Generalizing Patterns

One of the main themes of Mathematics is generalization. which means, from observing a specific situation you will try to jump to a more general case. To make this more clear, let us give a simple example in arithmetic. From our early years of school we know that 1+1=2  we can write this  also 1+1= 1\times 2,  and 2+2=4, 2+2=2\times 2. so we already we are observing a pattern in which if you add a number with itself it will be equal to twice itself. In the first case, this fact looks very simple and innocent, but in mathematics we appreciate the way we formulate and abstract it lot. In another way, this shifts our thinking from concrete to abstract, from locally to globally, from numerically to algebraically. so after we add some number to itself we get the general fact that:

n+n=2\times n

Of course, it’s not enough to leave it there — we need to show and justify that the above statement is true. The process of justifying any mathematical fact is called a proof, and here we do not do that.

Until now, we have only talked about math, and now the question, where does IGP come into play? We say that in IGP there are a lot of generalizations. That is, if you have one pattern, and you know the way to construct it ( the “construction line”), you can generalize that pattern, you can even give the mathematical terms of that generalization.

To make this more understandable, we will give two examples. and we will see how this happens.


if you look at the pattern above, which can be found on Eric Brouc’s Website, you can see it consists of 10 interlaced chains. Below you can see the steps of creating this pattern.

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