how may polygon arrangments does a semi-regular tessellation have?
a) exactly three, b) exactly two or three, c) exactly four, d) exactly two of four
To determine the number of polygon arrangements in a semi-regular tessellation, we need to understand what a semi-regular tessellation is.
A semi-regular tessellation is a pattern formed by regular polygons, where each vertex has the same arrangement of polygons around it. In other words, at each vertex, the same combination of regular polygons meet.
In a semi-regular tessellation, there are two types of vertices:
1. Regular Vertex: These vertices have the same arrangement of polygons around them. For example, at a regular vertex, you might have three squares meeting at each vertex.
2. Semi-Regular Vertex: These vertices have two different polygons meeting at each vertex. For example, at a semi-regular vertex, you might have a triangle and a hexagon meeting at each vertex.
Now, to answer your question:
- Option (a) states that a semi-regular tessellation has exactly three polygon arrangements.
- Option (b) states that a semi-regular tessellation has exactly two or three polygon arrangements.
- Option (c) states that a semi-regular tessellation has exactly four polygon arrangements.
- Option (d) states that a semi-regular tessellation has exactly two of four polygon arrangements.
The correct answer is option (d), exactly two of four.
To understand why, let's break down the possibilities:
1. At a regular vertex, there can be three polygons meeting. The polygons can be arranged as triangle-triangle-triangle, square-square-square, or hexagon-hexagon-hexagon. So, there are three polygon arrangements for regular vertices.
2. At a semi-regular vertex, there can be two polygons meeting. The combinations can be triangle-hexagon or square-octagon. So, there are two polygon arrangements for semi-regular vertices.
Taking both regular and semi-regular vertices into account, we have four possible polygon arrangements in a semi-regular tessellation: triangle-triangle-triangle, square-square-square, hexagon-hexagon-hexagon, and square-octagon.
Thus, the correct answer is option (d), exactly two of four.