# 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).

John F. Lindner

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).

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).

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.

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 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.