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Machines and Mathematics Play Together
June 9, 2017
Yoni Nazarathy
What are the building blocks of mathematics? Numbers, equations, coordinates and formulas? Or perhaps more abstract entities such as axioms, sets, definitions and theorems? I think that the answer depends on who you ask.
How about the building blocks of machines? Do these include design, operation, function and dynamics? Or maybe, cogs, bolts, shafts and some source of energy? Again there isn't a definite answer. Where am I going with this?
Can "mathematics" and "machines" play together? How?
One incredible way is through a mathematical framework called Exploding Dots. It is the main theme of the Global Math Project for 2017. If you haven't seen exploding dots yet, you must! You'll be blown away by the simplicity and elegance of this mathematical system. How is it related to machines though?
Exploding dots is an amazing math framework based on mathematical machines.
We'll get to exploding dots soon, but first lets think about this concept: mathematical machines. What is a mathematical machine? Here is one example:
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This neat illustration taken from Jason Wilkes's book, Burn Math Class, depicts a mathematical function as a machine. It operates on input s, and returns an output M(s). For this specific machine, throw in s=7 and get output M(s)=49. To test your understanding of this machine, consider this:
Can this "times self" machine produce negatives?
You may recall that a product of two numbers can be negative only if the numbers have opposite signs. This means that one of them is positive and the other is negative. Does this ever happen inside this machine? No.
Using machines to illustrate the concept of a function is a great pedagogical metaphor. You may now ask:
Are there other ways in which machines and mathematics play together?
Here are interactions that immediately come to mind:
Designing Machines with Math: Mathematics is often used to design, calibrate and optimize physical machines. Cars, computers and cranes are all products of intricate designs hinging on mathematical techniques. Such an engineering process is sometimes the bread and butter of a mechanical engineer or an electronics hardware engineer. Even bicycle design is based on mathematics. It all starts somewhere.
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The Basic Machines of Math: Some machines are used in elementary mathematics. These include protractors, various types of rulers and of course the drawing compass. You can actually have much fun with a ruler and a drawing compass. Such simple machines can drive so much thought. See for example this numberphile video featuring Zsuzsanna Dancso.
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Control Theory: There are also whole fields of applied mathematics that sprung from the invention of machines. For example the centrifugal governor invented more than two centuries ago motivated the field of control theory. It is an interdisciplinary field that combines mathematics and engineering, with a wide range of applications like airplane autopilots, space exploration and health care machines. I highly recommend a neat simple book on this topic: Feedback and Control for Everyone, by Pedro Albertos and Iven Mareels.
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Logic and Computer Science: Then you come to computers. Incredible machines, aren't they? The past century has witnessed the computing revolution. One of its fathers, Alan Turing, invented a mathematical abstraction of a computer. This mathematical model is now called the Turing Machine. Some may say it is the most serious mathematical machine invented to date. See this incredible video by Art of the Problem:
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Cellular Automata: There are also other forms of mathematical machines. For example, cellular automata have become very popular in recent years among artists, mathematicians and practitioners. Cellular automata allow "life-like behaviours" to be generated from a few simple mathematical rules. For example, see this neat animation:
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Machine Learning: This is all about making a computer carry out intelligent tasks by training it to "think softly". Can a computer look at an image of an animal and decide if it is a cat or a dog? Machine learning helps. Watch this neat Oxford Sparks video:
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So as you see, there is a vast playground for machines and mathematics.
nd now exploding dots!
Exploding dots is a neat conceptual framework for arithmetic, algebra and mathematical exploration. It allows you to feel mathematics in a visual and intuitive manner.
What aspects of math is exploding dots good for?
Exploding dots is great for understanding addition, subtraction, multiplication and division. It is great for understanding base representation of numbers. You don't have to think base 10 all the time. It is great for understanding how long division works. But that is just the start.
What else can you do with exploding dots?
Exploding dots allows you to understand how to represent polynomials just like numbers. Yes polynomials, those objects encountered in algebra class. It even allows you to explore division of polynomials. You can even create fractions and make up certain irrational numbers. Exploding dots provides you with intuition for all these things. Watch this introduction by James Tanton:
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So how are machines related?
The basic building block in exploding dots is a machine. Common machines explored in exploding dots are the 2-->1 machine, the 10-->1 machine, or if you are into base 4, the 4-->1 machine. Then there are more abstract machines, for example the x-->1 machine that helps with exploring algebra and polynomials. In fact, you can explore any M-->N machine that you wish.
How do these machines work?
The basic operation of the machines in exploding dots is simple. Consider for example the 2-->1 machine as in the video below. The machine has cells that contain dots. But dots can be stable or not. Put more than two dots in a cell and "Kaboom". They explode. The result is a single new dot born on the next consecutive cell for each explosion. That is it! The operation of the machine is very simple. Watch:
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So how can such a simple thing be used for mathematics?
Exactly! This is the beauty of exploding dots. It is a simple set of rules (mathematical machine) that allows you to carry out extremely powerful mathematics. Play around with the 2-->1 machine, the 10-->1 machine or the x-->1 machine and you'll understand so many things better. See this collection of free great resources.
There is so much more to say about exploding dots. There are many little and big lessons to be learned through the content of the global math project and the content of G'day Math by James Tanton. The nice thing about it all, is that hundreds of caring educators around the globe have joined the global math project. You'll surely hear more from us at One on Epsilon about Exploding dots as we near the global math week.
The global math week takes place on October 10-17, 2017.
In the meanwhile, we recommend that you register with the global math project and prepare to be blown away by exploding dots. Join the movement.
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