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

How do you turn a rubber band into a knot? What happens when you slice a Menger Sponge on a diagonal plane? What is the math behind juggling? In this video series, George Hart illuminates mathematical concepts and surprising hidden geometries that may be found in the world around us.

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, not just numbers in isolation.

How do long sword dances produce stable patterns of interwoven segments? What are the possible variations? Explore the mathematics of this traditional art form.

Why does a mixture of sand and colored sugar spontaneously separate when poured?

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.

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 mathematics.

The art exhibition at the annual Bridges Conference showcases a wide range of artworks inspired by mathematical thinking.

Musical chords naturally inhabit certain topological spaces, which show the possible paths that a composer can use to move between chords.

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.

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 jugglers to attempt. In addition, the connections between juggling and the algebra of braids provide another view of juggling.

A nice mathematical puzzle, with a solution anyone can understand, is to determine the direction a bicycle went when you come upon its tracks. The answer involves thinking about tangent lines, geometric constraints and the bicycle’s steering mechanism.

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 the radiolarian family.