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## Cycloid illusion is not really an illusion

Photo credit: brusspup

Recently I came across the following “optical illusion” on the normally good website I Fucking Love Science:

http://www.iflscience.com/brain/cycloid-optical-illusion-will-boggle-your-mind

I encourage you to read the article and think about the video (otherwise the rest of this post will be less than illuminating).

Supposedly, there is an “optical illusion” in the video because you think there’s a wheel when in “reality” the dots are moving linearly.

This is bullshit.

This is not an illusion. The dots are moving linearly, that’s true. But there is also a wheel. If this causes you cognitive dissonance, so be it. It is not a paradox, however. It is the case that some wheels, when spinning inside other wheels, have points on them which travel linearly.

I admit, this is a field of mathematics close to my heart. Much of my recent work has involved geometric phase, which has connections to Spirographs, epitrochoids and hypotrochoids as mentioned in the IFLS article. I have battered notebooks with over 500 pages of algebra devoted to such things. This is something I know something about.

One way to see that this isn’t an illusion at all is to watch it being drawn. Go to the following Spirograph applet:

http://www.personal.psu.edu/dpl14/java/parametricequations/spirograph/

Play around a little bit. Then create the following specific Spirograph (which is exactly the one in the IFLS “illusion”):
Radius1 = 60
Radius2 = -30
Position = 29
Velocity = 8
If you need to, CLEAR the picture, input the above parameters, and hit DRAW. Hit DRAW again to watch it all over again.

There’s no illusion. A circle is rotating inside another circle. Simultaneously, a particular part of that circle is traversing a straight line. This comes as a surprise to many people: there’s a frisson of incredulity from the idea that your motion can be simultaneously linear, but curved as well. But there’s an easy explanation. Your motion is different in different frames of reference.

Consider an ant on the wheel. With respect to the center of that wheel, he just orbits in a circular manner. But with respect to us, outside of the contraption, he moves back and forth along a single line. This is not an optical illusion. It’s the relativity of geometric shapes.  And I think that’s even cooler than some cognitive trickery.

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### 4 Responses

1. I agree with you and then some. The rolling of a circle inside one of twice the radius shows up in various funny places. One is this: A ladder slides while
keeping contact with both the floor and a vertical wall (all 2D). Like all motions in the plane it can be though of as rolling. What are the body and space curves for this motion? Answer: a circle whose diameter is the ladder and a circle centered at the origin with the ladder-length as radius. Another, related is a wooden office toy that has two blocks in two slots and a overlength rod with a handle connecting them. They move my each other in an interesting way while the connecting rod motion is the rolling of a disk (not seen) inside a circle (also not seen).

The supposed optical illusion is BBBBS! That some cycloids are straight lines is a cute interesting non-intuitive thing. But its not an optical illusion.

2. Matt, this “illusion” was posted again today on the “Why Evolution is True” website…not sure if you are a regular reader of that one. If not, you might find it interesting to join and share your observations above.

• I did see that. I noticed someone had a correction at the end.

3. The word “illusion” is used to describe one perspective; when 8 dots are moving then 8 dots are moving. When 8 dots are moving it will appear as if there is a circle within a circle. If you perspective is as you described then you can conclude that in two rotating gears you can extrapolate a central point back and forth along a radius/diameter (whatever). I’m using the illusion word.