Can I Please Have My Cat Back?
What Arnold Cat Map informs us about How To Teach Mathematics Effectively
Introduction
This post is on:
Why do people get disinterested in Mathematics quickly?
I take the fun game of:
Arnold Cat Mapping
to make my point.
Please do this for me:
1) Stretch the picture of the cat in the following way so that every x and y changes:
x = 2x + y
y = x + y
Notice that the picture has become larger.
2) For each x and y take the integer part off. For example if x becomes 2.35 then the new x is 2.35 - 2 = 0.35. If y becomes 1.50 then new y is 0.5.
That is, with a unit equal to the width of the square image, the image is sheared one unit up, then two units to the right, and all that lies outside that unit square is shifted back by the unit until it is within the square.
Now the question is:
If you keep doing this again and again (each iteration) - will the Cat image appear back again?
The miraculous answer is yes! After 348 mangling the original cat image appears!!
I was recently fascinated reading about Arnold’s Cat Map which is a transformation that can be applied to an image. The pixels of the image appear to be randomly rearranged, but when the transformation is repeated enough times, the original image will reappear!
In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name. It is a simple and pedagogical example for hyperbolic toral automorphisms.
If this is too technical do not worry. This post will explain why Arnold’s Cat Map is a beautiful example of How To Teach Mathematics effectively. Before you read any more, please try demonstration of Arnold’s Cat map here:
The transformation is as follows:
Shear the image in the both the x- and y-directions by a factor of 1.
Wrap the image back onto itself i.e. treat it like a torus (doughnut shape). This is a way of treating a 2D grid like it has no edges.
This transformation demonstrates some of the principles of chaos i.e. underlying order to an apparently random evolution of a system. Arnold's Cat Map is named after the Russian mathematician Vladimir Arnold, who discovered it using an image of a cat.
I clicked through each iteration (till 348) to see effect of cat mangling, I observed:
Cat ghosts
Many miniature cats (apparently)
I did not observe inverted cat.
Basics
So here is what happens with Arnold’s Cat Map:
You have an image in the Unit Square (a photo bound within x = 0,1, y = 0,1)
For each iteration, do this:
x = 2x + y
y = x + y
If x and y becomes more than 1 then make it less than 1 by subtracting the largest integer. For example if we start from (x = 0.75, y = 0.25) then new x, y will be:
x = 2 * 0.75 + 0.25 = 1.75 => 1.75 - 1 = 0.75
y = 0.75 + 0.25 = 1.0
Next iteration:
x = 2 * 0.75 + 1.0 = 2.50 => 2.50 - 2 = 0.50
y = 0.75 + 1.0 = 1.75 => 1.75 - 1 = 0.75
As we go iteration after iteration, pixels in the cat (confined within unit square) gets mangled over and over again.
Intuition
To provide you a solid feeling of the iterations, I have attached the dynamic GIF of a picture of a pair of cherries. The image is 74 pixels wide, and takes 114 iterations to be restored, although it appears upside-down at the halfway point (the 57th iteration). What you are observing is novel visualization of Poincaré recurrence theorem - Thanks to Vladimir Arnold.
In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state.
The Poincaré recurrence time is the length of time elapsed until the recurrence. This time may vary greatly depending on the exact initial state and required degree of closeness. The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume. The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems.
The theorem is named after Henri Poincaré, who discussed it in 1890.
Applicatiion
Steganography is the practice of representing information within another message or physical object, in such a manner that the presence of the concealed information would not be evident to an unsuspecting person's examination. In computing/electronic contexts, a computer file, message, image, or video is concealed within another file, message, image, or video. The word steganography comes from Greek steganographia, which combines the words steganós (στεγανός), meaning "covered or concealed", and -graphia (γραφή) meaning "writing".
The advantage of steganography over cryptography alone is that the intended secret message does not attract attention to itself as an object of scrutiny. Plainly visible encrypted messages, no matter how unbreakable they are, arouse interest and may in themselves be incriminating in countries in which encryption is illegal. Whereas cryptography is the practice of protecting the contents of a message alone, steganography is concerned with concealing both the fact that a secret message is being sent and its contents.
If we want to protect an image from tampering we can insert a smaller set of adjacent pixels (the watermark) at certain point of Arnold Cat Map. The watermark detection alogorithm continues the iteration till it detects the small watermark.
Realization
People often quit Mathematics education (and interest) early since often help is not available in the form of:
Gentle Introduction
Jargon Free Language
Examples
Visualization
Application
This post has all of the above elements - judge for yourself.
Postscript
If I were to provide you the following facts:
Two Tables make a Chair
Two Chairs make a Beer Mug
For a Beer Mug, given a Chair and a Table not in it, at most one Chair similar to the given Chair can be passed through the Table.
I then ask you to prove the following:
A set of at least three distinct Tables are CoChair if and only if, for every three of those Tables A, B, C with d(AC) greater than or equal to each of d(AB) and d(BC), the pub inequality d(AC) ≤ d(AB) + d(BC) holds with equality.
I am sure you will throw your hands up in the air - What nonsense is this??
What I did is substitute Table, Chair and Beer Mug for Point, Line, and Plane. Where Euclid had viewed the postulates of geometry as evident, and representative of physical space, Hilbert made no attempt to relate his axioms to anything, nor did he see the need to as a matter of general principle. In 1891 he had remarked:
"One must be able to say at all times instead of points, straight lines, and planes – tables, chairs, and beer mugs."
What counted was simply what the rules of logic allowed you to deduce from uninterpreted axioms.
This style of Mathematics is known as:
Formalist Style
This is similar to Instrument Rated Pilots - who can fly in complete darkness with the help of instrument panel. Formalist style mathematicians can do mathematics without attaching intuitive meaning to axioms.
In my humble opinion, formalist styled mathematics education causes early termination for many due to the sheer terror of flying blind. When Sir Andrew Wiles was asked about his journey on proving Fermat’s Last Theorem, he said this:
Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them.
Not many are as courageous navigating dark rooms and turning on lights one by one over seven years.
Vladimir Arnold resisted Formalist Style in Mathematics education. He gave us
Arnold’s Cat Map