# Geometry: Max Geometry

## Max Geometry

With the advent of the taxi-cab geometry, mathematicians learned that if they used a different method to measure the distance between two points, a new geometry was created. So they began to play around with other acceptable definitions of distance. Now there are more non-Euclidean geometries than you can shake a stick at.

Another interesting non-Euclidean geometry that has its origins in a new distance formula is the max geometry. In the max geometry, the notion of the distance between two points, say (a,b) and (c,d), is defined to be either ? a - c ? or ? b - d ? , depending on which is bigger. That's how it got its name: the max distance is the maximum of ? a - c ? and ? b - d ? . We usually write this distance formula as d_{Max} = Max{ ? a - c ? , ? b - d ? }.

Let's go back to the last problem: You have two points, one with coordinates (1,3) and the other with coordinates (4,7), and you want to know the max distance between these two points. Using the formula,

- d
_{Max}= Max{ ? 4 - 1 ? , ? 7 - 3 ? } = Max{3,4} = 4.

So the distance between the points with coordinates (1,3) and (4,7) depends on what is meant by distance. The distance between them ?as the crow flies? (that is, the Euclidean notion of distance) is 5. The distance between the two points according to the taxi-cab metric is 7, and the distance according to the max metric is 4. All three notions of distance have their uses, though the max metric is the most obscure, and is probably used by mathematicians more than any other group. It might be helpful to think of the max distance between two points as being the furthest distance in either the North/South or East/West directions.

##### Eureka!

The max distance between two points **(a,b)** and **(c,d)** is determined by the formula d_{Max} = Max{? a - c ? , ? b - d ? }.

Excerpted from The Complete Idiot's Guide to Geometry 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with **Alpha Books**, a member of Penguin Group (USA) Inc.

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