Sums of Powers in Discrete Mathematics: Archimedes Sums Squares in the Sand

Introduction

Figure 1. Portrayal of the mathematician Archimedes (3rd century BCE) by Italian painter Domenico Fetti (c. 1589-1623) (Source: Wikimedia Commons) 

In the third century BCE Archimedes made astonishing discoveries of precise areas and volumes, including the area contained within one turn of what we call an Archimedean spiral. His proof hinged on determining a "closed formula" for a sum of squares. Here we introduce and provide for instructors a student project based on Archimedes' original source analyzing a sum of squares, designed for courses on introductory discrete mathematics or calculus.

Figure 2. Archimedean spiral

Today we express Archimedes' spiral in polar coordinates as \(r=k\theta\), and boil the area down to calculating \(\int_{0}^{2\pi}\theta^{2}d\theta\). Then we either allow ourselves the late seventeenth century fundamental theorem of calculus and antidifferentiate, or argue this is equivalent to finding an area under the curve \(y=x^{2},\) and modernize earlier seventeenth century techniques by calculating a limit of approximating (Riemann) sums. To evaluate this limit, though, we will need to have a closed form polynomial for the sum \(\sum_{i=1}^{n}i^{\,2},\) or at least know that its leading term is \(n^{3}/3\). We can then write, as did Archimedes, that "the area bounded by the spiral and the straight line which has returned to the position from which it started is a third part of the circle ...." But how did Archimedes do this almost two millennia earlier? His approximations involved inscribing and circumscribing circular sectors around the spiral, and using the Greek method of exhaustion as a rigorous alternative to a modern limit. This led him to tackle the very same sum of squares.

Figure 3. Archimedes used circle sectors to approximate the area contained in one turn of the Archimedean spiral.

Thus began the epic search for closed formulas for sums of \(k\)-th powers \(\sum_{i=1}^{n}i^{\,k}\), which stretched from the Pythagoreans and Archimedes through many mathematicians, including Nicomachus, Aryabhata, al-Karaji, ibn al-Haytham, Fermat, Pascal, Bernoulli, and Euler, culminating in the discovery of the Bernoulli numbers and the Euler-MacLaurin summation formula, which enabled Euler to rise to fame by first guessing and then proving that the sum of the reciprocal squares is \(\pi^{2}/6\).

Figure 4.  Pictorial representation of Archimedes' analysis of a sum of squares

This quest for formulas for sums of powers was a primary theme throughout the evolution of discrete mathematics, and students may learn it from beginning to end through primary historical sources in [1, Chapter 1]. From two portions of that sequence we have created primary source project modules for students, the present one for an introductory discrete mathematics or calculus course, and the other, Figurate Numbers and Sums of Numerical Powers: Fermat, Pascal, Bernoulli, for a more advanced combinatorics course. Both projects are part of a larger collection published in Convergence, and an entire discrete mathematics course can be taught from various combinations selected from these projects. For more projects, see Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science.

Our project, Sums of Powers in Discrete Mathematics: Archimedes Sums Squares in the Sand, is ready for students, and the Latex source is also available for instructors who may wish to modify the project for students. The comprehensive "Notes to the Instructor" presented next are also appended to the project itself.

Notes to the Instructor

This project is for students of introductory discrete mathematics or calculus (Riemann sums). It is based on Archimedes' writing on sums of squares in the service of finding areas and volumes.

This project has no formal prerequisite other than basic algebra. The project is quite flexible, and the instructor can pick and choose from various activities offered. The full project can be completed within two class weeks or less, and for a shorter project the instructor may choose selectively. Students can work productively in groups on this project, with group or individual writeups. One excellent challenge the project provides is interpreting Archimedes' verbal descriptions of adding up many magnitudes, and manipulative rods or graph paper have been encouraged productively by some instructors.

The goal of the project is for students to learn many basic notations, techniques, and skills in the context of an historically and mathematically authentic big motivating problem with multiple connections to other mathematics. Hopefully this will be much more effective and rewarding than simply being asked to learn various skills for no immediately apparent application. Many of the techniques first introduced in a discrete mathematics or calculus course arise naturally as needed in this project, like reindexing summation notation, working with algebraic inequalities, and telescoping sums. Instead of separately covering various such topics and techniques, that class time can simply be spent on the project, and students will learn those things in the process. The project has the integrating theme throughout of the application of sums of powers to Riemann sums and area calculation, and vice versa.

Most of the ideas and knowledge in the project are acquired through guided discovery exercises, a number of which are open-ended, so the instructor should work through all the details before assigning any student work, and select carefully from the big picture if exercises are omitted. Students may need substantial guidance with some parts.

The project asks students to interpret and convert verbal descriptions into modern mathematical formulations, conjecture from patterns they generate, develop their mathematical intuition and judgement, and try proving their conjectures, i.e., putting students in the creative driver seat. The setting of sums of powers in the context of primary sources allows a richness of questions and interpretations, especially includes deep connections to geometry and the two-way interplay with calculus, as well as basic algebra and linear algebra, and a richness of proof techniques, including natural comparison of the efficacy of various proof methods.

Download the project Sums of Powers in Discrete Mathematics: Archimedes Sums Squares in the Sand.

Download the modifiable Latex source file for this project.

For more projects, see Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science.

Bibliography

[1] A. Knoebel, R. Laubenbacher, J. Lodder, D. Pengelley, Mathematical Masterpieces: Further Chronicles by the Explorers, Springer Verlag, New York, 2007.

Acknowledgment

The development of curricular materials for discrete mathematics has been partially supported by the National Science Foundation's Course, Curriculum and Laboratory Improvement Program under grants DUE-0717752 and DUE-0715392 for which the authors are most appreciative. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.