Regular Spherical Tessellations Exploration

Objective: Discover the five regular tessellations of the sphere.

Regular tessellations by triangles

1. Let's build a regular tessellation of the sphere by demanding that 4 equilateral triangles meet at each vertex.
1. What corner angles will each triangle have?
2. What defect will each triangle have?
3. What fraction of the sphere will each triangle cover?
4. How many such triangles will we need to cover the sphere?
2. Draw on a ball this regular tessellation of the sphere.
3. Answer questions A-D assuming three equilateral triangles meet at a vertex.
4. What are other possibilities for number of triangles meeting at a vertex? Do these give spherical tessellations?
5. Use Kaleidotile to view the regular tessellations you found in this section. What are the names of the flat versions?

Regular tessellations by polygons

6. Suppose a sphere is tessellated with regular quadrilaterals (four equal sides, four equal angles) so that three quadrilaterals meet at each vertex.
1. What corner angles will each quadrilateral have?
2. What defect will each quadrilateral have?
3. What fraction of the sphere will each quadrilateral cover?
4. How many such quadrilaterals will we need to cover the sphere?
7. Can four regular quadrilaterals on a sphere fit together around one vertex? More than four?
8. Can you tessellate a sphere with regular pentagons? Decide how many fit around a vertex and then answer A-D.
9. Can you tessellate a sphere with regular hexagons? With regular 7-gons?
10. Use Kaleidotile to view the regular tessellations you found in this section. What are the names of the flat versions?