Business Insider/Andy Kiersz

"Pythagorizing" the Flatiron Building

As part of their first anniversary celebrations, the Museum of Mathematics (MoMath) and about 500 math enthusiasts of all ages proved that New York's iconic Flatiron building is approximately in the shape of a very special type of right triangle.

"Pythagorizing the Flatiron" is the first in a planned series of "MathHappenings" to be run by MoMath.

The proof of the Flatiron building's right triangle nature is based on the Pythagorean theorem — the statement that for a right triangle with legs (shorter sides) of lengths a and b, and hypotenuse (long side) of length c, the sum of the squares of the two shorter lengths equals the square of the long length — a2 + b2 = c2 .

The Museum of Math flipped this idea around — if the lengths of the sides of a triangle, like the Flatiron building, satisfy the Pythagorean theorem, then the triangle must be a right triangle.

MoMath measured the sides of the Flatiron building in a unique way. People lined up around the three sides of the building, and MoMath workers and volunteers handed out lightsticks that the math enthusiasts held end to end. By counting while handing out the glowing toys, MoMath was able to estimate the length of the building's sides in terms of lightsticks.

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The shortest side of the building, along 22nd St and designated side A, measured 75 lightsticks. The longer leg of the building, going up 5th Ave and designated side B, measured 180 lightsticks. The longest side of the building, the hypotenuse C running along Broadway, had a length of 195 lightsticks.

Having found the lengths of the sides of the building in lightsticks, MoMath then projected on the side of the building the calculations that showed that the three sides do in fact match up with the Pythagorean Theorem, proving that the Flatiron is in fact a right triangle:

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Square the lengths of the two shorter sides: a2 = 752 = 5625 and b2 = 1802 = 32,400. Add those together and get a2 + b2 = 38,025. Then, square the length of the longest side: c2 = 1952 = 38,025 — you get the same number as the sum of the squares of the shorter sides: a 2 + b 2 = c 2. So, the Pythagorean theorem holds for the Flatiron building, proving that it is in fact a right triangle.

Right triangles whose sides all have whole number lengths are special. For most right triangles, at least one of the three sides will be an irrational number — a right triangle whose shorter sides both have length one will have a hypotenuse of length square root of 2 (since, again by the Pythagorean Theorem, 12 + 12 = 1 + 1 = 2, and 2 is, by definition, the square root of 2, squared).

Sets of three whole numbers that describe a right triangle, like 75, 180, and 195, are called Pythagorean triples, and they have been of interest to mathematicians since the time of the ancient Greeks.

This particular Pythagorean triple also has one other interesting property. All three of the numbers 75, 180, and 195 can be divided by 15 without leaving a remainder: 75 ÷ 15 = 5, 180 ÷ 15 = 12, and 195 ÷ 15 = 13. So, if we "rescale" the length measurements by performing this division on the three lengths, we get that the sides of the Flatiron building are 5, 12, and 13.

MoMath thus was very clever in choosing the date for the event — December 5, 2013, or 12/5/13, matching the sides of the building.

After making the measurements of the Flatiron building's sides and showing their Pythagorean relationship, MoMath projected a couple nice geometric proofs of the Pythagorean theorem onto the side of the building:

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This proof is based on representing the squares of the side lengths as being the areas of squares drawn around the triangle. MoMath then showed how one can cut up and move around the two smaller squares based on the legs of the triangle, so that they fit perfectly inside the large square based on the hypotenuse, showing that the two smaller squares put together have the same area as the larger square — proving that a 2 + b 2 = c 2.

Events like this are a delightful way for all kinds of people to participate in the elegance and beauty of mathematics. The Pythagorean Theorem is a core concept in our understanding of geometry, and in many ways, it defines the shape of our world. It was exciting to see this fundamental mathematical principle brought to life in a fun and interactive way.

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