For Pi Day, Pi equals Four?

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 of

the 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.

And there is actually an official(?) Pi Day organization with a 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.

After Medians Comes Nedians

On the day I wrote my last blog about some interesting properties of the medians of a triangle, I received a package of old Mathematics Teacher articles from Dave Renfro. One of the first I looked at was a January 1951 "Mathematical Miscellanea" edited by Phillip S. Jones. The article contained a contribution by John Satterly of the University of Toronto on a type of cevian that he called "Nedians". [My personal choice, since the "med" root is for the middle, would have just been to call them n-dians, but I'm sure that would have had a cultural backlash.]

For those who may not be familiar with the term "cevian", it refers to a segment in a triangle from a vertex to the opposite side (extended if necessary). Angle bisectors, medians, and altitudes are all cevians then, but a perpendicular bisector of a side would not be because it doesn't necessarily pass through a vertex. The name is in honor of Giovanni Ceva and was originated in France in 1888 and has spread from there.

Professor Satterly seems to have created the term "nedians" as a comparison for medians to describe a cevian that cuts the opposite side 1/n th of the way from one vertex to the next. [I shall use the notation 4-nedian for a nedian that cuts 1/4th of the way along the opposite side,] A median would be the 2-nedian.
In the image, triangle ABC has 3-nedians AD, BE, and CF where D is 1/3 of the way from B to C; E is 1/3 of the way from C to A, etc.

The intersections of the three Nedians of a triangle will form another triangle at their three points of intersection, called the nedian triangle. [JKL in the image].

Professor Satterly seems to have discovered several properties of the nedians, and their nedian triangles which I will give here; and then I have come up with several interesting properties of my own about them that I will add to this blog.

It is not too difficult using affine properties of a triangle to verify many of these.
Professor Satterly showed that the sum of the squares of the nedians would be times the sum of the squares of the sides of the original triangle. Notice that when n=2, this reduces to 3/4, the ratio given and proved for the medians.In any triangle, the sum of the squares of the medians is equal to 3/4 the sum of the squares of the three sides.
Students may wish to explore these properties by creating Sketchpad or Geogebra interactive models to confirm them, and then challenge themselves to prove them. Proving them is easier with a property of affine geometry. Every triangle is affine equivalent to every other triangle in the plane. Under affine transformations areas and lengths may change, but ratios of them are preserved... which means that you can choose any triangle... a right triangle or an equilateral triangle, to find a property about ratios of lengths or areas, and it will apply to any other triangle... and that is a BIG idea to lock away (I am not well schooled in the particulars of affine geometry, so if I have mis-stated that in some way, please advise).

Professor Satterly also stated that the Nedian triangle will have an area of \(\frac{(n-2)^2}{n^2-n+1}\) times the area of ABC. Note that for the median, or 2-nedian, the area diminishes to zero since the three medians intersect in a single point.

Professor Satterly suggested the term "backward nedian triangle" for a case in which the 1/n ratio went in the opposite order (Let D be 1/3 of the way from C to B insted). This can be eliminated if we simply allow any real number for the coefficient. Then the backward 3-nedian is just a 3/2-nedian in the regular order, and it seems that all his properties are still preserved. Notice that the areas of the 3 and 3/2-nedians are equal, but they are not congruent.

Exploring these constructions a little more, I came up with a few more properties that were not in the article. For example, the perpendicular distance from the three vertices of the nedian triangle to any side of ABC will equal the altitude of ABC to that same side. [I call these the sub-altitudes.]

Here the altitude is shown in bold red, and the three corresponding sub-altitudes are shown in dotted red. In the 3-nedian shown the distance from J to side B plus the distance from K to side B plus the distance from L to side B will equal the altitude from B to side B. A similar result exists for each altitude of the triangle. In addition, the three sub-altitudes will always partition the altitude in the same way. The shortest sub-altitude will be , while the next longer one will be and the longest will be .

It is also clear from the last statement that each of the three small triangles at the vertices of A, B, and C will be congruent. Their bases will each be 1/n of a base of the original triangle and their heights are so each of them is an equal fraction, of the original area of ABC. By similar reasoning we see that all three of the quadrilaterals will also have the same area.

I also observed that that each nedian is partitioned into three parts whose lengths, in order from the vertex to the opposite side, are .
For the 3-nedians in the image, for example, the nedian CF is partitioned so that CJ is 3/7of CF; JL is 3/7 of CF; and LF is 1/7 of LF. A similar partition holds for the other two 3-nedians AD and BE. In a 4-nedian, the partitions would be 4/14, 8/13, and 1/14.

I'm gonna call that a good days work...please advise of typo's or just plain bad math... and thanks Dave, for another stimulating journal article.

A Guest Blog (Rant?) from Dave Renfro

I mention Dave a lot here because his regular supply of interesting journal articles from today back 200+ years has been a major source of my continuing education in the last few years... Dave sent me a copy of a recent set of remarks he had written about recent posts related to Pi day. Actually Dave calls it a "rant" but we should all keep in mind that he is NOT referring to MY pi-day post (That's right, isn't it Dave.....DAVE? Talk to me Dave?)
Anyway, here is Dave's guest post, and I greatly appreciate him letting me be the on-line voice for his ideas this time.
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I've mostly tried to not read anything about pi day, but in two cases I failed, with predictable results.
In the March 8-12 issue of "The Chronicle of Higher Education" there is an article about pi day titled "A Small Number With a Big Following". At one point in the article I saw the following comment:

"pi--so tiny (it's closer to three than four) yet random and infinite (as far as anyone knows)."

A question immediately occurred to me. Given the readership audience, which consists mostly of academic Ph.D.'s, why say something as meaningless and silly as this?

"random" -- What does that mean? I guess when computers calculate the digits of pi they're just randomly guessing what the digits are and then someone comes along and checks (how, nobody knows) to see if the digits are correct. Also, there are plenty of clearly non-random-looking sequences of arithmetic operations that generate pi, such as 4 - 4/3 +
4/5 - 4/7 + 4/9 - ... Unlike the CHE writer, some of us don't think the only way to represent a real number is by (integer) + a/10 + b/100 + c/1000 + ...

"infinite" -- If they mean the decimal expansion is infinite, then so too is the expansion of 1/3, the expansion of 1/7, etc. Maybe they mean that pi can't be described in a computable way, like Chaitin's constant "omega", except everyone knows pi is computable in very simple ways (as far as computability theory measures of complexity go), or do they?
Perhaps not everyone can type "pi" into Google and skim the pi Wikipedia page (the top hit when I tried this).

I think I know what they wanted to say, which was that no one knows whether, for each n, all possible n-digit strings appear in the decimal expansion with the same limiting frequency, although all computer explorations into the digits of pi seem to suggest this. Of course, they might want to express it a little less mathematical than this, but still
get the point across. As for saying "infinite", that's silly and a waste of words. Besides, it's automatic if you say something along the lines of what I just said, not to mention that everyone learns in 10th grade geometry (probably in 7th or 8th grade math books by now) that pi is irrational. Personally, I think knowing pi is irrational and what that means regarding decimal expansions is at least equivalent to knowing some of the literary references and schools of thought that get mentioned in their other articles without batting an eye. But you see, it's O-K in educated circles to say "economic determinism", but you don't ever want to say "irrational number".
The other article I saw showed up on March 12 at the CNN internet news
page:
http://www.cnn.com/2010/TECH/03/12/pi.day.math/index.html
The article is titled "On Pi Day, one number 'reeks of mystery'", and surprisingly it seems to be pitched at a higher mathematical level (probably 8th or 9th grade) than the Chronicle of Higher Education article (about 6th or 7th grade). Here is an excerpt from the CNN article:

***********************************

Mathematicians know that pi is irrational -- it cannot be represented as one number divided by another -- and transcendental, meaning it is not algebraic. That means, theoretically, that its digits will continue on indefinitely without ending in repetition -- in other words, the digits won't suddenly continue infinitely as 5s after 3 trillion digits (Pi's digits were calculated out to a record 2.7 trillion places in December
by French computer scientist Fabrice Bellard).

That also means, mathematicians theorize, that any string of numbers you can imagine is somewhere in pi -- for instance, look for your birthday. Coincidentally, "360," the number of degrees in a circle, occurs at digits 358 to 360. (Pat here..how cool, I did not know that... I know that it's a coincidence, but I love it)

***********************************

Right off the bat, in the first sentence, we have something any good middle school student would question -- isn't pi equal to pi divided by 1, and hence pi can be represented as "one number divided by another"? O-K, so the editor was asleep on that part. Let's continue. What's with this "theoretically" part? The digits of any irrational number (it's
worded as if you need to know that pi is both irrational and transcendental to conclude this, which is also an editorial oversight)continue indefinitely without being periodic, period. Then, in the next paragraph, we have another editorial flop. It's written as if the fact that pi is an irrational number (and maybe also the fact that pi is a transcendental number, an ambiguity we're left to figure out on our own)might mean that pi contains every finite string of digits, which of course isn't true -- plenty of irrational (and even transcendental)numbers have this property and plenty don't. Mathematicians theorize that pi might have this property, and even the much stronger limiting frequency property of these digit strings that I mentioned earlier, but to say that mathematicians theorize this on the basis of pi being irrational is extremely misleading. I think the author just wanted to write "This also means" because it sounded like a good transitional phrase, without worrying about what the phrase actually meant, and apparently the editor didn't worry about what it actually meant either.

And finally, what's up with saying "transcendental" means "not algebraic"? Does the author really think anyone who doesn't know what a transcendental number is will be helped by saying this is a number that isn't algebraic? I found this especially puzzling in view of the fact that in practically every single news article I've ever come across in which the term "light year" is used, the author seems compelled to state that a light year is the distance that light travels in 1 year (and then the author usually gives the equivalent in miles), and yet here "algebraic number" is thrown in without comment. I'd be willing to bet almost anything that far more people know what a light year is than what an algebraic number is. If I were editing the article I would have suggested saying something like
"although mathematicians have known that pi is irrational since the late 1700s, and transcendental--a certain extreme way that a number can be irrational--since the late 1800s, to this day no one knows . . ."There's no need in an article like this to define transcendental, but one should probably use the word since it's so well connected with pi
that it would seem strange to knowledgeable readers to not use the word.
If all this careful language analysis sounds unfair, ask yourself if being this sloppy with language usage would be accepted in an article about a bank robber (oops, I mean an alleged bank robber) or in an article about world affairs. No, it wouldn't. But it's O-K in math for some reason, and why this is allowed without much criticism is rather curious for a society that is so science-math-technology based.
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I hope I got all that right, and if not, it was almost certainly my cutting and pasting that created the problem. Thanks Dave, for bringing a little class to my blog... (hey, since I wrote part of this, does this qualify as co-publishing??? I'm ready to stretch the rules where needed...)

And Dave, If you ever have something else to share with my (somewhat limited) audience, I would love to host you again.

A Serendipitous Coincidence? The First-Ever Pursuit Problem.

Just reading through some old copies of the Mathematical Spectrum from my great source of mathematical periodicals, Dave Renfro. Intrigued by a couple of posts about a problem from the ancient Chinese Chiu Chang Suan Shu or Jiuzhang suanshu (Nine Chapters of Mathematical Art...about 150 BC) submitted by David Singmaster.

The problem: "A water weed grows 3 feet on the first day, and its growth on each succeeding day is half that on the preceding day. A reed grows 1 foot on the first day and its growth on each successive day is twice that of the preceding day? When are they of equal size?"

Go ahead, stop reading for a minute and try to solve it because I give an answer (actually two different ones) below and I don't want to spoil the fun.

The interesting thing to me, was the two letters of solution. My pre-calc students came through exponential growth and decay a few chapters ago, so they would approve of the first solution that was submitted. It suggested that we assume that the height of the water weed was growing according to the exponential function hw= 3 (1.5)^d-1. The reed would reach a height of hr=3^d-1. Setting these equal we would find the heights are equal when d= log₂6; or at about 2.585 days. They also pointed out that the mutual heights would be 5.705 feet.

Simple, quick, and "Wrong" according to the next commenter. They pointed out that a careful reading of the problem stated that the water weed "growth on any succeeding day is half that of the preceding day", would meant that it grew 3 feet the first day, and then successively it would grow 3/2, 3/4, 3/8 ... feet on each day.. to find the height we should sum this geometric series. So he suggests the height of the water weed would be hw=. This would result in the weed being somewhat shorter after each day than in the previous solution. In the same way, the reed should, on successive days, change by 1, 2, 4, 8 etc feet, so it would have a height of 2^d-1 after d days. Setting these equal we see that the two plants will both reach a height of five feet (exactly) after d= log₂6; or at about 2.585 days.???? WAIT, that sounds familiar....where have I ... Oh YEAH!!!, that was the answer to the problem done the first way? WOW, what a lucky coincidence........ Well, NO... the good Professor who posed the problem stepped up to assure us that, in fact, if you used those same two approaches to similar problems they would always reach equal heights at the same time... (can you prove that???)

Try for yourself. I assumed similar means that the shorter grows at r times its previous days amount, and the taller at 1/r times the previous day. Try a few. In fact, it is frequently the case that the second method gives an integer solution, and when it is not, it seems pretty easy to adjust the growth on the first day of the two plants to make it come out an integer. If we call the first day growths W and R, then the solution works out to log_r(rb/a); where r is the common ratio of the plant whose growth each day is increasing.

Dr Singmaster points out that this ancient text solves both quadratics and cubics, as well as systems of equations (including indeterminate systems with more unknowns than equations) using the "modern" method of elimination. It also is the earliest known text to have used negative numbers, and includes the rules for all the arithmetic operations. It is also the oldest know source of chase or pursuit problems, such as this one. This is problem 11 of the 7th chapter. I had the good fortune to sit with my beautiful Jeannie as the only two non-Chinese speakers at a discussion about the text at the Needham Research Institute in Cambridge by a group of English and Chinese experts. The amount I was able to take in is a credit to the patience of the gracious hosts.