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<spacer> <spacer> Taxicab Geometry > The Basics

The heart of the contrast between Euclidean geometry and taxicab geometry is the change to the distance formula. As an example, the Euclidean distance between the points (1,1) and (4,5) is

However, the taxicab distance between the two points is

The difference between these two distance calculations is best seen visually. Figure 1 illustrates the difference. In Euclidean geometry, the distance between the points can be found by traveling along the line connecting the points. Inherently in the calculation you use the Pythagorean Theorem to compute the distance.

FIGURE 1: Visualizing distance in Euclidean and taxicab geometry.
The Euclidean distance is 5; the taxicab distance is 3 + 4 = 7.

In taxicab geometry, the usual explanation is that to travel from one point to the next you must use only horizontal and vertical streets as if traveling by taxi in a perfect city. For the points given, the Euclidean distance according to the Pythagorean Theorem is 5 but the taxicab distance is simply the sum of the distances in the horizontal and vertical directions: 3 + 4 = 7.

In general, the only time you can expect the Euclidean and taxicab distance to agree is when both points lie along a horizontal or vertical line (Figure 2).

FIGURE 2: The Euclidean and taxicab distances are equal when
both points lie along a horizontal or vertical line.

The City-Dweller's Shortest Distance

Anyone who has walked in a city center area where they are constrained to walk on sidewalks has likely noticed the concept of "closer" is different than normal. As an example, in Figure 3, as the crow flies the courthouse is closer to the post office (5 units) than the theater is (6 units). But, for one confined to walk the city streets, the theater is closer to the post office (6 units) than the courthouse is (7 units).

FIGURE 3: A city-dweller's view of shortest distance is different than a crow's:
the theater is closer to post office than the courthouse.

The Distance Impact

The concept of distance has far-reaching effects in a geometry. Everything from angles to conic sections to area and volume are in some manner dependent on the meaning of distance. Having changed the distance formula for Euclidean geometry and seen some of the basic effects, one should expect the ripple effect to continue deep into other concepts. The remainder of this website looks at different aspects of taxicab geometry to understand their nature and often contrast them with Euclidean geometry.


[1] Krause, Eugene F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry, 1975.
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