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.

# 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.

### 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: Long Sword Dancing

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

### Mathematical Impressions: Spontaneous Stratification

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

### 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.