Square Root of 2

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

The poem-collage pair appears in the Bridges 2021 online gallery http://gallery.bridgesmathart.org/exhibitions/2021-bridges-conference

Sarah Glaz

 Sarah Glaz
 University of Connecticut, Storrs,
 Connecticut, USA 
Mark Sanders

Mark Sanders 
Rushden, Northamptonshire,   

History, Mathematics, Poem, Collage


In the 5th century BCE, Hippasus of Metapontum discovered the existence of irrational numbers. Metapontum was a Greek city located on the gulf of Tarentum. It was part of Magna Graecia, the coastal area of southern Italy colonized by Greece, which also included the city of Crotona, where Pythagoras (580 - 500 BCE) settled after he left his native island of Samos. In Crotona, Pythagoras established the secret philosophical society known as the Pythagorean Brotherhood. Hippasus was a member of Pythagorean Brotherhood. The society followed strict rules of conduct and a common lifestyle and philosophy.   


A fundamental belief of the Pythagorean Brotherhood was that whole numbers underlie all natural phenomena. Whether in music, or astronomy, or philosophy, the central position of "number" was everywhere evident. This belief led them to undertake investigations into the properties of numbers and to the discovery of many interesting mathematical results.  Greek mathematics contributed to the discipline its most fundamental principle -- the requirement that mathematical results are validated by proofs. The Pythagoreans were among the first to prove many mathematical truths that were in common usage in the ancient world, like for example, the so-called Pythagoras Theorem. They also discovered and proved numerous new results in geometry and Number Theory.  Many of their discoveries were kept secret and when shared with outsiders, they were presented as a common accomplishment of the entire brotherhood. 


Hippasus discovered that square root of 2 is an irrational number, that is, he proved that square root of 2 cannot be expressed as a ratio of two whole numbers.  Pythagoras Theorem applied to a right-angled triangle whose sides are 1 unit in length, yields a hypothenuse whose length is equal to square root of 2. Thus, square root of 2 is a number arising as a measure of length of a line segment. The discovery that such a number is not a ratio of whole numbers created a crisis of enormous magnitude for the Pythagoreans. On one hand, it invalidated many of their geometric proofs, which relied heavily on the assumption that lengths of line segments were rational numbers; and on the other hand, it shattered their deeply held belief in the supremacy of whole numbers as the underlying principle of the universe. In addition, Hippasus breached their most sacred rules of conduct, he revealed his discovery of the irrational number square root of 2 thereby breaking his oaths of both secrecy and individuality. For his sins, legend has it, he was thrown overboard during a sea voyage.


The crisis of incommensurable quantities, as the discovery of the irrational numbers came to be called, increased with the realization that Pythagoras Theorem yields an unlimited number of irrational numbers. It fell to Eudoxus of Cnidos (408 - 355 BCE) to resolve the crisis by introducing a theory of proportion that corrected all the invalid proofs. In the end, the discovery of the irrational numbers turned out to be one of the greatest contributions the Pythagoreans made to mathematics.


The poem "Square root of 2" plays with the imaginary possibility that Hippasus' murder occurred before he made his discovery public. How did the future find out about the irrational numbers and their history?  To answer this, use the power of your imagination. The poem's stanza line count follows the decimal expansion of square root of 2 to five decimal places. This poem was first published in Sarah's  poetry collection Ode to Numbers (Antrim House, 2017).


The collage "Square root of 2"  is a square  with a diagonal labeled by the decimal expansion of square root of 2 to twenty-seven decimal places. The figure of Houdini on the left-hand side, bound in chains and padlocks, reflects the laws of secrecy by which the Pythagorean Brotherhood bound itself; the "Sshhh..." face, on top right, represents the 5-year vow of silence taken by newcomers to the brotherhood in order to prove their commitment to the cause; the ammonite represents the diagram which shows the construction of irrational numbers according to Pythagoras Theorem; and the gloved hand carefully holds these secrets. For  more details,  see Mark's "Dissecting Square Root of 2."


Square Root of 2 poem and collage

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