Exploring Regular Edge-Edge Tessellations of the Cartesian Plane and the Mathematics behind it
Mathematics and Art Activity - Basic Plane Tessellations with GeoGebra
![Image](https://beta.geogebra.org/resource/BKQGrFaH/5ClQFC4ZNBRJpKEX/material-BKQGrFaH.png)
Internal and External Angles of a Regular Polygon
Symmetry
Defining a Tessellation and a Regular Tessellation of a plane
Naming convention for Regular Plane Tessellations
Consider the example of an edge-edge plane tessellation in Figure 1. Although all the polygons are regular, there are more than one type of polygon which that are used to tessellate. This makes this a non-regular tessellation (or tiling) of the plane.
A vertex is a common point where sides (edges) of polygons meet. The configuration of a vertex is the sequence of polygon orders that exist around it. Normally these orders are given in a sequence starting with the lowest order. The vertex configuration of each vertex in the tiling shown in Figure 1. is 3.3.4.3.4 as each vertex is surrounded by two equilateral triangles, a square, another equilateral triangle and finally a square.
Clearly the vertex configuration of each vertex of a regular tessellations of the plane will be identical.
Figure 1.
![Figure 1.](https://beta.geogebra.org/resource/uNeWZmUH/eTkxCk6YKd1J8hYQ/material-uNeWZmUH.png)
Equilateral Triangle 3.3.3.3.3.3 Tiling
![Equilateral Triangle 3.3.3.3.3.3 Tiling](https://beta.geogebra.org/resource/rkjDm9SD/S0H141vsQWS8ugVq/material-rkjDm9SD.png)
Square 4.4.4.4 Tiling
![Square 4.4.4.4 Tiling](https://beta.geogebra.org/resource/gGEhD6DU/MYO8JOLYyIkOKwy5/material-gGEhD6DU.png)
Hexagon 6.6.6 Tiling
![Hexagon 6.6.6 Tiling](https://beta.geogebra.org/resource/NmUmsXDH/xi7EMnRfEoYrdFUN/material-NmUmsXDH.png)
Pentagons does not tessellate the plane - open uncovered spaces will always exist.
![Pentagons does not tessellate the plane - open uncovered spaces will always exist.](https://beta.geogebra.org/resource/wER99BFX/xY0RG5P4J7D2rYzX/material-wER99BFX.png)