Calculus in Context as a Pyramid of Oranges

The setting is the crowded Mercat de la Boqueria in Barcelona, by reputation one of the best fresh food markets in Europe. Scores of fruit vendors display varieties of apples, pears, peaches, plums, lemons, apricots, cherries, grapes, grapefruit, and oranges in but the few square meters of area that each has available. In competition with each other, they vie for the attention of housewives and tourists who press ahead from stall to stall.

One of the merchants has arranged his Valencia oranges in an eye-catching display. His fruit is large, round, polished, beautiful, and the tall pyramid of their arrangement glows rich in color, light, and shape. It is striking and draws much attention. The child of a customer is about to reach for an orange near the bottom of the stack, … but our story now turns to an analysis of the pyramid. The merchant’s attractive display had been constructed very carefully. The pyramid has a triangular base of 12 oranges to a side. From each side of this triangle rises one of the sloping triangular faces of the pyramid.

Each triangular face starts with its row of 12 and continues with rows of 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 oranges as it ascends. An addition of these numbers tells us that each of the three faces of the pyramid is a triangle with 78 oranges. A simple addition informs us that there are 78 + 78 + 78 = 234 oranges visible on the rising surface of the pyramid. But this count is too simple. Counted once for each triangular face, the orange that presides over the pyramid from its perch at the very top is included three times, and each of the 11 oranges positioned below it along the three slanting edges of the pyramid is each taken twice. To correct for the multiple count, 2 has to be subtracted for the orange at the top and 11 for each of the three edges. It follows that a total of 234 – (2 + 3 · 11) = 234 – 35 = 199 oranges are visible on the three ascending triangular sides of the pyramid. A similar count for each of the 12 horizontal triangular sections of the pyramid, would have informed us that our merchant’s pyramid consists of 364 oranges in all. In addition to the 199 oranges visible on the pyramid’s surface, there are another 165 oranges that form the smaller interior pyramid that holds up the visible array from below. It might come as a surprise that so many, namely 199 of the 364 or about 55%, of all the oranges of the tower are visible on its surface.

An important issue is the stability of the pyramid. How did it remain stable during the time it was assembled? Wouldn’t the oranges on the three sides of the triangular base have been forced outward by the collective weight of the oranges above them … leading to the collapse of the growing stack? Could billiard balls have been arranged in the same way? Our merchant had been careful to always place a flatter part of an orange at the bottom. With this strategy and the force of friction between adjacent fruit, the lower sections of oranges remained in place as the higher tiers were added. This would not have been so for billiard balls. Very round and very smooth, they would not have provided sufficient resistance to these outward thrusts. Only a triangular rack of the sort that is standard equipment at every pool table could have kept the lowest triangle of balls (and hence the entire scheme) tightly in position. As a precaution, our fruit merchant had placed a similar barrier around the perimeter of his display.

The story of the pyramid of oranges provides a sense of what Calculus in Context is all about. Of course, the setting of the story is not the Mercat de la Boqueira in Barcelona and the subject is not a pyramid of oranges. Instead, the setting is a carefully selected flow of mathematical ideas with a focus on those that underlie calculus. But the oranges are a metaphor. Their arrangement into the layers of the pyramid gives a sense––simplistic certainly, but a sense nonetheless––of their relationships to each other and how they rise to form a structure that is beautiful. But there is more than beauty. The ideas provide incisive insights into and unlock a storehouse of information about a broad range of disciplines of science, engineering, and beyond.

The first four chapters of Calculus in Context present the basics of elementary geometry, trigonometry, algebra, and coordinate geometry. This discussion of the building blocks which calculus deploys––think of them as the oranges of the bottom tier of the pyramid––­­­­is framed and illustrated by the approaches to geometry and astronomy of ancient Greece (Euclid, Archimedes, and Ptolemy, for instance) and––even more importantly––by the mathematical methods that informed the scientific revolution that produced the Sun-centered model of the motion of the planets (as Copernicus, Galileo, and Kepler carried it out). Chapter 5 draws a brief, basic calculus course from the pages of Leibniz and Chapter 6 adds fundamental insights and applications from the work of Newton. Both Newton and Leibniz had a good intuitive sense of their calculus, but were not able to give it a rigorous foundation. In current context, this is a plus. It means that what emerges is the unvarnished essence of the matter without the elements of a theoretical substructure that was added later. Part I of the text is a complete essential calculus and its precalculus preliminaries. While the subject is framed by the historical context, the notation and the selection of topics is current. The scientific revolution reached its culmination in the latter part of the 17th century with the publication of Newton's Principia and its explanation of the planetary orbits with a combination of fundamental laws of motion and force and the strategies of calculus. A discussion of the central aspects of this challenging work concludes the first part of the text.

Part II of the text starts by engaging calculus again, this time much more deeply and fully. Chapter 7 consists of a comprehensive treatment of the essentials of differential calculus and Chapter 9 does the same for integral calculus. A sense is provided how the mathematicians of the 19th century reformulated the fundamental concepts, moved the function concept to center stage, and provided calculus with the logical foundation that gave validity to its assertions. Given the applications in later chapters, exponential, log, and hyperbolic functions are added to the collection of the library of functions. A course may well start with Part II and use the discussions of Part I––especially Chapters 1 and 4––for reference and review. After the presentation of the basics of calculus, Part II turns its attention to applications. At this point, an instructor’s strategy vis-a-vis the topics presented in this text should shift more into a “pick and choose” approach (much like the shoppers at the mercado).  If the power of calculus to inform is to be illustrated convincingly and beyond such trivial (or contrived) situations as “the farmer and the fence,” then explanations of some of the very basic aspects of science and engineering that calculus informs need to be included. The fact is that the explanation of such central concepts as acceleration, force, momentum, torque, moments of inertia, and the properties of lenses require core ideas of calculus. Chapters 8 and 10 apply calculus (differential in the first case and integral in the second) to mechanics (in particular to statics and dynamics), optics, architecture, the geometry of the pseudo-sphere, planetary motion, and ballistics. Chapter 11 provides the fundamentals of differential equations and then turns to applications. The focus is on first- and second-order differential equations, complex numbers, power series, and applications to motion with air resistance and the action of shock absorbers. Chapter 12 starts with a study of polar functions and then completes the agenda of the text by returning to Newton's orbital mechanics and applying the calculus of polar functions to reach its important conclusions. The various applied segments of the text do not build on each other but are independent. Whether they are selected or not is entirely up to the intentions and preferences of the instructor.

Expressed with the metaphor of language, this book combines the vocabulary and basic grammar of calculus with much of the poetry and the stories that have been written with it.  Another metaphor, if a bit superficial, is the Mercat de la Boqueira. The rich display of tasty fruit is the mathematics and, in particular, calculus in its simplest and most complex forms. Like the displays––see the image on the first page––­mathematics is infinite in terms of its configurations, juxtapositions, interconnections, and colors. The growers, the transporters, the vendors and also the shoppers are the mathematicians and those who apply and use it. They build the displays, arrange, rearrange, and add to them, select carefully from them, and take what they need away to consume them.   

 

Alexander J. Hahn is a professor of mathematics at the University of Notre Dame. He is the author of Basic Calculus: From Archimedes to Newton to Its Role in Science and Mathematical Excursions to the World’s Great Buildings and Calculus in Context: Background, Basics, and Applications.