How do you read/make sense of the sphere diagrams shown here: http://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups
What do the yellow shaded areas represent? What are the red triangle/arrows? Why are the curves drawn like so?
Can you explain how to read one for - say - a platonic solid like the dodecahedron?
The yellow areas are fundamental domains.
The red curves taken together form a collection of great circles which cuts up the sphere into a collection of different fundamental domains which, taken together, form a uniform tiling of the sphere.
Added For example, the dodecahedron has 12 pentagonal faces, each of which can be subdivided into 10 right triangle fundamental domains, by slicing the pentagon up using 10 segments that connect the center of the pentagon to each of the five vertices and to the midpoints of each of the five edges. Each of those 10 triangles has a right angle at the corner where it meets a side midpoint, and an angle of $\frac{2 \pi}{10}$ at the corner where it meets the midpoint of the pentagon. Altogether there are $10 \cdot 12 = 120$ of these fundamental domains which fit together to form a uniform tiling.
Then, rather than viewing these 120 triangles on the dodecahedron itself, project them outwards from the center of the dodecahedron to a sphere in which the dodecahedron is inscribed. Then you do this, each of those 120 triangles will still have a right angle (where 4 of them meet), and still have an angle of $\frac{2\pi}{10}$ (where 10 of them meet), as well as an angle of $\frac{2\pi}{6}$ (where $6$ of them meet, at the point on the sphere which is the projection of a vertex of the dodecahedron).