This week the #puzzle is: Can You Rile and Grace the Polyhedron? #geometry

An alien—more specifically, an Eridian—named “Rocky” needs to pass a solid xenonite crystal to his human friend in a neighboring spaceship. The crystal is shaped like a regular tetrahedron, as shown below, and all its edges have length 1.

To safely transport the crystal, Rocky needs to create a long cylindrical tunnel between the spaceships, and then orient the tetrahedron so that it fits through the tunnel. It’s okay if the crystal fits snugly inside the tunnel—in this case, it can slide along without any friction.

What is the minimum possible radius for my tunnel so that the crystal will fit through it?

And for extra credit:

Next, Rocky wants to transport a solid crystal shaped like a regular dodecahedron to his human friend. As before, each edge has length 1.

This time, the long tunnel between the space ships can be any right prismatic shape, not necessarily a cylinder. Once again, Rocky needs the crystal to fit through the tunnel, and it’s okay if that fit is snug.

What is the minimum possible cross-sectional area for the tunnel so that the crystal will fit through it?

Can You Rile and Grace the Polyhedron?

Intermission

The results from Q1 are in, and the finest of Fiddlers, who each solved all 24 puzzles, are: 👑 Adnan Haque 👑 from New York 👑 David Kong 👑 from Toronto, Canada 👑 Josh Silverman 👑 from Glen Head, New York 👑 Lise Andreasen 👑 from Valby København Danmark 👑 Peter Ji 👑 from Madison, Wisconsin 👑 Seth Cohen 👑 from Concord, New Hampshire

Woohoo!

Solution, possibly incorrect:

Desmos

My only option this week is to guess. I guess the circle shown on Desmos. Radius 0.5.

Given more time, I would have coded a tetrahedron, that could rotate, so that I could test different options.

And for extra credit:

I don’t even have a guess.