Discovering “New” Methods Through Visual Patterns

I had a go at creating a “Visual Pattern” after having so much fun attempting Simon Gregg’s pattern from Fawn Nguyen’s website.

It looked like this:

Can you see how it grows? Is there a rule to how it is growing or is it random?

Stage 4, 5 and 6 might make it easier, but no guarantees:

If you want to figure it out for yourself, don’t scroll past this point.

Warning: There’s a solution past this point!

Ok, here was my thinking when I created this – was it the same as how you saw it?

For each triangle, I would first add another triangle to its left side, then its right side, and then its bottom side. What happens if it’s upside down or on its side you ask? Well, each new triangle must also follow this rule, therefore the side it is joined by is its “left side”. See below for a more visual explanation:

Then, I found something quite interesting when I started looking at how it was growing. This is undoubtedly an existing method, but it was new to me, so it might just be new to you. Here’s a video of my solution:

If you don’t want to watch it, here’s what I did:

State the pattern

Find the difference of each stage (rate of change)

Find the difference of the differences (the rate of change of the rate of change)

Then, I realised what was going on. I had just found the values of the first and second derivatives!

From then, I just used the values I had already found to find the unknown constants in f’(x) and f”(x).

Pretty cool.

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