Geometry Activities Suggested by the Taxicab Plane (1/31/2005)

prepared by:

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451

Email: malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

The consequences of using taxicab distance rather than euclidean distance are surprisingly varied in light of the fact that at the axiomatic level the two geometries differ only in that euclidean geometry obeys S-A-S (side angle side) as a congruence axiom for triangles and the taxicab geometry does not.

It is an interesting approach to understanding the consequences of this seemingly small difference between the two geometries to take common ideas in euclidean geometry and look at what is the counterpart of these ideas in the taxicab world.

Example:

1. Everyone knows what a euclidean circle of radius 1 about a point looks likes. What does a taxicab circle of radius one look like?

2. Everyone knows that the (locus) collection of points equidistant from two distinct points in euclidean geometry is a line which is perpendicular and goes through the midpoint of the segment joining the two points. What does the locus of points equidistant from two distinct points in taxicab geometry look like?

By way of practice and to make sure you understand fully the ideas involved, determine the points in the taxicab plane which are equidistant from:

a. (0, 0) and (2, 0)

b. (0, 0) and (2, 2)

c. (0, 0) and (2, 4)

d. (0, 0) and (4, 2)

Even though scholars have to a great extent looked into the ways that concepts in the taxicab plane differ from analogous concepts in the euclidean plane, it is still interesting in practice for those who have not seen these results to do it for themselves. Furthermore, these analogies and differences have not always been charted as fully as they might be and there may be interesting new ideas and phenomena to pursue by this approach.

Here is a sample of some such questions to pursue.

1. What do the conic sections look like in the taxicab plane?

Comment:

In the euclidean plane there are two equivalent approaches to the conics (hyperbola, parabola, ellipse). One approach uses foci (ellipse and hyperbola) and focus and directrix (parabola) and the other approach is based on the concept of eccentricity. In the euclidean plane these two approaches lead to the same geometric objects. Is the same true in the taxicab plane?

2. In euclidean geometry given a point P and a line l that does not pass through P there is a unique point on l whose distance to P is a minimum. Furthermore, there a nice "formula" for the distance between the point and the line.

What is the situation in taxicab geometry for finding the distance between a point and a line in the taxicab plane.

Eugene Krause's book Taxicab Geometry (available in a Dover Press edition) investigates this question.

3. What are the isometries (distance preserving transformations) in the taxicab plane?

Comment:

I posed this question many years ago and at that time the answer was open. Doris Schattschneider wrote an article in the American Mathematical Monthly giving the solution.

Here is a sample of questions that to the best of my knowledge have not been specifically studied.

1. In euclidean geometry if one is given a triangle, it has a unique circumcircle and a unique incircle. The circumcircle of a triangle is a circle which passes through all three vertices of the triangle. The center of the circumcircle is located at the point of concurrency of the medians of the triangle. The triangle will also have a unique incircle which is located at the point where all three of the angle bisectors of the triangle meet.

What is the situation regarding circumcircles and incircles in the taxicab plane? More specifically, does every triangle have a circumcircle? Does every triangle have an incircle? If a triangle does have a circumcircle is it unique? If a triangle does have an incircle is it unique?

2. In euclidean geometry the perimeter bisectors of a triangle which pass through the vertices of the triangle are concurrent (i.e. pass through a single point). What is the situation for taxicab geometry?

3. Given a point P in a triangle or on its boundary is there always a perimeter bisector through P? How many perimeter bisectors might P have passing through it?

4. A fascinating theorem in euclidean geometry is Ceva's Theorem which gives conditions under which lines through the vertices of a triangle are concurrent. Is there an analog for Ceva's Theorem for the taxicab plane?

5. Can you develop a theory of area in the taxicab plane?

6. In the euclidean plane if one rotates a segment about a point its length does not change. Similarly, translating a segment or reflecting the segment in a line does not change its length. What happens in the taxicab plane?

7. In the taxicab plane is it true that if two lines are parallel that the lines are equidistant? (What is the definition of two lines being equidistant?)

8. In the taxicab plane are the base angles of an isosceles triangle congruent? (ans. No! Give an example.) Can you determine under what conditions in the taxicab plane the base angles of an isosceles triangle are congruent?

9. Consider the following theorem in euclidean geometry: If l is a line and Q is a point on l but P is a point which is not on l, then PQ ≤ PR for all the points R on line l if and only if the line through P and Q is perpendicular to l. Verify that this theorem is not true in the taxicab plane.

10. Does the "Pythagorean Theorem" hold in the taxicab plane?

Useful observation?

In the euclidean plane although all lines are alike in various situations one may have to look at horizontal and vertical lines in a different way from lines with slopes which are not 0 or undefined. In the taxicab plane one may want to look at the behavior of lines which are vertical and horizontal, lines with slopes 1 and -1, and lines with slopes other than 0, undefined, 1, and -1. (This observation with regard to the taxicab plane is the result of insights obtained when one looks at the question of the points equidistant from two points. The answer depended on the slope of the line segment determined by the two points in question.)

References:

Brandley, M., Square circles, Pentagon, Fall, 1970, p. 8-15.

Brisbin, R. and P. Artola, Taxicab trigonometry, Pi Mu Epsilon Journal, 8 (1985) 89-95.

Byrkit, R., Taxicab geometry - A Non-Euclidean geometry of lattice points, Math. Teacher, 64 (1971) 418-422.

Gardner, M., The Last Recreations, Springer-Verlag, 1997.

Golland, L., Karl Menger and taxicab geometry, Mathematics Magazine, 63 (1990) 326-327.

Iny, D., Taxicab geometry: another look at conic sections, Pi Mu Epsilon Journal, 7 (1984) 645-647.

Krause, E., Taxicab Geometry, Dover, New York, 1986.

Laatsch, R., Pyramidal sections in taxicab geometry, Math. Magazine, 55 (1982) 205-212.

Mertens, L., A fourth dimensional look into taxicab geometry, J. of Undergraduate Mathematics, 19 (1987) 29-33.

Moser, J. and F. Kramer, Lines and parabolas in taxicab geometry, Pi Mu Epsilon Journal, 7 (1982) 441-448.

Reynolds, B., Taxicab Geometry: An Example of Minkowski Space, dissertation, (unpublished), St. Louis University, 1979.

Reynolds, B., Taxicab geometry, Pi Mu Epsilon Journal, 7 (1980) 77-88.

Schattschneider, D., The taxicab group, Amer. Math. Monthly, 91 (1984) 423-428.

Sheid, F., Square circles, Math. Teacher 54 (1961) 307-312.

Sowell, K., Taxicab geometry - A new slant, Mathematics Magazine, 62 (1989) 238-248.