# Regular Spherical Tessellations Exploration

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**Objective:**
Discover the five regular tessellations of the sphere.

### Regular tessellations by triangles

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

### Regular tessellations by polygons

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