Archimedean Exploration

Objective: Understand the role of the requirement that the vertex configuration be the same at every vertex.

1. Note that the requirement that all vertices are of the same type is rather important. If we consider (3,6,3,6) in combination with (3,3,6,6) for instance, then not requiring the vertices to all be of the same type would allow us to construct infinitely many tessellations using just triangles and hexagons.

The easiest way to see this is to think of the semi-regular (3,6,3,6) tessellation as being made up of horizontal strips.

The horizontal strips can then be stacked in many different ways.

(A)

(B)

Which one is the semi-regular tessellation? Why is the other one not semi-regular?

2. Why does this imply that there are infinitely many different tessellations using triangles and hexagons if we allow multiple vertex types to appear? Explain carefully. Draw some examples to illustrate your argument.

3. This tessellation fails to be semi-regular for two different reasons. What are they?

Handin: A sheet with answers to all questions.