of great interest and practical application, as well as periods of marginalization. It received renewed attention in 1975 when Eugene Krause, a Mathematics professor at the University of Michigan, published a detailed book on the subject entitled “Taxicab Geometry: An Adventure in Non-Euclidean Geometry.
The General Equation for Taxicab Distance In Euclidean Geometry, the distance between two points A and B can be derived based on the Pythagorean Theorem: d(A,B) = squareroot ((xA-xB)2 + (yA-yB)2 ). In the example at left, this formula yields the “crow flies” measurement of 7.81 cm, based on Squareroot(52 + 62). However, in Taxicab Geometry, the distance formula was redefined by Minkowski distance as: d(A,B) = |xA-xB| + |yA-yB|. Thus, the Taxicab distance in this example is 11 cm, based on 6 + 5. In other words, the distance is defined as the sum of the horizontal and vertical distance of the two points. This is the minimum distance a taxicab would need to travel to reach point B from point A, if all streets are only oriented horizontally and vertically. In the following pages we’ll see how this Non-Euclidean measure of distance can be applied to practical problems and result in Taxicab definition of geometric figures, such as circles and ellipses. Not surprisingly, our understandings these figures must change.
Taxicab Circles In Euclidean Geometry, a circle represents a series of points equidistant from a single point or center. If we apply the Taxicab distance to the definition of a circle, we get an interesting shape of a Taxicab circle. For example, the set of points 3 units away from point a (1,1) is outlined at left. The dotted line provides an example of a distance of 3. Note the figure appears to be a square. This definition of a Taxicab Circle provides a basis for addressing many practical applications Taxicab Geometry. For example, in the following section we explore Taxicab Treasure Hunt, a website (http://www.learner.org/teacherslab/math/geometry/shape/taxicab/) for determining the location of a hidden treasure. The solution is based on the intersection of respective Taxicab Circles.
Taxicab Treasure Hunt The object of the treasure hunt game is to find a hidden treasure located somewhere in the mythical city outlined to the left. You first define a starting point that I have stated as Third Avenue and Dogwood Street. The game then tells me my distance from the treasure - in this example. It states the distance in 3 blocks. Thus I know the answer consists of the locus of points a taxicab circle of 3 blocks from Third and Dogwood. That circle is outlined at the lower left (note the entire circle does not appear since it is outside the bounds of the defined city). I then selected Fifth and Elm (a point along the prior Taxicab Circle). It informs me that I am 4 blocks from the treasure. Through a second thicker circle now 4 blocks from my new location, I know the solution must be at the intersection of the 2 circles (see the middle figure below). Therefore, the possible solutions are Second and Fir, First and Elm, or Fourth and Birch. I then selected Fourth and Fir, and found out I was 2 blocks away. I drew a third (thin) Taxicab circle 2 blocks from the new location and found the solution is First and Elm, the intersection of the 3 circles.
Taxicab Ellipses In Euclidean Geometry, an ellipse is defined as the set of all points the sum of whose distance from two given points is a fixed distance. Again, applying the Taxicab concept of distance to this definition of an ellipse yields an interesting Taxicab Ellipse. For example, let A equal (-2, -1) and B equal (2, 2). What is the locus of points where the sum of the distance from these points is 9? The answer is outlined at below.
More Applications - A One problem outlined in Krause's book is the following... There are 3 high schools in Ideal City. Fillmore at (-4, 3), Grant at (2, 1), and Harding at (-1, -6). Draw in school district boundary lines so that each student in Ideal City attends the high school nearest his home. The first thing one must do is construct boundaries between each pair of schools. Since Fillmore and Grant are 8 blocks apart, the locus of equally distant points between them is the blue line. Since Fillmore and Harding are 12 blocks apart, the locus of equally distant points between them is the black line. And since Grant and Harding are 10 blocks apart, the locus of equally distant points between them is the red line. Since there are 3 schools the actual dividing lines are determined by the thicker set of lines. Therefore, Grant would serve those living in the yellow area, Harding would serve those in the gray area, and Fillmore would serve those in the teal area. Follow on question... Burger Baron wants to open a hamburger stand equally distant from the 3 high schools. Where should they locate it? Answer at (-2, -1) as it 6 blocks from each.
More Applications - B Another problem outlined by Krause... Ajax Industrial Corporation wants to build a factory in Ideal City in a location where the sum of its distances from the railroad station C = (-5, -3) and the airport D = (5, -1) is at most 16 blocks. For noise control purposes a city ordinance forbids the location of any factory within 3 blocks of the public library L = (-4, 2). Where can Ajax build? The solution to this problem is outlined by the thicker lined figure outlined at upper right. This is clearly an ellipse solution with a wrinkle involving the library. Interestingly, and suggested by the book, we use a Euclidean circle not a Taxicab circle for the Library cut-out. Why? Sound does travel the way the crow flies. Therefore, a 3 block Euclidean cut-out which is larger than the potential Taxicab cut-out (see dotted line) is appropriate.
Resources and Bibliography • Biography of Hermann Minkowski http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Minkowski.html • Hermann Minkowski and Taxicab Geometry http://www.mzt.hr/mzt/hrv/informacije/publi/casopisi/c-teh/kog/koga5.htm • Krause, Eugene. Taxicab Geometry: An Adventure in Non-Euclidean Geometry. Dover Publications, Inc. New York. 1975. • Taxicab Tresure Hunt http://www.learner.org/teacherslab/math/geometry/shape/taxicab/ • Why Taxicab Geometry? http://cgm.cs.mcgill.ca/~godfried/teaching/projects.pr.98/tesson/taxi/why.html