Many shapes in our world are irregular and complicated. The boundary of a lake or stream can be curved and winding. A stock market graph can be highly erratic, with jumps and dips of various sizes. The path of a fly ball is a smooth arc affected by gravity and wind resistance. Clouds are filled with cotton-like bumps and ripples. Measuring our world requires us to confront a variety of irregular shapes, ranging from graceful curves to sharp, jagged edges.
Measuring Irregular Shapes
Irregular shapes can be measured with a technique called discrete approximation, a powerful method that provides the foundation for calculus and a means by which computers perform calculations. Making a discrete approximation involves representing a continuous quantity through a collection of distinct pieces. We live with such approximations every day. For example, a movie reel is a collection of picture frames, shown to us rapidly on a screen to give the impression of a continuous flow of events. Computer screens and laser printers represent images by a collection of small, tightly packed cells (called pixels), joined together to give the impression of a continuous image. If one looks closely at a computer or television screen, the small cells can become noticeable.
Measurement with discrete approximation involves dividing an irregularly shaped object into a collection of smaller pieces whose measurement is more manageable. For example, consider an overhead view of a lake, whose boundary rests upon a square grid (see figure below). Every square that contains
a portion of the lake is completely shaded. The resulting collection of shaded squares is a discrete approximation of the lake, much like a collection of pixels providing a digital image of the lake. The lake's area can be computed by simply adding up the areas of the shaded squares. The larger grid in figure (a) results in a sizable overestimate of the lake area, but the smaller grid in figure (b) shows a notable decrease in the approximation error. Further refinement of the grid leads to increasingly better approximation of the lake area, much like how a digital image becomes sharper with an increasing number of pixels.
In a similar manner, the figure below shows a discrete approximation of a curve using a collection of line segments. The curve might represent a portion of a roller coaster track, a path through snow-covered hills, or the trajectory of a helicopter. The length of each line segment is determined by applying the Pythagorean theorem to a standard right triangle construction. Using this relation, the length of each line segment is equal to . The curve's length is approximated by the sum of the lengths of the line segments. Using an increasing number of smaller line segments gives an increasingly better representation of the curve, just as a regular polygon appears more like a circle as the number of polygon sides increases.
Approximating irregular shapes by collections of squares or line segments may seem unsophisticated and cumbersome. Such a measurement approach, however, is intimately connected with the tendency of curves to look simpler, indeed straighter, as we magnify our view of the curve. This fact has a profound implication for measurement, allowing for the definition of a slope at any point along a curve. Consider, for example, the graph of the parabola y = x 2 shown in figure (a) on the next page. Let us focus on the ordered pair (1, 1), and keep enlarging the region around this point using the zooming capabilities found in graphing calculators and other graphing software [see (b) and (c)]. The closer we zoom into a point on the graph, the straighter the graph looks, much like the curved surface of Earth can appear flat from close range. By the eighth magnification, the graph looks effectively linear and a two-point slope calculation makes sense. Using the two points in the eighth-zoom window, the slope of the graph is calculated to be
The eighth-zoom calculation gives a slope of 2 to within three decimal places, and with enough magnification, a slope calculation gives this result to within any number of decimal places. With the zooming visualization as an aid, the notion of approximating irregular shapes by small line segments or squares should seem more reasonable. Upon magnification, complicated shapes tend to look increasingly like the simple shapes upon which measurement is based.
Nature shows many examples of small building blocks in the fabric of its construction, like cells for organisms, atoms and molecules for substances, and photons for light. The mathematical technique of breaking down complicated objects into simpler pieces has both historical context and modern relevance. Using the idea of a limit, calculus offers a method for shrinking simple measurement pieces and making approximation error disappear. The result is a robust mathematical language applied by mathematicians and scientists since the late-seventeenth century in a wide variety of contexts, from direct measurement of physical quantities to the creation of equations that describe physical processes.
The modern digital revolution is intimately connected with the discrete representation of sound, images, and information. Computerized processes are founded upon discrete packages of information (called bits), and rapidly increasing technological power has enabled realistic, high-quality discrete representations of our world through computer graphics, compact discs, digital cameras, laser printers, and so on. Simpler building blocks for complicated objects are not only part of nature's fabric, they provide an increasingly pervasive means by which we measure, describe, and model the complexity of our world.
see also Calculus; Limit.
Beigie, Darin. "Zooming In on Slope in Curved Graphs." Mathematics Teaching in the Middle School 5 (2000): 330–334.
Hughes-Hallett, Deborah, et al. Calculus, 2nd ed. New York: Wiley, 1998.