### 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, not just numbers in isolation.

### 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.

### 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 mathematics.

### Mathematical Impressions: Art Imitates Math

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

### 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.

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

### Mathematical Impressions: Bicycle Tracks

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.

### 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 the radiolarian family.

### Mathematical Impressions: Attesting to Atoms

Can you combine simple observations and mathematical thinking to show that atoms exist?

### 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: The Surprising Menger Sponge Slice

The Menger Sponge, a well-studied fractal, was first described in the 1920s. The fractal is cube-like, yet its cross section is quite surprising. What happens when it is sliced on a diagonal plane?

### Mathematical Impressions: Printing 3-D Models

George Hart describes in this video how to create physical models of mathematical objects, surveying some examples of surfaces and polytopes.

### Mathematical Impressions: Shell Games

A video explaining how some seemingly complex patterns on sea shells can be created by simple, one-dimensional, two-state cellular automata.