A Poem-Collage Project
Poem by Sarah Glaz  with Collage by Mark Sanders

 

Sarah Glaz  University of Connecticut, Storrs,  Connecticut, USA    Website

Mark Sanders  Rushden, Northamptonshire,    UK  Website

History, Mathematics, Poem, Collage  
   

Diophantus (ca. 200 - 284 CE) was a Greek mathematician who lived in Alexandria, Egypt. Considered the "Father of Algebra," he authored a work of great genius, Arithmetica. Innovative in both subject matter and approach to problems, Arithmetica paved the way to the development of symbolic algebra, and to the advancement of the area of mathematics called number theory.

 

Diophantus lived during a dark and turbulent time in the history of Alexandria. The city with its legendary institutes of learning, the Museum and its magnificent library, began its decline several hundred years before. The Golden Age of Greek Mathematics initiated by Euclid ended in the 2^nd century BCE when anarchic conditions brought on by political strife and frequent ethnic and religious clashes contributed to the decline of original scholarship at the Museum. A further blow was delivered by the Roman conquest of the city and the establishment of a military regime in 30 BCE. Shortly before Diophantus' birth, during the Kitos Wars, a large part of the city was destroyed. The situation deteriorated further during Diophantus' lifetime. In 215, Emperor Caracalla visited the city and, insulted by a satire directed at him by the city's inhabitants, ordered the slaying of all the city's youth capable of bearing arm. Alexandria features prominently in both poem and collage. The main thoroughfare of Alexandria, the Canopic Way, mentioned indirectly in the first few lines of the poem, is depicted in the collage below and to the right of the central map segment. Canopic Way stretched from the Sun Gate in the east (represented by William Blake's figure, from "The Sun at His Eastern Gate," holding high a copy of Diophantus' Arithmetica), to the Moon Gate in the west (represented by the Moon Tarot card, signifying the light of the imagination, which guides the spirit through times of darkness).

 

Almost nothing is known about Diophantus as an individual. The meager information we possess, comes from a riddle in verse, circulating since the 4th century CE, which, when translated into modern algebraic notation gives rise to the equation appearing in the center of the collage. Solving this equation, we obtain that Diophantus' age at his death was x = 84. The mystery surrounding Diophantus' person expressed in the poem and its title, is reflected in the collage by casting Diophantus as the masked figure descending to earth on the wings of the Sumatran Swallowtail butterfly, Papilio Diophantus.

 

Greek mathematics' main focus had been on geometry. Even when solving algebraic equations, the methods employed were geometrical. In contrast, Diophantus' Arithmetica was a treatise in algebra and number theory. Six out of its thirteen books survived either in the original Greek or in Arabic translations. The treatise, comprising of 189 problems with their solutions, solved algebraic equations by abstract algebraic means. Moreover, it introduced various algebraic symbols rather than describing the equations with words alone, as done by its predecessors. One of the main contributions of this approach was the beginning of symbolic algebra, which will not proceed to the next step in its development till late 15^th century. In addition, the problems and their ingenious solutions provided, through the ages, a constant source of inspiration for generalizations that became a driving force behind the development of number theory.

 

The most spectacular example of this kind, the so called "Fermat's Last Theorem," is a conjecture posed by the 17^th century French mathematician, Pierre de Fermat, in response to Problem 8 from Arithmetica, Book II (cited in the poem), stating that:

 

No integer solutions x, y, and z, exist to the equation xⁿ + yⁿ = zⁿ, if n is larger than 2.

 

The conjecture was solved in the affirmative in 1997 by British mathematician, Andrew Wiles. In the over 300 years that passed since Fermat's conjecture and Wiles' proof, there had been numerous attempts to prove the conjecture, some with partial success. More important, many mathematical tools and deep theories were developed as a result of these attempts. This is where the true contribution of Diophantus' problems lie. Without them a large chunk of modern mathematics would simply not exist.

For more details see Mark's Dissecting Diophantus.