Bounded Distance

The distance function dista defined for points on the plane via the stereographic projections has a property usually unexpected of distance - it's bounded. For many reasons in mathematics it is often convenient to work with bounded functions, distances in particular. There is a simple mechanism to convert a given distance function D(x, y) into (in a sense, equivalent) a bounded distance function d(x, y). The approach is indeed simple: let d(x, y) = D(x, y)/(D(x, y) + 1). I'll show that if D satisfies the axioms of distance, then so does d. It is obvious that when distance is measured with the function d, no two points may be farther away from each other than 1. In other words, the unit circle defined with distance d contains the whole of the plane. This is true regardless of the location of its center.

The first and second axioms hold trivially for d if they do for D: D(x, y) = 0 iff d(x, y) = 0. Also, D(x, y) = D(y, x) implies d(x, y) = d(y, x).

Let's then verify the triangle inequality d(x, z) ≤ d(x, y) + d(y, z).

One thing to note is that the function f(a) = a/(a+1) is monotone increasing. From here, if, for example, D(x, z) ≤ D(x, y) then also d(x, z) ≤ d(x, y), in which case the triangle inequality is immediate. This is also true if D(x, z) ≤ D(y, z). Therefore, we can assume that D(x, z) > D(x, y) and D(x, z) > D(y, z). Thus we obtain:

  d(x, z) = D(x, z)/(D(x, z) + 1)
D(x, y)/(D(x, z) + 1) + D(y, z)/(D(x, z) + 1)
< D(x, y)/(D(x, y) + 1) + D(y, z)/(D(y, z) + 1)
= d(x, y) + d(y, z).

Although D is not bounded whereas d is, when one is close to 0 so is the other.Thus both distances induce the same notion of nearness in the sense that sets closed in one are also closed in the other and vice versa.

There are many functions that, like f(a) = a/(a + 1), help produce new distance functions. For example, square, cubic and other root functions have this property. These functions are monotone, but monotonicity alone is not sufficent to ensure the required property. For example, f(a) = a2 does not lead to new distance functions. (I thank Professor Frank Plastria for the example and for pointing out an error in an earlier version of this page.)

In any event, in any metric space there is a great variety of distance functions.

Consider, for example, the set of real numbers with the usual distance D(r,s) = |r - s| and the function g(r) = r. Then dg is a distance function that automatically satisfies the metric axioms. In particular, (a + b)a + b, for non-negative a and b, implies the triangle inequality:

  |r - t||r - s| + |s - t| .

There is another way of forming new distance functions from a given one. Let f be a 1-1 function from a set X into a metric space Y with the metric function D. X becomes a metric space if we introduce the distance as df(x, y) = D(f(x), f(y)). Since in reality the distance is measured between images f(x) and f(y), df is indeed a distance function.

I was reminded of various distance functions when reading of the distance effect described in The Number Sense by S.Dehaene. In Chapter 3, the author describes a series of experiments where a subject was presented with flashing pairs of digits such as 9 and 7 and had to point out the larger of the two. I cite:

 

This elementary comparison task was not as easy as it appeared. The adults often took more than half a second to complete it, and the results were not error-free. Even more surprising, performance varied systematically with the numbers chosen for the pair. When the two digits stood for very different quantities such as 2 and 9, subjects responded quickly and accurately. But their response time slowed by more than 100 milliseconds when the two digits were numerically closer, such as 5 and 6, and subjects then erred as often as once in every ten trials. Moreover, for equal distance, responses also slowed down as the numbers became increasingly larger. It was easy to select the larger of the two digits 1 and 2, a little harder to compare digits 2 and 3, and far harder to respond to the pair 8 and 9.

Let there be no misunderstanding: The people that Moyer and Landauer tested were not abnormal, but individuals like you and me. After experimenting on number comparison for more than ten years, I still have yet to find a single subject who compares 5 and 6 as quickly as he or she compares 2 and 9, without showing a distance effect. I once tested a group of brilliant young scientists, including students from the top two mathematical colleges in France, the Ecole Normale Supérieure and the Ecole Polytechnique. All were fascinated to discover that they slowed down and made errors when attempting to decide whether 8 or 9 was the larger.

Nor does systematic training help. In a recent experiment, I attempted to train some University of Oregon students to escape the distance effect. I simplified the task as much as possible by presenting only the digits 1, 4, 6, and 9 on a computer screen. The students had to press a right-hand key if the digit they saw was larger than 5, and a left-hand key if it was smaller than 5. One can hardly think of a simpler situation: If you see a 1 or a 4, press left, and if you see a 6 or 9, press right. Yet even after several days and 1,600 training trials, the subjects were still slower and less accurate with digits 4 and 6, which are close to 5, than with digits 1 and 9, which are further away from 5. In fact, although the responses became globally faster in the course of training, the distance effect itself -- the difference between digits close to 5 and far from 5 -- was left totally unaffected by training.

With the assumption that it's easier to tell apart more distant numbers, we are looking for a distance function for which the distance between, say, 5 and 10 is greater than the distance between 55 and 60 which, in turn, exceeds the distance between 105 and 110.

On the positive real axis, let again D(x, y) = |x - y| be the standard distance between real numbers measured as the absolute value of their difference. Consider df, where f is the square root function. Fix a and let y = x + a. Then df(x, x + a) is a decreasing function of x. You may want to verify that, as x becomes larger, the distance between x and (x + a), for a fixed a, becomes smaller. The same is true for other functions, the cubic root in particular. I am not qualified to speculate, but the latter seems to be the most promising, given the theory developed in the book that our brain converts digital information into analog before processing it further. Our brains being 3-dimensional, cubes must pop up somewhere.

References

  1. S.Dehaene, The Number Sense, Oxford University Press, 1997

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