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Generating Pythagorean Triples: A Mini-Primary Source Project in Number Theory for Mathematics Majors, Elementary Teachers and Others

Author(s): 
Janet Heine Barnett (Colorado State University - Pueblo)

In his commentary on Euclid's proof of the Pythagorean Theorem, Proclus (c. 411 - 485 CE) asserted (Proclus, pp. 339—340):

There are two sorts of right-angled triangles, isosceles and scalene. In isosceles triangles you cannot find numbers that fit the sides; for there is no square number that is the double of a square number, if you ignore approximations, such as the square of seven which lacks one of being double the square of five. But in scalene triangles it is possible to find such numbers, and it has been clearly shown that the square on the side subtending the right angle may be equal to the squares on the sides containing it. Such is the triangle in the Republic, in which sides of three and four contain the right angle and five subtends it, so that the square on five is equal to the squares on those sides. For this is twenty-five, and of those the square of three is nine and that of four sixteen. The statement, then, is clear for numbers.

Certain methods have been handed down for finding such triangles, one of them attributed to Plato, the other to Pythagoras.

Although their actual discovery is now hidden in the long shadow of history, the two methods for generating Pythagorean triples described by Proclus remain of interest in number theory today. Proclus’ simple description of how to generate Pythagorean triples using each method offered no speculation about how these algorithms may have been discovered. The mini-Primary Source Project Generating Pythagorean Triples provides students the opportunity to explore how these methods might have become known through an intriguing theory related to the Greek notion of a gnomon.

With etymological roots in common with the English words gnostic, agnostic and ignorance, the literal meaning of the Greek work gnomon is "that which allows one to know.'' In astronomy, a gnomon is the part of a sundial that casts a shadow, thereby allowing one to know the time. In the ancient world, a vertical stick or pillar often served as the gnomon on a sundial. The term gnomon was also associated in ancient Greek architecture with an L-shaped instrument, sometimes called a 'set square,' that was used for the construction of (or 'knowing of') right angles.


A hand-carved table-top analemmatic sundial.
By John Carmichael (Own work) [Public domain], via Wikimedia Commons.

By Xofc - Own work, GFDL, https://commons.wikimedia.org/w/index.php?curid=4937786,
licensed under the Creative Commons Attribution-Share Alike 4.0 International, 3.0 Unported, 2.5 Generic, 2.0 Generic and 1.0 Generic license.

Within Greek mathematics, Euclid used the term gnomon to refer to the plane figure formed by removing any parallelogram from a corner of a larger similar parallelogram. Figure 1 shows the L-shaped gnomon associated with a square. Eventually, Greek mathematicians began to also use the term to refer to the increment between two successive figurate numbers. The square number gnomon diagram shown in Figure 2 is suggested in particular by certain passages in the Physics in which Aristotle (c. 384 - c. 322 BCE) discussed the mathematical beliefs of Pythagoras and his followers.


Figure 1


Figure 2

The student project Generating Pythagorean Triples itself begins with an exploration of the primary source excerpt from Proclus above, followed by a brief introduction to the mathematical concept of a gnomon. It then returns to Proclus’ description of the two methods for generating Pythagorean triples which he attributed to Pythagoras and Plato respectively. For each of these two methods, students are presented with a series of tasks. These include numerical tasks based on Proclus' purely verbal descriptions of the two methods, algebraic tasks in which those verbal formulations are translated into symbolic formulas, and geometric tasks that connect the numerical and algebraic formulations of the method in question to gnomons in a figurate number diagram.

Two versions of the project are available, one somewhat more open-ended than the other. Beyond some basic arithmetic and (high school level) algebraic skills, no mathematical content pre-requisites are required in either version. This mini-PSP can thus be used with a wide range of students for whom the study of number theory is part of the curriculum.

Both versions of the project Generating Pythagorean Triples are ready for student use, subtitled The Methods of Pythagoras and of Plato via Gnomons (pdf file) and Gnomonic Explorations (pdf file) respectively. The latter version is the more open-ended, exploratory of the two versions. A set of instructor notes offering practical advice for the use of the project in the classroom is appended at the end of each version. These notes include more detail about the differences between the two versions of the project and the suitability of each for various student audiences. The LaTeX source of the complete project is also available from the author by request.

This project is the fourth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection includes eight PSPs for use in courses on number theory.

Acknowledgments

The development of the student project Generating Pythagorean Triples has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation's Improving Undergraduate STEM Education Program under grant number 1523494. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily represent the views of the National Science Foundation. The author also wishes to thank George W. Heine III for creating the gnomon diagrams in Figures 1 and 2.

References

Proclus. A Commentary on the First Book of Euclid's Elements. English translation by Glenn R. Morrow (1970). Princeton: Princeton University Press.

Janet Heine Barnett (Colorado State University - Pueblo), "Generating Pythagorean Triples: A Mini-Primary Source Project in Number Theory for Mathematics Majors, Elementary Teachers and Others," Convergence (November 2017)

Dummy View - NOT TO BE DELETED