The Discovery That Transformed Pi

Source: The Discovery That Transformed Pi, Veritasium, 18:40, uploaded 2021-03-16, category Other / Unclear, playlist index 1479.

Veritasium opens with the old way of calculating $π$: draw a polygon inside a circle, draw another around it, and squeeze the circle’s circumference between their perimeters. The method works, although every extra decimal place demands a much larger polygon. Isaac Newton finds a different route by extending a familiar algebraic pattern into cases where the usual rule says it should stop.

Pizza, circumference, and area

The video begins with a pizza because the geometry is easier to see there. Cut off the crust and lay it across identical pizzas. Its length covers a little more than three diameters, which gives the relation between a circle’s circumference and its diameter: the circumference is $π$ times the diameter.

The same constant appears in the area. Cut a pizza into very thin radial slices and arrange them alternately to form an almost-rectangle. Its width is the radius, and its length is half the original circumference, which is $π R$. The area therefore becomes $π R^2$. A unit circle has area $π$, a fact Newton will use when he turns a series into an area calculation.

Archimedes and the polygon method

The obvious bounds are easy to construct. A regular hexagon with side length one fits inside a circle of diameter two. Its perimeter is six, so the circle’s circumference is larger than six and $π > 3$. A square around the circle has perimeter eight, which gives $π < 4$.

Archimedes improved the bounds around 250 BC by repeatedly bisecting the polygon’s sides. He compared an inscribed polygon and a circumscribed polygon, so one perimeter stayed below the circle’s circumference whilst the other stayed above it. At the 96-gon, he had placed $π$ between 3.1408 and 3.1429. The calculation required repeated square roots and the conversion of those roots into fractions, yet the result was remarkable for its time.

The method then travelled through Chinese, Indian, Persian, and Arab mathematics. In the late sixteenth century, François Viète doubled the number of sides again and calculated the perimeter of a polygon with 393,216 sides. Ludolph van Ceulen spent 25 years calculating the perimeter of a polygon with $2^{62}$ sides, roughly 4.6 quintillion sides, to obtain 35 correct decimal places. He had those digits carved into his tombstone. Christoph Grienberger surpassed him with 38 places.

The precision had ceased to serve a practical need. Alex Kontorovich, the mathematician who appears in the video, describes it as a display of mathematical strength. The polygon method had become a way to show how much calculation a person could carry through.

Pascal’s triangle beyond integer powers

Newton found the faster method in 1666, whilst he was 23 and quarantined at home during an outbreak of bubonic plague. He was studying expressions such as $(1+x)^2$ and $(1+x)^3$. Multiplying the terms gives coefficients that follow a pattern. Those coefficients are the rows of Pascal’s triangle.

Pascal’s triangle has appeared in many mathematical cultures. To construct it, put a one at each edge and add neighbouring numbers to produce the row below. The row gives the coefficients for the matching power of $(1+x)$. A general formula for those coefficients is the binomial theorem, which rigorously describes the finite expansion for every positive integer power.

Newton’s important move was to test the pattern outside those powers. The binomial theorem had been built for a positive integer $n$, since $(1+x)^n$ meant multiplying $(1+x)$ by itself $n$ times. Newton substituted $n=-1$ anyway. The result is the infinite series

The extension could have been nonsense. Newton checked it by multiplying the series by $(1+x)$. Every term cancels its neighbour and the leading one remains, so the product is one. The series therefore behaves as $1/(1+x)$ within the range where it converges.

This also extends Pascal’s triangle above its usual top row. A row of alternating positive and negative ones adds to zero beneath it, and the same pattern continues for negative integer powers. If the negative signs are ignored, the numbers form the same arrangement as the ordinary triangle, rotated on its side.

Newton then tried fractional powers. Setting $n=1/2$ turns $(1+x)^n$ into $\sqrt{1+x}$ and produces another infinite series. The idea suggests a continuum of Pascal’s triangles, with a separate pattern for every possible power. Newton uses the half-power to calculate $\sqrt{3}$: write $3=4-1$, pull out the square root of four, and substitute $x=-1/4$ into the series for $\sqrt{1+x}$. The terms shrink quickly enough to give an accurate result with simple arithmetic.

The circle as a source of a series

The half-power becomes useful for $π$ because the upper half of a unit circle is the curve

This gives Newton two descriptions of the same object. The circle has a geometric area, and the square-root expression has an infinite series. Integrating the curve from $x=0$ to $x=1$ gives a quarter of the unit circle, so the result is $π/4$. Integrating the series term by term is also straightforward because integrating $x^n$ means increasing the power by one and dividing by the new power. Evaluating the resulting series at one gives a way to calculate $π$ to arbitrarily high precision with fractions.

Newton adds a further improvement by integrating only from zero to one half. The terms in the circle’s series contain increasing powers of $x^2$. Substituting one half makes each new term shrink by an extra factor of one quarter, so far fewer terms are needed for a given accuracy.

The smaller integral has a geometric value of its own. The area under the curve from zero to one half consists of a 30-degree sector, with area $π/12$, and a right triangle whose base is one half and whose height is $\sqrt{3}/2$. Hence

Equating this area with the integrated series and rearranging gives Newton’s accelerated formula for $π$. The first five terms produce 3.14161, which the video says is wrong by only two parts in 100,000. Matching van Ceulen’s 35 decimal places would take about 50 terms. A calculation that once occupied years could then be completed in days.

The method and its limit

The polygon construction disappeared from serious calculations because Newton had changed the kind of work involved. His method did not simply ask for a larger amount of the same arithmetic. It found a pattern in finite algebra, tested the pattern beyond its stated range, connected the resulting series to the area of a circle, and then chose an interval that made the terms fall quickly.

The video treats this as a lesson about mathematical practice. The obvious construction can remain useful for finding the first bound, although it can also become an inherited habit that hides a better representation. Newton’s result depends on extending a rule with care. The multiplication check supports the negative-power series, while convergence still limits where the series can be used. The insight matters because it changes the cost of the calculation, not because every pattern survives every extension.

Limits

The historical episodes, figures, quotations, and technical explanations in this note follow Veritasium’s English captions and description. The video features Alex Kontorovich, who is credited as a professor of mathematics at Rutgers University and a distinguished visiting professor for the public dissemination of mathematics at the National Museum of Mathematics. The description credits Derek Muller and Alex Kontorovich with the writing and names three works as references. I have not independently checked the historical details, the quoted precision figures, or the source’s account of Newton’s reasoning against those underlying works.

The video compresses several ideas that require qualifications in a mathematics text. The binomial expansion for non-integer powers depends on convergence, and the speed of the resulting calculation depends on the chosen form and interval. The note follows the source’s explanatory route, including its claim that 50 terms would match van Ceulen’s precision, rather than presenting a new derivation or an independent numerical audit.

Further reading / references

• Arndt and Haenel, Pi Unleashed
• Dunham, Journey through Genius: The Great Theorems of Mathematics
• Borwein, “The Life of π: From Archimedes to ENIAC and Beyond”