Thursday, November 17, 2011

The Diagonal Paradox

Shown here is a square with edges of length 1. The two orange edges will sum up to 2. Additionally, the length of the diagonal can be calculated using the Pythagorean Theorem. You know, A² + B² = C². That one. This gives √2 ≈ 1.41 for the diagonal.

Start carving out a stairway as shown in the pictures below. Realise that the total length of those orange lines will remain 2 no matter how many steps you chose to make!

From left to right, there are more (smaller) steps each time. But hang on a second: that orange line is quickly starting to look like the diagonal. Would it also approach 1.41 in length? If you made infinitely many steps, would its length turn out to be exactly √2?
No, not so, on both accounts.

The same apparent contradiction can be seen when ‘proving’ that pi is not ~3.14159, as we all thought, but actually 4. Let's see how that goes.
Draw a circle of diameter 1. Since π is the ratio of any circle's circumference to its diameter, this circle's circumference is indeed π. Draw a square around it (perimeter 4), and start removing corners. As before, the perimeter remains the same. Keep on going! Soon the resulting perimeter will closely follow the circumference of the circle.

Does it follow that π = 4? That would have simplified some calculations, but of course it's nonsense. These guys are still very much spot on. So why then does the perimeter not approximate the actual pi?

To find out, zoom in. Cutting more corners? Zoom in further. If you look closely enough, the corners are always still there and you won't ever get rid of their length. So the point is, it might look like a curve, but it simply isn't.

The left-hand image here demonstrates what it could look like when you zoom in. But to approximate a circle, the shape that would have been logical is as shown on the right. Does it still seem strange that their length differs so much?

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