Some of My Favorite Things

An Aperiodic Tiling

Three convex tiles discovered by Robert Ammann circa 1975 can only tile the plane nonperiodically. These tiles are related to Roger Penrose's famous fat & skinny rhombs (and kite & dart).

A Flexible Polyhedron

It was long thought that all polyhedra composed of rigid faces and hinged edges are themselves rigid.

However, in 1977, Robert Connelly found a counterexample, and soon afterwards Klaus Steffen found the following simple polyhedron with only nine vertices that flexes without self-intersection or distortion of its faces (while preserving its volume).

An Amazing Puzzle

Ernö Rubik's 1974 masterpiece.

A seemingly impossible mechanism; how does it not fall apart? A silent challenge; one knows immediately what needs to be done.

A Beautiful Riddle

In 2000, Erich Friedman posed the following variant of a 1969 Hans Freudenthal problem:

I have secretly chosen two numbers between 1 and 9 (inclusive), and I have separately told their sum to John and their product to Kelly, both of whom are completely honest and logical.

Kelly says: I don't know the numbers.
John says: I don't know the numbers.
Kelly says: I don't know the numbers.
John says: I don't know the numbers.
Kelly says: I don't know the numbers.
John says: I don't know the numbers.
Kelly says: I don't know the numbers.
John says: I don't know the numbers.
Kelly says: I know the numbers.
John says: I know the numbers.

What are the numbers?

A Remarkable Shape

A gömböc (pronounced “gømbøts”) is a nonunique convex three-dimensional homogeneous body with just one stable (and one unstable) point of equilibrium.

Its existence was conjectured by Vladimir Arnold in 1995 and proven by Gábor Domokos and Péter Várkonyi in 2006.