- Edge-to-edge tilings.
- Can the plane be tiled with equilateral triangles, meeting edge-to-edge?
- What about other regular
*n*-gons?

- Non edge-to-edge tilings.
- Can the plane be tiled by equilateral triangles, where not all of them meet edge-to-edge?
- Where none of them meet edge-to-edge?
- What about the same questions for squares?
- For other polygons?

- Tiling with polyhedra.
- Can space be tiled with regular tetrahedra, meeting face to face?
- What about cubes?
- What about other polyhedra?

- Tilings with congruent tiles.
- For which triangles is it true that you can tile the plane with multiple congruent copies of itself?
- For which quadrilaterals?
- For which pentagons?
- For which hexagons?
- For the above questions, in which cases can this be done so that the same number of polygons meet at each vertex also?

- ``Escher-like'' tilings.
- Modify some of the polygons in the above tilings to get more complicated shapes for which the plane can be tiled with congruent copies.

- A planar cluster is a group of regular polygons (not necessarily all of
the same type) meeting edge-to-edge tightly around a common vertex in
the plane.
- Make a list of all possible clusters.
- How do you know you have them all?
- Which clusters can be extended to a tiling of the plane so that the same cluster (i.e., the same number of polygons of each type, meeting in the same cyclic order) occurs around each vertex? Draw a good picture of each such tiling.
- Determine possible coordinates of the vertices of each of these tilings.

- A space cluster is a group of regular polygons (not necessarily
all of the same type) meeting edge-to-edge around a common vertex, but
such that the sum of the interior angles is less than , so
that the cluster is not planar.
- Make a list of all possible clusters.
- How do you know you have them all?
- Which clusters can be extended to the boundary of a polyhedron so that the same cluster (i.e., the same number of polygons of each type, meeting in the same cyclic order) occurs around each vertex? Draw a good picture and/or make a model of each such polyhedron.

Wed Nov 4 12:13:22 EST 1998