Hopping Midpoints and Mathematical Snowflakes

Let me start this off by saying that if you’re reading this blog and haven’t ever checked out the Youtube channel Numberphile, you need to go there right away and start bingeing their videos. That’s what I’ve been doing for the last few days, and it’s given me tons of cool new puzzles to consider.

Here’s one:

Naturally, after watching this video I wanted to try this out for myself. Here you see the pattern arising beautifully from the randomness:

Serpinski

I urge you to think hard about why the Sierpinski triangle would arise from something as simple as randomly hopping between midpoints. It’s very non-obvious, and although I have a few ideas, I’m still missing a clear intuition.

I also made some visualizations for other shapes. I’ll show some of them, but encourage you to make predictions about what pattern you’d expect to see before scrolling down to see the actual result.

First:

Square

Instead of three points arranged as above, we will start out with four points arranged in a perfect square. Then, as before, we’ll jump from our starting point halfway to one of these four, and will continue this procedure ad infinitum.

What pattern will arise? Do you think that we’ll have “missing regions” where no points can land, like with the triangle?

Scroll down to see the answer…

(…)

(…)

(…)

Square

Okay! So it looks like the whole square gets filled out, with no missing regions. This was pretty surprising to me; given that three points gave rise to a intricate fractal pattern, why wouldn’t four points do the same? What’s special about “3” .

Well, perhaps things will be different if we tweak the positions of the corners slightly? Will any quadrilateral have the same behavior of filling out all the points, or will the blank regions re-arise? Again, make a prediction!

Quadrilaterals

Let’s see:

4 TemplesCross4 Double Temples4 Rolos

Okay, now we see that apparently the square was actually a very special case! Pretty much any quadrilateral we can construct will give us a nested infinity of blank regions, as long as at least one angle is not equal to 90º. Again, this is fascinating and puzzling to me. Why do 90º angles invariably cause the whole region to fill out? I’m not sure.

Let’s move on to a pentagon! Do you think that a regular pentagon will behave more like a triangle or a square?

Pentagon

Take a look…

Pentagon

And naturally, the next question is what about a hexagon?

Hexagon

Hexagon

Notice the difference between the hexagon and all the previous ones! Rather than having small areas of points that are never reached, it appears that suddenly we get lines! Again, I encourage you to try to think about why this might be (what’s so special about 6?) and leave a comment if you have any ideas.

Now, I because curious about what other types of patterns we can generate with simple rules like these. I wondered what would happen if instead of simply jumping to the average of the current point and a randomly chosen point, we built a pattern with some “memory”. For instance, what if we didn’t just look at the current point and the randomly chosen point, but also at the last chosen point? We could then take the middle of the triangle formed by these three points as our new point.

It turns out that the patterns that arise from this are even more beautiful than the previous ones! (In my opinion, of course)

Take a look:

Triangle

Threes Triangle

Square

Threes Square

Pentagon

Threes Pentagon

Hexagon

Threes Hexagon

Heptagon

Threes Heptagon

I’ll stop here, but this is a great example of how beautiful and surprising math can be. I would have never guessed that such intricate fractal patterns would arise from such simple random rules.

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