Polygons, Products, and Pi

Math lovers widely celebrate March 14 as Pi Day. Written as 3/14, the date matches the first three digits of pi—the constant ratio between a circle’s circumference and its diameter. One can go even further on that day, celebrating by the minute (3/14, 1:59) or the second (3/14, 1:59:26) to hit more digits. But no subdivision of time can cover all of pi’s digits, for they go on forever, never falling into a repeating pattern. In other words, pi is irrational: it cannot be precisely expressed as a fraction. 

This means that one can never obtain the exact circumference of a circle from its diameter, so history is full of attempts to approximate circles with more measurable figures. One obvious choice is a regular polygon. The great Archimedes (c. 287-212 BC), who worked for the king of Syracuse, realized that by increasing the number of sides, either an inscribed polygon (written inside the circle) or a circumscribed polygon (written outside) could get as close to the area of the circle as one wished. He used this fact to prove that a circle’s area is equal to half the product of its radius and its circumference. 

The modern expression of this formula is . Archimedes did not state it in this way, but he was clearly interested in what we now know as pi, and his polygon technique was useful for approximating its value. The perimeter of a circumscribed polygon is always greater than a circle’s circumference, and the perimeter of an inscribed one is always less, so one can use the perimeters to obtain over- and under-estimates for pi. Archimedes managed, with much complicated angle bisection, to compute the perimeters of two 96-sided polygons. This gave him .

Several centuries later, Liu Hui—a Chinese mathematician about whom nothing is known besides his name, his state of residence, an ancestor, and two mathematical works—came up with Archimedes’s ideas independently. In 263 AD he wrote a commentary on the classic text Jiuzhang suanshu (Nine Chapters on the Mathematical Art). That work had used 3 as a crude estimate of pi; Liu pointed out that this was actually the ratio of a hexagon’s perimeter to its diameter. Polygons with more sides would get closer to the true ratio. While he only used inscribed polygons, he proposed a more elegant and general method for successively doubling the number of sides.

As shown in the figure above, let AB be the side of the polygon and OB the radius of the circle. Apply the Pythagorean Theorem to get OG, and then subtract it from the radius to produce CG. Finally, a second use of the Pythagorean Theorem will give the value of CB, the new polygon’s side length. This may be the method used by the astronomer Zu Chongzhi (429-500) to compute , which remained the best worldwide estimate for over a thousand years.

The culmination of the polygon technique would come from François Viète (1540-1603). Viète was a lawyer who served as an advisor to two successive kings of France, breaking several notable Spanish codes. His real fame, however, comes from his mathematical work, published on the side under the Latinized name Franciscus Vieta. He was the first mathematician to use letters for writing unknown and constant values, and could thus manipulate equations much more easily than his contemporaries. Though most of Viète’s specific terminology is no longer used, it was crucial in the shift to symbolically written algebra.

In 1593, Viète published Variorum de rebus mathematicis responsorum liber VIII (eighth book of various responses regarding mathematical things), among which was a chapter on Archimedes’s inscribed polygons. Viète recognized that the ratio from a circle’s area to that of any polygon would be irrational, but he wanted to “philosophize more freely” about it. As shown below, he let BC be the diameter of a circle, BD be the side of an inscribed regular polygon with n sides, and BE be the side of an inscribed regular polygon with 2n sides. He drew DC, calling it the apotome of the n-sided polygon. After this came a dense paragraph of geometrical reasoning, in which he showed that the ratio of the n-sided polygon to the 2n-sided one was the same as the ratio from DC to BC. Viète then simply stated that when the diameter of a circle has length , the apotome of an inscribed square has length , the apotome of an inscribed octagon has length , the apotome of an inscribed hexadecagon has length , and so on “in a constant progression towards infinity.”

Dividing all these values by 2 produces a series of ratios between the polygon’s areas; multiplying the ratios together produces the ratio between the area of the inscribed square and the area of the surrounding circle. Those two areas are and respectively, so the full formula is 

This is commonly known as Viète’s formula. (Confusingly, Vieta’s formulas are completely unrelated and deal with patterns in polynomial expansion). One can of course rearrange the expression to get

This was the first exact value for pi, as well as the first infinite product of any sort. It is true that Viète himself did not have a particularly modern understanding of values at infinity. He characterized a circle as a “vague” polygon made up of an infinity of triangles; this is much farther from the definition of a limit than Archimedes’s observation that the polygons could get arbitrarily close to a circle. Nonetheless, such a precise description of an infinite expression was innovative. Viète’s formula is worthy of far more than just Pi Day celebration.

How Viète wrote nested square roots

Sources:

https://www.wikipedia.org/

Berggren, Lennart, Johnathan M. Borwein, and Peter Borwein, eds. Pi, a source book. New York: Springer-Verlag New York, 2004.

Moreno, Samuel G, and Esther M. García-Caballero. “On Viète-like formulas.”  Journal of Approximation Theory 174 (October 2013): 90-112. https://www.sciencedirect.com/science/article/pii/S0021904513001159#s000050

Stewart, Ian. Significant Figures. New York: Hachette Book Group, 2017.

Suzuki, Jeff. Mathematics in Historical Context. Mathematical Association of America, 2009.

Image Credit:

https://www.researchgate.net/figure/A-diagram-to-show-the-approximation-of-a-circle-by-a-polygon-improving_fig13_301363216

https://liuhuimathmatician.wordpress.com/2014/04/03/liu-huis-pi-algorithm/

https://www.sciencedirect.com/science/article/pii/S0021904513001159#s000050 (Edited by Hudson Green)

https://commons.wikimedia.org/wiki/File:Vi%C3%A8te%27s_formula.png