Wednesday, July 13, 2016 ... /////

Circle or square? Sorcery in the mirror

When squares look into a mirror, they see themselves as circles, and vice versa. Is it possible?

You bet. How was this video created? Isn't it just a product of photoshopping much like this fruit-based symbol of supersymmetry?

No, you can actually create these hybrid objects with a 3D printer. The rest of the blog post is full of spoilers.

Well, this 3D printing guy tells us how the object works.

What it actually looks like in 3D is a truncated cylinder whose base is a "squircle", a compromise between a square and a circle. Imagine that we parameterize the shape of the base contour in polar coordinates, by a function $r(\phi)$. And the height of the object which determines how the cylinder is cut (it is not cut by a flat knife) is chosen to be $h(\phi)$ in such a way that when you look at it from one particular angle similar to 45°, the upper boundary looks like a square. When you look at it from the opposite side, also at 45°, it looks like a circle.

Is it possible to find $r(\phi)$ and $h(\phi)$ that obey these two conditions?

It's not shocking that the answer is Yes. We have two functional $f(\phi)$ conditions – it looks like a square; it looks like a circle from another direction – for two functions $r(\phi),h(\phi)$. So you should expect that a solution exists. But you don't have much freedom here. You should better create both the top view as well as the terrain properly, otherwise it won't look like a square or a circle.

Here is one URL and another with pictures revealing what the objects actually look like from different angles – and you may buy the models for \$10 or so.

The video embedded at the top has over 6 million views and this Japanese guy's trick has been named as one of the best illusions of 2016 – by someone. ;-)

For a mathematically skilled reader: Can you actually write the equations for $r(\theta),h(\theta)$ and solve them? Numerically? Or even analytically?

Ultimate hint: The upper boundary is simply the intersection of one circular cylinder extended in one 45° direction, and a rectangular tube/cylinder in the other 45° direction. So everything is solvable.

A cool bonus exercise: Can you design a shape that looks e.g. like Hillary Clinton from one side and as Donald Trump from another side? You may even color the interior by two independent colors per "point". Just create teeth whose angle is parallel to the direction of sight for the unwanted color. Things like that are surely possible. For example, giraffes may look like an elephant from another angle:

The precise trajectories of these black ropes must be very particular in order to interpolate between the two animal species. Quantum gravity – as achieved by string theory – somewhat analogously interpolates between the low-mass quantum black hole behavior that should match the list of elementary particles; and the high-mass black holes that should match general relativity.