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I Spy with my Little Pi

March 14, 2016

Pie: such an integral piece of American culture that has inspired ideas of prosperity and quite a few idioms. Ironically, though, pie has been around for a lot longer than the United States, dating back all the way to Ancient Egyptians and Greeks, who made them not for their delicious taste but for their reliability. However, we aren’t talking about that kind of pie today.

The Art of Pi

Where does mathematical pi come from? This constant is the relation between the diameter and circumference of a circle, first calculated by the Ancient Greek mathematician, Archimedes.

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Detail, Melchor Pérez Holguín, Virgin of the Rosary, Late 17th-Early 18th century, Dallas Museum of Art, gift of Mary de la Garza-Hanna and Virginia de la Garza and an anonymous donor.

For thousands of years, the simple circle has inspired art and philosophy from all over the world. In religious art–from Buddha to Jesus to Apollo–circles as halos adorned the heads of the divine and sacred. On the other hand, circles and other geometric shapes became prominent in early Islamic art because of an opposition to creating figures, since they could be construed as idolatrous. These circles became part of exquisite Islamic architecture, like in the immense arches, domes, and designs of the Hagia Sophia.

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Detail, Folio of a Qur’an, 1409 AD, The Keir Collection of Islamic Art on loan to the Dallas Museum of Art.

The Math of Pi

There is one place in which our old pal pi comes to play in the world of math and circles–the beloved radian. These are often addressed in pre-calculus and paired with the similarly adored unit circle. These concepts are often rushed with little explanation. Why can’t the world be fine with degrees? Don’t they just serve the same purpose as radians?

There is a reason for the existence of radians. They are an alternative that not only measure an angle, but the correlating arclength. One radian is the angle made when you wrap the radius along the circumference of a circle. We can visualize the internal relationship of radians as follows:

radian

We have a circle on the xy-plane with some angle that can be seen starting from the positive x-axis to the red line. To see this concept, we can find the arclength that is highlighted by referring to our well-known equation to find the circumference (circumference = 2πr).

Let us take the circle as the unit circle and angle as π/2. From inspection and prior knowledge, we see that the arclength is ¼ of the entire circumference. In order to find this measure, we would calculate a fourth of the circumference (so arclength = ¼ circumference = ¼ 2πr = πr/2). We see that the arclength = πr/2 with a radius of 1 (due to the unit circle) and we see that – in this case – an angle of π/2 has an arclength of π/2.

This works even without the unit circle! If our radius is 2, 5, or 1,000! Knowing that the arclength of this angle is πr/2 means that we know that it is π,  5π/2, and  500π respectively.

Activity

Here’s a familiar activity: to show pi in the real world, you can take any circular object, string, ruler, and scissors. Take your string, wrap it around your item once, and cut it so that both ends tightly meet. Measure your string and the diameter. Using our handy dandy formula, the circumference = 2pi r = pi diameter. With what we have, pi should be equivalent to circumference / diameter. Take your measures and see just how close to pi you can get!

Just a tip: make sure your string doesn’t have much give as when using it to measure, its stretch will distort your calculations.

A way to see the wonderful radian–using the same materials as before–is to measure your object’s radius and cut an equal length of string. See how many times this length will fit along the circumference. You should find that 2π – or 6.283 – pieces will cover it just nicely.

Personally, though, we think everyone should celebrate the journey of the circle with a generous slice (or two) of your favorite pie and a trip down to the DMA.

Kennedy Schleicher and Nikki Li
Teen Advisory Council Members

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