by Jonathan Tennenbaum

Our pedagogical discussions concerning the problem “incommensurability” in Euclidean geometry demonstrated, among other things, that the shift from linear to plane, or from plane to solid geometry cannot be made without introducing new principles of measure, not reducible to those of the lower domain. Thus, the relationship of the diagonal to the side of a square can only be constructed in plane geometry, and is inaccessible — except in the sense of mere approximations — to the mode of measurement characteristic of the simple linear domain (i.e., that embodied in “Euclid’s algorithm”). In the following discussion, we propose to explore that change from a somewhat different standpoint.

I choose, as a point of departure for this exploration, the issues posed by any attempt to compare the areas of various plane figures. The famous problem of “squaring the circle” falls under this domain. But I propose, before looking at that, to start with something much simpler. For example: How can we compare the areas of arbitrary polygons, by geometrical construction? Or, to start with, take the seemingly very simple case of rectangles. Let’s forget what we were taught — but do not know! — namely the proposition that the area of a rectangle is equal to the product of the sides. (Actually, even if the assumptions of Euclidean geometry were perfectly true, the proposition in that form is either false or highly misleading: an AREA is a different species of magnitude, distinct from all linear magnitudes.) In the interest of making discoveries of principle, let us resolve to use nothing but geometrical construction.

Experimenting and reflecting on this problem, the insightful reader might come to the conviction, that the problem of the relationships of area among rectangles of different shapes and sizes, pivots on the following special case: Given an arbitrary rectangle, how to construct “many” other rectangles having the equivalent area. Or perhaps even to characterize the entire manifold of rectangles of area equivalent to the given one.

The first line of attack, which might occur to us, were to find a way to cut up the given rectangle into parts, and rearrange them somehow to form other rectangles. Should we admit any limitation to the shapes and numbers of the parts? To avoid a bewildering bad infinity of options, let us focus first on what would appear to be the “minimum” hypothesis, namely to divide the given rectangle into congruent squares (i.e., squares of equal size). A bit of reflection shows us, that such a division is only possible for the special case, that the sides of the given rectangle are linearly commensurable (i.e., are multiples of a common unit of length). So, for example, if the sides of the given rectangle are 3 and 4 units long, respectively, then by cutting the rectangle lengthwise and crosswise in accordance with divisions of the sides into 3 and 4 congruent lengths, respectively, we obtain a neatly packed array of 12 congruent squares. We discover, that it is possible to rearrange those squares to obtain five other rectangles: 4 by 3 (instead of 3 by 4), 2 by 6, 6 by 2, 1 by 12, and 12 by 1 (i.e., six in all counting the original one, or three if we ignore the order of the sides).

Experiment further. If we start, for example, with a square, and divide the sides into five congruent segments, we obtain 25 congruent squares. The “harvest” of rectangular rearrangements is disappointingly small: all we find is the long, skinny 1 by 25!

Carrying out such simple experiments, the attentive reader might detect a number of potential pathways of further inquiry. One of these would be to ask, for a given total number of congruent squares, how many different rectangles can be formed as arrangements of exactly that number of squares? We can then organize the number into species or classes, according to the resulting number of rectangular arrangements (or “rectangular numbers” as the Greek geometers called them). The class of numbers for which only one rectangular arrangement is possible (disregarding the order of the sides) are known as “prime numbers.” After these, we have a class of numbers with exactly two rectangular arrangements, such as 6, 10, 14, 15, 21, etc. (The otherwise mind-destroying game of “Scrabble” might be put to good use, by employing the wood squares for experiments.)

For the present purposes, however, we would like to construct as many different rectangles as possible out of the original one. We note, that the number of rectangles generated from any given division of the rectangle is very narrowly bounded, and certainly does not include all geometrically constructible ones. How to obtain more? If we stick to the method of division into squares, the only option is to increase the number of divisions. So, for example, we can bisect the unit length in our 3 by 4 rectangle, obtaining a division into 6 times 8, or 48 squares. This raises the total number of rectangles obtained by rearrangement to 10 (5 not counting the order of the sides). By repeated such subdivisions, we might hope to increase the density of population of rectangles so generated, whose areas are all equivalent to the area of the original rectangle. It might be interesting to see how the population grows, as we add new divisions.

But, should we be satisfied with this approach? Aren’t we plunging into a “bad infinity” of particulars? Is there no way to obtain an overview of the whole domain? And remember, our geometrical domain is not limited to linear commensurability of sides. Indeed, a bit of reflections suggests, that for EVERY given segment, there must exist a rectangle, whose area is equivalent to the given one, and one of whose sides is that length. How might we construct such a rectangle?

For a glimpse at a higher bounding of our problem, try the following construction: Take the rectangles constructed from any given rectangle by divisions into squares and rearrangement, as above, and superimpose them by bringing the lower left-hand corners into coincidence and aligning the sides along the vertical and horizontal directions. What do you see?

The Curvature of Rectangular Numbers, Part II

The general task, posed in last week’s discussion, was to generate the manifold of rectangles whose areas are equivalent to a given rectangle. The initial tactic chosen, was to divide the given rectangle into an array of congruent squares, and rearrange them into rectangles of different dimensions, but equivalent area. It became clear, that this tactic only yields a discrete “population” of rectangles (“rectangular numbers”), whose number depends on some characteristic of the number of divisions chosen. On the other hand, if we arrange the resulting rectangles in such a way, that their lower left-hand corners coincide, and their sides are lined up along the horizontal and vertical axes, then a hidden harmony springs into view: the upper right- hand corners of the rectangles, so arranged, appear to describe a HYPERBOLA, or at least a hyperbola-like curve. The idea suggests itself, that the discreteness of dividing and rearranging parts to form individual rectangles, is bounded from the outside by a higher continuity (ordering), whose presence reveals itself in the hyperbolic “envelope” of the rectangles.

To proceed further, let us change our tactic, concentrating on the idea, that there must exist a PROCESS of TRANSFORMATION which generates the entire manifold of equivalent-area rectangles and hyperbolic “envelope” at the same time. We might adopt the attitude, that any pair of rectangles of equivalent area expresses a kind of INTERVAL within the implied “hyperbolic” ordering of the whole.

With this in mind, start with any given rectangle, and consider the following approach. If we triple the length of the rectangle, keeping the width the same, then we obtain a rectangle whose area is clearly equivalent to three times that of the original one. If we then reduce the width of the new rectangle to one-third of its original value, while keeping the length unchanged, then the area of the resulting rectangle (with three times, the length, but one-third the width of the original) will clearly be equivalent to the original rectangle’s area. In fact, we might verify that equivalence in the former, discrete manner, namely by dividing the original rectangle lengthwise into three congruent rectangles, and then rearranging them to obtain the new one. In the same way, we could quadruple the length of the original rectangle and reduce its width to one-fourth, and so on. Obviously, nothing prevents us from applying the same procedure with ANY factor (i.e. not only 3 or 4), or from reversing the roles of “length” and “width” in this procedure.

At this point, something might occur to us, which allows us to “jump” the gap between the discreteness of our former procedure, and the underlying ordering of the problem. Up to now, we have considered as primary a process of multiplying or dividing lengths or widths by some integral number. But now we realize, that the crux of the matter, lies not in this duplicating or dividing up, but rather in the relationship of “INVERSION” between the transformation applied to the length and the transformation applied to the width. This suggests a new approach, which does not depend upon whole-number relationships at all.

Thus, take any rectangle with length A and width B. Now imagine A prolonged to ANY ARBITRARY LENGTH X. Those two lengths, A and X, define an interval. Evidently, what we must do, is to “invert” that interval with respect to B! In other words, construct a length Y, for which the interval (proportion) “Y to B” is (in relative terms) congruent to the interval “A to X”.

The required construction can be approached in many different ways. For example, generate a horizontal line, and erect a perpendicular line at some point P. Starting from P, lay off a vertical line segment PQ, whose length is equivalent to X, and determine a point R between P and Q, such that PR is equivalent to the length A. Next, chose an arbitrary point S, lying to the left of P on the horizontal line, and construct a vertical line segment ST whose length is equivalent to B. Now, generate a straight line through the points T and Q. Leaving aside the case, where that line happens to be parallel to the horizontal axis, the line through TQ will intersect the horizontal axis at some point O. Finally, generate a straight line through O and R. That straight line will intersect the vertical line ST at some point U. Reflect on the relationship formed, relative to “projection” from O, between the line segments on the two vertical lines from P and S. Evidently, the interval of PR to PQ (i.e. A to X) is congruent to the interval of SU to ST, the latter being equivalent to B. Thus, SU gives us the value Y for the required “inversion” of the transformation from A to X. In other words, the transformation of A to X, and the transformation from B to Y are inversions of each other, and the rectangle with sides X, Y will have the equivalent area to the rectangle with sides X and Y.

Consider the case, in which the value of X is changing, and observe the manner in which the positions of O and U vary in relation to X. The hyperbolic envelope is already implicit.

Those skillful in geometry will be able to devise essentially equivalent constructions, which make it possible to generate the hyperbolic envelope and the entire array of equi-area rectangles at the same time. Just to give a brief indication: Start with a rectangle, whose sides A and B lie on vertical and horizontal axes. Let O and M denote the lower left-hand and lower right-hand corner-points of the rectangle. Generate any ray from O, with variable angle, which intersects the upper horizontal side of the rectangle, at a point P. Prolonging the right vertical side of the rectangle upward, the same ray will intersect that vertical line at some point, Q. Now draw the vertical line at P and the horizontal line at Q. Those two lines intersect at a point R. Now examine the relationship of the rectangle with upper right-hand corner R and lower left-hand corner O, to the original rectangle. Examine the motion of R as a function of the angle of the ray from O.

For those who feel the compulsion to scribble algebraic equations, now is the time to kick the habit! The whole point here is to think GEOMETRICALLY. The notion of “geometrical interval” supercedes that of discrete arithmetic relationship…