Imagine a world where circles are square and π is equal to 4. Strange as it seems, we live in such a world: urban geometry is determined by the pattern of streets in a typical city grid and distance “as the crow flies” is not the distance that we have to travel from place to place.
Suppose the streets run north-south and east-west in a regular grid. Then the distance a taxi must travel from one point to another is the number of streets east or west plus the number north or south. This leads to a new metric, or measure of the distance between two points (x1,y1) and (x2,y2).
While an urban grid has discrete intervals between streets, we do not have to restrict our thoughts to the lattice of crossing points. Thus, we define the “Taxicab metric” as
d1 = |x2-x1| + |y2-y1|
It is also known by other names: mathematically it is the L1 norm; popularly, it is the Manhattan metric. It is in contrast to the usual Euclidean or L2 metric
d2 = √ [(x2-x1)2 + (y2-y1)2 ]
From the definition of distance we can build up an entire geometric structure, and the Taxicab geometry (TG) arising from d1 is quite distinct from the familiar geometry of Euclid in some very surprising ways.
A family of geometries using metrics based on Lp norms was first considered by the Russian-born German mathematician and physicist Hermann Minkowski. Further developments were made by the American mathematician George Birkhoff. The special case of taxicab geometry has some advanced applications, but it is primarily studied in recreational mathematics.
Squaring the Circle
In Figure 1, the Euclidean distance from A(1,5) to B(8,9) is d2 = √ [(72 + 42 ) = 8.06. Here we have used Pythogoras’ Theorem. The taxicab distance is d1 = 7 + 4 = 11.
Figure 1: Distance between point A (1,5) and B (8,9) in the taxicab metric is 11 units. The square at lower right is actually a circle of radius 3 centred at D.
A circle is defined as the set of points that are equally distant from a given point (the centre), the distance being the radius of the circle. In the Taxicab metric, circles are shaped like squares with sides oriented 45° to the axes. Figure 1 above shows a circle of radius 3 or diameter 6, centred at point D(7,3). Each straight section is of (TG) length 6, so the circumference is equal to 24.
We define π to be the ratio of the circumference of a circle to its diameter. In Euclidean geometry, π = 3.14159 … . In taxicab geometry, we are in for a surprise. For the circle centred at D(7,3),
π1 = ( Circumference / Diameter ) = 24 / 6 = 4.
This can be shown to hold for all circles so, in TG, π1 = 4. Thus, we have
Circumference = 2π1r and Area = π1 r2
where r is the radius.
Some more Remarkable Aspects of TG
Taxicab geometry satisfies all the basic axioms of Euclidean geometry, as expressed systematically by David Hilbert, with one exception. The so-called SAS congruence axiom fails to hold. This posits that two triangles having two sides and the included angle equal are congruent, or equal in all respects. For TG, two triangles may have two sides and the included angle equal and yet by quite distinct.
We define the magnitude of an angle as the length of a circular arc between the sides divided by the lengths of those sides. In the Euclidean domain this ratio is π/2 for a right angle and 2π for a complete circle. Looking again at Figure 1, each straight part of the circle centred at D has arc-length 6. Since the radius is 3, the angle subtended by each straight section is 2. Thus, in TG, a right angle has size π1/2 = 2 TG radians and there are 2π1 = 8 radians in a full circle.
A triangle may be equilateral and yet have unequal angles. In Figure 2 we have a triangle with vertices at (-3, 2), (1, -2) and (3, 4). All sides are of length 8, and two of the angles, A and B, are 1.5 (TG) radians, but angle C is 1 (TG) radian. Note that the sum of the three angles is π1 = 4. This is true for all triangles.
Figure 2: An equilateral triangle with unequal angles.
In Euclidean geometry, the angles at the base of an isosceles triangle are equal. This result, known as Pons Asinorum, the Bridge of Asses, is a fundamental theorem, taught to generations of students and often regarded as a benchmark of mathematical talent. In TG, it is no longer true. It has been suggested that the taxicab has rendered the donkey obsolete.
In TG, an arbitrary angle may be trisected or, indeed, divided into any integral number of equal parts. This is in complete contrast to the Euclidean case.
Pythogoras’ Theorem does not hold in TG. We may consider, for example two triangles:
Triangle T1: Vertices at (0, 0), (0, a) and (a, 0)
Triangle T2: Vertices at (0, 0), (-a, a) and (a, a).
Both triangles have right angles at the origin. For T1, the lengths of the sides are {a, a, 2a}. For T2, they are {2a, 2a, 2a} (T2 is actually an equilateral triangle with a right angle!). For both T1 and T2, the hypotenuse is of length 2a; clearly, no unique relationship between the hypotenuse and the other sides can hold.
Finally …
In conclusion, taxicab geometry provides a delightful contrast to the standard Euclidean geometry studied for thousands of years. It is very useful as a teaching aid and has great pedagogical value, serving to shine a new light on the nature of Euclidean geometry.
TG provides a range of striking and counter-intuitive results, most of which can be demonstrated by mathematical reasoning that is completely elementary. And it has real-world relevance: the taxi fare you pay depends on d1, not on the Euclidean distance!
ThatsMaths 