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Freaky Dot Patterns

This post is inspired by the last post of one of my favourite youtube educators, Numberphile.

I think the video is wonderful and mesmerizing. The basic idea of the video is sliding two sheets of paper identical in all aspects including the patterns on them, except one of the papers is a normal one and the other, transparent. The relative motion, linear and otherwise, gives rise to impressive patterns.

The dots on the papers are either random or regular. So it can be simulated using a regular or random 2d grid of points. This can be easily simulated and I am trying to reproduce the patterns shown by Tadashi Tokieda. Quoting from his personal page, “One of his lines of activity is inventing, collecting, and studying toys—simple objects from daily life that can be found or made in minutes, yet which, if played with imaginatively, exhibit behaviors so surprising that they intrigue scientists for weeks…his fellowship project is to make his collected toys (now numbering nearly a hundred) permanently and widely available by creating texts, photos, movies, web resources, and durable and aesthetic models. The toys, in turn, provide both inspiration for fresh research in applied mathematics and elegant entertainment. He hopes to invent new toys and improve old ones during the fellowship.” This reflects the intent of the video, which is to explore the rather complex patterns produced by something that is a simple as random dots on parchment.

I try to reproduce all the patterns that are shown in the video. I will illustrate all the basic steps for the first patters, after which the same protocol will be used for all the subsequent demos.

Lets take the same first step as Tadashi.

Rotation

To begin with, we need a set of random dots as shown below.

original

We then take the same pattern and rotate it by an angle (2 degrees in this case), to get the following pattern.
rotated

Now, we superimpose these two ‘sheets’, with the regular dots on the bottom and the rotated dots on top.baseline

We start to see the concentric circle patterns, just as in the video. Now all we need to do is to slide the ‘top’ (read: transparent) layer, in the ‘X’ direction (parallel to the horizontal or parallel to the keys in your keyboard). I limit the X displacement to within the range of -0.05 to 0.05.

We can clearly see that as the relative motion happens in the ‘X’ direction, the ‘center’ of the circular pattern appears to move normal to it. Tadashi Tokieda explains the reason for this in the video, so we shall not dive into it.

Scaling

For the next trick, we still stick to the random dots, but the second layer of dots is shrunk (by 4%) relative to the first layer. When we superimpose these 2 layers, we get a interesting radial pattern, as shown below.

021

Now we try to rotate the two layers relative to each other, but limiting the range of rotation to within +/- 10 deg. We see the mesmerizing spiral, just as in the video.

This brings an end to the use of random dots. We now experiment with regular 2d grid-styled dots.

Regular Squares

The grid for the regular squares look like this.

001

As always, we rotate the top layer past this layer (between 0 and 90 deg). This gives rise to more intriguing patterns called the Moire Patterns.

Other Regular Shapes

When we do a similar animation of regular triangles, we get a hexagonal pattern.

And we wrap it up with circles.

Summary

Though the final intent of Tadashi’s research interest is developing potential new toys, the motivation of this particular post was having fun with regular and random dots and reveal any interesting pattern their self interaction might have. And it did. There were some very interesting Moire patterns and some more of the unexpected kind. Its really astounding in what ways simple dots can interact with itself. Thanks to Numberphile and Tadashi Tokieda for this wonderful video.

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