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


 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 yoyo) and then try it.


 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 nperson
"dance" that shows all n! permutations.




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


 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."


Juggling
 Learn to juggle and learn about "siteswap
notation."
 Understand Shannon's Juggling theorem.


 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.


 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.


 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.


 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.


 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 1storder 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.


 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.


 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.


 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.


 Make a large geometric construction using one of the
templates here: (1),
(2),
(3).
 Design your own giant mathematical object to build.
