Summary

Created for George Mason University Math 401: Mathematics Through 3D Printing

Based on the work of Kokichi Sugihara's ambiguous cylinders and other optical illusions, this object will show two different images when looked at straight-on and reflected off of a mirror. This object needs a supported base so that it stands up in the correct orientation. The illusion only works when viewed from above at a 45 degree, with a mirror backing it.

The illusion is based on the fact that our eyes do not see three dimensional objects. Instead, we only see the two dimensional projection of the world onto a plane normal to our viewing direction. Our eyes are slightly separated from each other to give the perception of depth, but at significant enough distances the light from an object to each eye will approach becoming parallel. This means we can approximate human vision as being orthogonal for the purposes of the calculation.

The illusion is created by creating two curves. We project one curve onto the plane representing the audience's point of view, and project a second curve onto a plane on the other side of the mirror, representing the reflection. By solving a system of equations for when these two curves intersect, we can create this illusion.

A less mathematically intensive way, and the technique in which this object was created, is to use geometric intersections instead of projections. Simply create two curves and position them at right angles to each other. Linearly extrude them both, and then take the intersection. This object will look like the first curve when viewed from a 45 degree angle, and look like the second curve when reflected in a mirror.

I printed this on an Ultimaker S7 with a raft and judicious supports, at 2x scale. It took just under an hour to print.

For further learning about ambiguous shapes and 3D-printed optical illusions, check out
Sugihara's Website: http://www.isc.meiji.ac.jp/~kokichis/Welcomee.html
Printable Templates for Ambiguous Cylinders:
https://divisbyzero.com/2016/07/06/make-a-sugihara-circlesquare-optical-illusion-out-of-paper/
https://divisbyzero.com/2016/10/02/two-more-impossible-cylinders/