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'Small Steps' printed from https://nrich.maths.org/
Well done Robert of Madras College, St Andrew's, Scotland and
Andrei of School No. 205, Bucharest, Romania for your solutions to
this problem.
In both parts of this question we consider the limiting case of a
process which is repeated infinitely often and things are not what
they might seem to be.
(a) In a square $ABCD$ with sides of length 1 unit a path is drawn
from $A$ to the opposite corner $C$ so that all the steps in the
path are either parallel to $AB$ or parallel to $BC$ and not
necessarily equal steps. If we draw paths of this sort putting in
more and more and more steps the length of the path is
always the same."
The steps parallel to $AB$ together must stretch all the way across
from $A$ to $B$ and the steps parallel to $BC$ together must
stretch all the way up from $A$ to $D$. Irrespective of the number
of small steps, A point moving on any path of this type moves a
total of 1 unit parallel to $AB$ and a total of one unit parallel
to $BC$, hence a total of 2 units altogether. With more and more
steps the path gets closer and closer to the diagonal so you might
expect the length to converge to $\sqrt 2$. Surprisingly the length
is always 2 units and not even close to $\sqrt 2$ units.
(b) Now consider the graphs of $y={1\over 2^n}\sin 2^nx$ for $n=
1,2,3, ...$ and $0\leq x \leq 2\pi$. As $n$\ tends to infinity the
graphs oscillate more and more and get closer and closer to the $x$
axis. We have to prove that the length of the curve from $x=0$ to
$x=2 \pi$ is the same for all values of $n$. The hint says we don't
need to calculate the length of the path here and we should think
about scale factors.
The graph of $G_n:\ y={1\over 2^n}\sin 2^nx$ from $x=0$ to $x=\pi$
is similar to the graph of $G_{n-1}:\ y={1\over 2^{n-1}}\sin
2^{n-1}x$ from $x=0$ to $x=2\pi$\ but scaled down by a linear scale
factor of 1/2 so $G_n$ is half the length of $G_{n-1}$. However
$G_n$ is repeated twice periodically between $x=0$ and $x=2\pi$ so
the two pieces together have the same length as $G_{n-1}$.
This shows that all these graphs on $0\leq x \leq 2\pi$ have the
same length although as $n\rightarrow \infty$ the graphs get closer
and closer to the $x$ axis so you might suppose that the length
converges to $2\pi$. Surprisingly the length is always the same and
much more than $2\pi$.