A classic puzzle

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[billh]

Archimedes seemed to be aware of this problem: the step curve is not convex. He restricted himself to only using convex curves to do the approximation for calculating arc lengths like the circumference of a circle. He used the following axiom which he explicitly stated: If we have two convex paths from P to Q, one inside the other, then the inside one is the shorter.

[paullovekamp]

I actually asked my high school calculus teacher this very question. I now understand why she couldn't explain it to naive little me - It's a much deeper question than it appears at first glance !! Thank you Michael Penn, you've cleared up a quandary I've never understood for 50+ years.

[blub]

In my head I've always explained it like that: The number of points that are exactly on the line grows with 2^n - this means that in the limit there will only be countably infinite points correct. The lines consist of uncountably infinite points and therefore there will always be an uncountably infinite number of points missing - even in the limit n->infinity.

[maurobraunstein]

Doesn't fn(t) converge to f(t) uniformly, though? For any e > 0, there exists an N such that ||f(t) – fn(t)|| < e for all t in the interval. What doesn't converge is the derivative. I'm not sure what that's called.

[richardbrewer]

Basically the approximations are only ever taking smaller and smaller pieces of the vertical and horizontal lines, so always have a total of 2. It is not approximating closer and closer to the hypotenuse.

[zh]

There is a similar "proof" that pi = 2 using semicircles. 1 big semicircle = 2 half-size semicircles drawn along the diameter = 4 quarter-size semicircles and so on, until you have arbitrarily many tiny semicircles whose length is still equal to pi.

[bobh]

I learned something new at about the 2:20 mark. 16 x 1/2 is 2.

[ingiford]

"Taxi cab" metric, d((0, 0) to (1, 1)) is 2, and this shows many of the ways to measure it by taxicab metric

[tetsi]

15:15 every time a mathematician or math teacher says something like "we all agree that $X" I feel like sitting in front of a magician telling asking me to confirm that they have a normal deck of cards... I know I'm getting tricked (in a sense) but I've not really understood yet how. 😀

[coolbikerdad]

The limit line is a curiosity. While it contains the same points as the diagonal, and only those points it in fact an object with fractional dimension. Bigger than 1-D and less than 2-D. Another example is the boundary of a Koch curve at its limit.

[ScandGeek]

Another way to finish the calculations is to calculate
||fn'(t)-f(t)||.

At every point, the vector fn'(t) is either (1, 0) or (0, 1), while the vector f'(t) is always (½, ½). Hence, ||fn'(t)-f'(t)||² = (½)²+(-½)² = ¼+¼=½, so ||fn'(t)-f'(t)|| = 1/sqrt(2).

The derivatives clearly then do not converge, so we cannot assume that the lengths converge

[eiseks]

Next: Let's prove why 1.1 is not equal to -34567

[DeJay]

It's because the approximating path started off longer and will always be a longer path, even if it goes from A to B just like the straight line AB and in the limit it's very close to it, even with infinite iterations it's a 2D path that does up and down, never straight.

[sciphyskyguy]

This is why the measured length of a curved coastline isn’t the same value when you use rulers of different lengths to measure it.

[michaelneuman]

The approximations only seem like approximations because the deviations from the diagonal get smaller each time. However, they also get more numerous, therefore...

[tashquantum]

me when im working over a field other than F_2

[crsmtl]

Why rad two ain’t two! or Why the straight line is the shortest! Great job, Coach!

[maxvangulik]

Since each iteration is just 2 copies of the same thing at half the size, it's like repeatedly multiplying and dividing by 2. This reminds me of a famous sum that doesn't converge to a single value but has been given one anyway.
1-1+1-1+1-1+1-1+... converges either to 1 or 0 depending on the parity of the number of terms. Since we multiply and divide by 2 each iteration, that's like adding and subtracting 1 in the exponent. In this case, you'll always end up with 2, but note that if you scale the triangle down without duplicating it, the length will be 1, or 2^0.

The sum shown above has been assigned a value of 1/2 by the analytic continuation of the dirichlet eta function, so raising 2 to the power of that value will give 2^(1/2), or the square root of 2.

That's why the limit is sqrt(2) even though every partial sum is 2.

[berndhutschenreuther]

In taxicap geometry it is indeed 2, just as you showed.

[QoowedeDgikQmZG]

suddenly thinking out, the shortest path between two points, is not always equal to the fractal sum