Saturday, February 18, 2012

Put the “ah” back in arithmetic…

Maths that makes you go "Hmmm"
Mathematics often puzzles me.  Sometimes it also makes me wonder.  We get accustomed to the Fibonacci sequence and the Golden Ratio appearing in nature so often that we cease to register just how remarkable this actually is and sometimes fail to ask ourselves why this should be.  

Sometimes it takes something new to bring us to a fresh realisation that the presence of some forms of mathematics in nature is not only unexpected but  possibly unexplainable.  Some unexpected occurrences of  mathematics are truly remarkable.  The following clip provides a worthy example.

It is taken from the BBC series “Why Maths doesn’t add up” and features
features Marcus du Sautoy,  Professor of Mathematics at Oxford and Simonyi Chair for the Public Understanding of Science with actor, comedian and maths fall guy  Alan Davies. Early in the clip we see a graph first devised by Bernhard Riemann showing the random distribution of prime numbers. In this excerpt  the pair see what happens to the vibrations of a sphere of quartz when the quartz is struck with a ball bearing.    Quite unexpectedly the two graphs are startlingly similar.

As du Sautoy says in the clip, the similarity of the graphs is so striking that “it can not be a co-incidence”.

So if it isn’t a co-incidence...what is it?

Perhaps in pondering such “oddities” we will also remember that the “common” occurrence of the Fibonacci sequence and the Golden Ratio in nature are also worth of contemplation.  The unique Vi Hart has a series of her manic videos about the Fibonacci sequence which is well worth a diversion.

I believe it is this sense of wonderment and intrigue that we need to impart to our students as much as computational skills and procedural understanding.

Sierpinski Surprises
One of the wonderful things about mathematics is that it is frequently surprising.  Often relatively simple things reveal unexpected hidden depths - and do so with minimal probing.  One such aspect of mathematics is the Sierpinski triangle (aka Sierpinski gasket).  
The Sierpinski triangle was named after a Polish mathematician who explored the concept around 1915,  although it is not true that he “discovered” it as the basic shape apparently appears in art work dating from some centuries before.

The Sierpinski triangle is a self similar set - a pattern that can be made larger or smaller indefinitely while maintaining the same pattern,  in other words, it is a fractal.  Fractals are not just found on the pages of maths books - which is one of the reasons that they are so fascinating.  Fractals are found in many places in nature - from snowflakes to certain leaves, from ferns to forked lightning, even, most unexpectedly, in broccoli. Making Sierpinski triangles is something that even relatively young students can do. Start with an equilateral triangle. Inscribe that triangle with an inverted copy of itself (or, more simply, make an upside down triangle inside the first one). And keep repeating. (One set of instructions for use with young mathematicians can be found here.)  The pattern created, as depicted above, is surprisingly pleasing for what is essentially a mathematical process.

Creative types have added colour, and combined them to make them even more so as shown here.  So, the Sierpinski triangle is interesting and visually attractive in its own right...but there are some fascinating attributes that do not immediately meet the eye.  (Source)

The other famous triangle in mathematics is Pascal’s triangle. It has become customary to credit the discovery of this triangle to Blaise Pascal although the concept was known well before
him.  In Pascal’s triangle the two numbers above a cell are added to create the  number in the next row and the process repeats.  Now, this is where things get interesting.  If we shade only the odd numbers in Pascals triangle we get … something VERY similar to Sierpinski’s triangle - and the similarity strengthens as the larger the triangle continues.   

Why might this be?
Sierpinski and Chaos
There is a fascinating game invented by Michael Barnsley called “The Chaos Game” - which, ironically, proves that order can come where chaos is expected.
To “play” this game grab a sheet of paper and mark three points of an equilateral triangle.  Label the points A, B and C. Make a mark at any random point on the paper - call it X. Use a die - numbers 1 & 2 relate to point A, 3 & 4 to point B and 5 & 6 to point C. In our example imagine you roll a 1 (which therefore indicates point A).  Measure half the distance from P to A and make a new mark. This is the next point. Roll again and mark the point half way between the relevant point indicated by the die and the last point obtained. Repeat and repeat...and repeat.  The longer you “play” the game the more strongly a pattern emerges - and that pattern is the Sierpinski triangle!  
When I first read about this years ago I wasted hours “playing” the game in an attempt to test it. It proved to be true. These days actually playing the game is not strictly necessary as there are any number of interactive versions of the game available on the Internet.  One can be found here - I recommend that you try it for yourself.  Concentrate on your sense of wonderment as the patten unfolds. This is the feeling that we want our students to have when they explore mathematics.   Incidentally, this “game” has been called the “Creationist’s worse nightmare” - for those who like their maths flavoured with mysticism - and the site that uses the term explains why the pattern works well.
Counter-intuitive results such as that produced by the Chaos game can intrigue students - which leads to engagement...which leads to learning.
Tower of Sierpinski?
For me personally this is perhaps the most perplexing of the unexpected appearances of the Sierpinski triangle.
The Towers of Hanoi is a popular game / puzzle where the player is required to shift a number of disks from one of three “posts” to another of the three available and reassemble them in order with the largest disk at the bottom and the rest of the disk sitting on the bottom disks in order. It is a surprisingly simple yet engrossing game. 

An interesting thing occurs when the moves leading to a solution of the Towers of Hanoi are graphed.
(source)The resemblance of this graph to the Sierpinski triangle is startling. It is worth pausing a moment to think about this. The Tower of Hanoi is a mental experiment, a “game”  devised by a human.  It is not a “naturally occurring”  phenomena such as a fractal snowflake or a symmetrical fern leaf - it is totally the product of human imagination.  Yet the solution to this totally invented  game, when graphed, has a strong resemblance to the Sierpinski triangle - which is a fractal.  It could be argued that Sierpinski triangle is also the product of the human mind - yet this does not diminish the sense of surprise when the link between the two concepts is established.  Why should this link exist? The sense that there is some intriguing connection between two such different things is tantalising.  In mathematics the enjoyment is in the exploration and the discovery.  Sometimes no answer is much more satisfying than a clinical definition that puts everything in its place.A slice of Biblical pi

Author Alex Bellos has been rightly praised for his fascinating book “Alex’s Adventures in Numberland”. It is a book full of mathematical gems presented in an accessible style not normally associated with mathematics books - which possibly explains its success.  One of the chapters is devoted to the study of pi and the people who have pushed back the boundaries of this number - a dry topic transformed into a engrossing read.  

For those who need a reminder - pi is the ratio of a circle’s circumference to its diameter.  As Bellos puts it ”...if you take the diameter of a circle and curve it around the circumference, you will find that it fits just over three times.”  It is that “just over” bit that makes pi interesting.  The “just over” bit is actually an irrational fraction - meaning it never ends  - it simply goes on forever.  The accuracy of pi has been calculated to a ridiculous degree - billions of digits. According to Bellos, manufacturers of precision instruments only need an accuracy of four decimal places so the quest for a more and more accurate figure for pi is no longer driven by any practical reasons.  However, early methods of calculation were ingenious and the chapter in Bellos’ book provides an entertaining overview of many of them.
The chapter contains this discussion of pi in, of all places, the Bible. 
“...A line in the Bible reveals a situation in which pi is taken a 3: ‘Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about’ (I Kings 7:23)”.
Thus the Bible is on shaky ground from a mathematical perspective.  After a very brief discussion on some squeamish “explanations” from devote believers on what the bible may have meant when giving that description Bellos throws in an absolute gem.
“A mystical explanation is much more enticing: due to the peculiarities of Hebrew pronunciation and spelling, the word ‘line’, or qwh, is pronounced qw. Totting up the numerological values of the letters gives 111 for qwh and 106 for qw. Multiplying three by 111/106 gives 3.1415, which is pi correct to five significant figures.
Today we would call this  numerology - but the practice has a long history. Traditionally the practice of ascribing numerical values to words was known as gematria and was wide-spread amongst the ancient Greeks.
I’m not sure what I find hardest to believe - that someone in Biblical times not only knew pi to five significant figures and could hide it via gematria in a passage to be decoded by the enlightened...or that purely random chance and an accident of linguistics has delivered pi to five significant figures - in a passage describing a circle.
Irrational numbers like pi can be tricky to understand.  A certain Dr Edward J. Goodwin came up with a … novel  … “solution”.  The good doctor decided that pi,  featuring all those uncooperative numerals after the decimal  point, was simply too complicated and decided that pi should be LEGISLATED to be 3.2.  He would grant to the state of Indiana a legal right to use his version of pi = but every one else would have to pay him a royalty when using his version of pi!  Legislation to introduce this actually made it to the Indiana legislature.  More of the story can be read here

What has this mathematical musing have to do with education? Two things. As teachers we don’t always have the answers - and we should admit this to our students - not necessarily in relation to the concepts discussed here but in general terms. The second is that the simple fascination of the unknown and the delight of exploration that is associated with mathematics should be shared with our students. Again, not necessarily these concepts, but with mathematics in general.  We should present mathematics as a subject full of fascination to be explored rather than allow it to be reduced to a collection of rote exercises, algorithms and meaningless formulae.
The worth of a word:
Gematria is an ancient belief. For most of us it is as valid as the notion of a flat earth or that the earth was a disk sitting on the back of a turtle floating through space.  However, if you want to calculate the numeric value of certain words or names then clicking here will take you to  a site that may be useful as well as amusing.  The site also provides words with the same numerical value as the text you input.

Have fun.

NB: This post is a combination of three earlier posts all with the theme of mathematics that makes you wonder. If you enjoyed this post you may find this link to my maths page which features other mathematical posts that go well with a cup of coffee.
Credits:Sierpinski triangle image link: 
Pi graphic:
Pascal/ Sierpinki triangle link:


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