Mathematical Impressions: The Golden Ratio
The golden ratio is greatly hyped, partly for its beautiful mathematical properties but also for nonsensical reasons. Distinguishing between the two requires understanding that mathematics is about structures and relationships,…

Mathematical Impressions: The Bicycle Pulling Puzzle
If you pull straight back on the lower pedal of your bicycle, will the bike move forward or backward? This classic puzzle has a surprising twist.

Mathematical Impressions: Change Ringing
Change ringing, in which a band of ringers plays long sequences of permutations on a set of peal bells, is a little-known but surprisingly rich and beautiful acoustical application of…

Mathematical Impressions: The Mathematics of Juggling
Juggling has advanced enormously in recent decades, since mathematicians began systematically investigating the possible patterns of non-colliding throws. As part of this research, many new possibilities have been discovered for…

Mathematical Impressions: Curved and Straight?
The elliptic hyperboloid is a beautiful quadratic surface that is “doubly ruled,” meaning that the surface, although curved, contains two straight lines through each point.

Mathematical Impressions: Spontaneous Stratification
Why does a mixture of sand and colored sugar spontaneously separate when poured?

Mathematical Impressions: Knot Possible?
The mathematics of knot theory says that a simple loop and a trefoil are fundamentally different knots. But is that all there is to the question?

Mathematical Impressions: Symmetric Structures
It is an unexplained fact that objects with icosahedral symmetry occur in nature only at microscopic scales. Examples include quasicrystals, many viruses, the carbon-60 molecule, and some beautiful protozoa in…

Mathematical Impressions: Making Music With a Möbius Strip
Musical chords naturally inhabit certain topological spaces, which show the possible paths that a composer can use to move between chords.

Mathematical Impressions: Goldberg Polyhedra
Because of their aesthetic appeal, organic feel and easily understood structure, Goldberg polyhedra have a surprising number of applications ranging from golf-ball dimple patterns to nuclear-particle detector arrays.