Activities to Go Along with

Mathematical Impressions

George W. Hart

I made a series of Mathematical Impressions video essays for the Simons Foundation.  Each shows something cool that I think is interesting to think about.  Any of these videos could be the starting point for a personal math project or a group activity at a math club.  So I am listing here some open-ended ideas of things to do that could go along with each video.  On the page with each video there are some references that may be useful to get you started.

If you try any of these out, let me know what you come up with.  And if you have other activity ideas I can add to any of these videos, please let me know.

The Golden Ratio

  • Read claims in books and online about the golden ratio and assess which you can justify as valid.

Curved and Straight

  • Make a hyperboloid using shish-ka-bob skewers and rubber bands.
  • Make a hyperboloid that can twist, using two disks and string.
  • Make one large enough to walk through.

Long Sword Dancing

  • Invent some popsicle-stick bombs.
  • Can you give rules that predict when they are stable?
  • Using yard-sticks (or meter-sticks) try the dances shown.

Spontaneous Stratification

  • Try it!
  • Experiment with varying materials to better understand the process.

The Bicycle Pulling Puzzle

  • Challenge your friends to predict what will happen with their bicycle, then try it.
  • Challenge your friends to predict what will happen when you pull the thread on a spool as shown (or pull the string of a yo-yo) and then try it.

Change Ringing

  • Play all the permutations of three, four, then five notes on a piano or other instrument.
  • Work in a group of n people, each with a sound or word, to produce all n! permutations.
  • Work in a group of n people to do the n-person "dance" that shows all n! permutations.

Bridges 2013 Art Exhibition

Mobius Music

  • Experiment with the free ChordGeometries software available here for visualizing spaces of musical chords.

Goldberg Polyhedra

  • Examine the dimple patterns on different brands of golf balls.
  • Make paper models of the first half dozen possibilities.
  • Read the paper cited with the video and solve the open problems about "paths."


  • Learn to juggle and learn about "site-swap notation." 
  • Understand Shannon's Juggling theorem.

Bicycle Tracks
  • Look for real bicycle tracks and figure out which way the bicycle went.
  • Consider whether a bicycle can make a single track, like a unicycle, other than the special case of a perfectly straight track.  Then look up the paper Can a bicycle construct a unicycle track? by David Finn.

Symmetric Structures

  • For each 3D symmetry group, make a list of the things you find around you that have that symmetry.
  • Construct your own "sculpture" for each 3D symmetry group.

Attesting to Atoms
  • Calculate the angle you expect to find between faces of an octahedron, rhombic dodecahedron, or pyritohedron. 
  • Obtain some of the crystals shown, measure their dihedral angles, and compare them to what you expect.

Knot Possible?

  • Cut a bagel, an inner tube, or other toroidal object to form a trefoil (in two fundamentally different ways).
  • Read about torus knots and see what other ones you can make physically.
The Surprising Menger Sponge Slice
  • Cut a cube of potato or cheese to reveal a regular hexagonal cross section.
  • Combine 20 cubes of wood, foam, or something, to make a 1st-order approximation to the Menger Sponge and slice it to reveal its special cross section.
  • Learn about a 3D modeling program (e.g., Blender or OpenScad) or ray tracing program (e.g., PovRay) to generate your own models and see their cross sections.
3D Printing Mathematical Models
  • Use the software shown to generate some algebraic surfaces or Seifert surfaces.
  • Explore Mathematica, Maple, or other software that can make other families of mathematical models
  • Find a local 3D printer or use an online 3D printing service bureau to physically produce your models.
Regular Polylinks
  • Make the two paper models shown: four triangles and six squares.
  • Make the regular polylink of six pentagons from paper.
  • Explore other materials for making polylinks, e.g., wood or basketry.

Shell Game
  • Simulate some of the rules shown, using two colors of checkers, beads, etc.
  • Invent your own rules and see what patterns they lead to.
  • Write a program which can follow your rules more quickly and accurately than you can.
  • Examine real shells (or online photos) and classify their pattern by which rule generates them.
Geometry of Spaghetti Code
  • Make a large geometric construction using one of the templates here: (1), (2), (3).
  • Design your own giant mathematical object to build.

Also, I have videos on my YouTube page, which are less formal, but may also encourage interesting projects and activities.