Manhattan Metric 2

In the previous post we saw use of the program Galton that maps out on city streets how far you can travel in 10 or 20 minutes. Displayed on a rectangular array of streets and avenues, square or rectangular regions develop, as walking is constrained to follow the paths of the gridded streets.

The image above is a Google Earth view of the parking lot of an office building in Maryland. Commuters have parked their cars to enter a building just off the image at the lower left. They must follow perpendicular paths and walk between the cars and/or along the lanes to enter the building. But to minimize the distance of the walk, most have parked along lines of equal distance from the bottom left according to the Manhattan or city block metric. A few stragglers don't fit this pattern, perhaps wanting to protect their cars from door dings or just get a little extra exercise. But the prominent pattern in the image above is one quadrant of the rectangular 'circle' of the city block metric.

 This line graphic from Taxicab Geometry.

Manhattan Metric

Urbica is a design firm specializing in urban data analysis. They have developed a program called Galton that graphs, for a few select cities, how far you could walk in 10 minutes (in dark blue) or 20 minutes (in lighter blue). The map of Manhattan above shows those regions for a walk originating at Broadway and 42st Street. As you walk NYC you are, for the most part, constrained to travel the grid of avenues and streets. Of course, you cannot travel as the crow flies. If you could, these regions would be concentric circles with a perimeter an equal (Euclidean) distance from your start. But walking the streets, your distance is measured by the city block metric (also known as the taxicab metric or more appropriate here the Manhattan metric). This measures distances constrained along perpendicular avenues and streets. Plotting points of equal distance with this metric from would result in the roughly rectangular (or diamond-shaped) regions shown above. Since the streets and avenues are not equally spaced and obstacles can block our travel, we don't see perfect square or rectangular regions. By the Manhattan metric, circles become squares.
Via Maps Mania.

Next week, we will see directly the results of minimizing the distance traveled in such constrained walking.