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Friday, March 14, 2014

Pi day - on the ratios of similar shapes

Today, March 14, or 3/14, is celebrated by some, especially by schoolteachers, as pi day. The reason is that 3.14 is an approximation for pi. Why isn't 3.14 pi?

Pi is the ratio between the circumference, the distance around, and the diameter, the distance across, of a circle.

Let's consider a seemingly unrelated topic. Suppose you think about someone you are very familiar with, someone you have seen many times in current photos. Perhaps yourself, perhaps someone else. Can you recognize that person in a small photo, as well as a large one? Of course, unless the photo is so small that you can't make out details at all. Why is that so? The photos are different, yet they are all of the same person, and they are similar. Similar in what way? Several ways, such as eye color, perhaps, but they are similar in many ratios. The distance between that person's eyes, in the photos, has the same ratio to the distance from one corner of the mouth to the other, in every photo. The distance from one corner of the left eye to the tip of the nose has the same ratio to the distance from one corner of the right eye to the tip of the nose, in every photograph. You haven't measured these distances, or calculated the ratios, but if these, and many other ratios, weren't the same in each photograph, you would start to think that some of the photographs weren't of the same person. (For a person's face, we need to allow some leeway for whether they are smiling, the angle that the photo was taken from, and the like, but the ratios need to be close to the same, or the same, for the photo to look natural.)

One more example. Consider squares. Are all squares the same size? No, of course not. But all squares have some ratios that are identical, or they wouldn't be squares. Most obviously, the ratio between any one side of a square, and any other side, is 1. The sides are all identical. Another ratio of all squares is the ratio between any side and the distance from one corner to the opposite corner. That ratio can be determined by the Pythagorean Theorem, which tells us that the sum of the squares of the shorter sides of a right triangle is equal to the square of the hypotenuse, the longest side. From that, the ratio of the distance across a square, from one corner to an opposite corner, to a side of that square, is equal to the square root of 2. Always, for all squares.

So a person's picture always has the same ratios, or nearly so, between dimensions. A square always has exactly the same ratios between various dimensions. Circles, too, are all similar, regardless of size, meaning that the ratios of their dimensions are the same. What ratio? What dimension? The ratio between the distance around any circle, the circumference, to the distance across the circle, through the center, which is the diameter, is always the same. And that ratio is pi.

It so happens that pi (and also the square root of 2) are irrational numbers. That is, even though they are ratios, they cannot be exactly expressed as ratios between whole numbers. Pi is nearly 22/7, but not exactly. Pi is nearly 3 and 14/100, but not exactly.

For much more on pi, see the Wikipedia article.


Unknown said...

Fascinating to think about the consistency of our natural world!
One caveat I might offer: the circumference of a 'pie' appears smallest for the person who enjoys the crispy, flakiness of the crust the most. Conversely, the diameter of the same 'pie' is indirectly proportional to whether said 'flaky crust devourer' has already eaten the entire circumference of the 'pie'.

Martin LaBar said...

Thanks, unknown. I certainly hadn't considered that aspect!

atlibertytosay said...

I've always been baffled as to why there isn't a last number of Pi.

Once you got down to the subatomic level - wouldn't there be a last possible measurement?

I mean if you did this like a normal math statement creating known values - why can't it be figured out from one point on the circle to the next?

Martin LaBar said...

I'm mostly relying on mathematicians, which I'm not, to answer this, atlibertytosay.

See this Wikipedia article for some proofs of the irrationality of pi:

It's not the only irrational number, by a long shot. As the post mentioned, the square root of 2 is also irrational, as are the square roots of 3 and 5. In fact, there is an infinity of irrational numbers.

It is true that pi can be measured so that the measurement is very close to the actual value. It can also be, and has been, calculated to thousands of decimal places. But it cannot be exactly expressed as the ratio of two whole numbers.