A while back Dave Renfro graced this blog with a guest post. One of his former students was reading and related a "paradox" that Dave had shared in one of his classes at Central Michigan. He wrote:
"A person starts at (0,0) and wants to travel to (1,1) but can only move in right angles. If they travel one unit over then one up, they travel 2 units. If they go .5 units over, then .5 up, then .5 over, then .5 up, they travel 2 units. If they continue to decrease the distance traveled before turning by a factor of .5, to the point where the distance traveled before turning is at about 0 (limit as d -> o and as # of turns -> infinity) then the distance traveled suddenly becomes sqrt (2). "
I was reminded of this a few days ago when a post at Equalis used a similar approach to "prove" that pi = 4. This close to Pi day, I thought I would circulate it in case any classroom teachers out there needed one more topic for the day of celebrations they had planned. Here is the image from the post... see the whole discussion here:
The anaysis answer to the why involves the rather complex idea of semi-continuity. I will leave that back in the capable hands of Dave Renfro who sort of started al this, so here are some remarks (slightly edited) he wrote about the topic.
"..consider the (0,0)-to-(1,1) diagonal of the square whose vertices are (0,0), (1,0), (0,1), and (1,1) to be the limit curve, and use a sequence of ever smaller and more numerous
horizontally and vertically oriented staircase curves that approach the diagonal. The diagonal has length
sqrt(2) and each of the staircase curves has length 2, and hence the limit of the lengths of the staircase
curves is 2. The fact that the staircase curves are not graphs of functions is unimportant, since we can
simply rotate things so that the diagonal is along the x-axis. I believe I've read somewhere that this example
made a strong impression on Lebesgue when he was an undergraduate student (mid to late 1890s), and it likely played a small role in stimulating Lebesgue's later work in real analysis, such as the "Lebesgue integral"
(google the phrase if not familiar with its importance), among other things."
"...the limit of the lengths is greater than the length of the limit. In general, it is always true (subject to very mild restrictions on what a "curve" is) that the limit of the lengths is greater than or equal to the length ofthe limit. Thus, while arc length is not a continuous function in this setting, it is "lower semicontinuous"
at each curve. To consider what continuity and semicontinuity of the "arc length function" means,
we need to have in mind a domain space of curves and a "distance between two curves" notion (or at
least, a topology on the set of curves), which I'll skip in order to focus on what semicontinuity means."
"The notions "lower semicontinuous at x=b" and "upper semicontinuous at x=b" are two halves of the notion
"continuous at x=b" in a way that is similar to how "left continuous at x=b" and "right continuous at x=b"
are two halves of the notion "continuous at x=b". The former (semicontinuity) makes use of the two
sides (below and above) on which the outputs can be located, while the latter (unilateral continuity)
makes use of the two sides (left and right) on which the inputs can be located. Even more precisely, we
could consider what it means for a function to be right lower semicontinuous at x=b, or any of the other three
combined possibilities . . .Specifically, if f(x) is defined on an open interval containing b, then f(x) is lower semicontinuous at x=b means that, for each sequence x_n approaching b such that LIMIT[f(x_n)] exists, we have f(b) less than or equal to LIMIT[f(x_n)]. Requiring, instead, that f(b) be greater than or equal to LIMIT[f(x_n)] gives the notion "upper semicontinuous at x=b".
OK, If you still think of pi as about 3.1415.... you can find a nice screen wallpaper background for the day from John Hanna's web page.
For those who know it goes on and on, here is a pretty fact for Pi Day....
31415926535897932384626433832795028841 is a prime number. BUT, It’s also the first 38 digits of pi. Ok, So if you stopped after some arbitrary number of digits of Pi, what is the probability that those digits will form a prime number? ... It's prime for one digit, and for two, but not for three, four or five... go on, your turn....
From a post a few years ago...And on the "statpics" blog, Robert W. Jernigan, Professor of Statistics at American University, posted some notes on the First Published Random Walk. Turns out it was by John Venn in 1888, only fourteen years after the first copyrite date of "Life on the Mississippi." And the randomizing device??? The digits of Pi... Here is the image from Venn's classic "The Logic of Chance" :
Some years after I wrote this I found a twitter link to a random walk of Pi with 100 billion digits, and lots of nice graphics, enjoy.
web page ...and (surprise, surprise!) they will sell you stuff.. like Pi shirts and coffee cups and clocks... and you can get a little count-down gadget like mine..it's free.