# Exploring Regular Edge-Edge Tessellations of the Cartesian Plane and the Mathematics behind it

- Author:
- Werner Olivier

## Mathematics and Art Activity - Basic Plane Tessellations with GeoGebra

**Goal:**To enable Maths educators to use GeoGebra to understand some of the mathematics that supports the construction of regular plane tessellations.

**Relevant Maths Keywords**

**and Concepts:**Tessellation or Tiling, Euclidean Plane, Regular shape, Irregular shape, Regular Polygon, Interior angles, Exterior angles, Congruent shapes, Regular Tessellation, Irregular Tessellation

**1.**

**Pre-Knowledge for exploring basic plane tessellations:**

**Polygons**are 2-dimensional shapes. They are made of straight lines (edges), and the shape is "closed" (all the lines connect up in vertices).

**Regular Polygons**are polygons that are equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

**You should also know that:**· a whole turn around any point on a surface is 360° · the sum of the internal angles of any triangle = 180° · the sum of the internal angles of any quadrilateral = 360° · the sum of the external angles of any polygon=360° (one whole turn) · the sum of the interior angles of a n-sided regular polygon = (n -2) × 180° · how to calculate or measure the interior angles of regular polygons

## Internal and External Angles of a Regular Polygon

**Example:**The sum of the internal angles of a regular Hexagon (n=6) is (n - 2) × 180°=(6 - 2) × 180°=720°.Hence the internal angles of a regular Hexagon is 720°/6= 120°.

## Symmetry

**isometry**is an action (movement) in the plane that preserves size and shape. There are three basic types of isometries that present symmetry of a figure in a plane.

**Types of Symmetry:**(a)

**Reﬂectional**

**symmetry.**An object has reﬂectional symmetry if you can reﬂect it in a way such that the resulting image coincides with the original. Hold a mirror up to it, its reﬂection looks identical. (b)

**Rotational symmetry.**An object has rotational symmetry if it can be rotated about a point in such a way that its rotated image coincides with the original ﬁgure before turning a full 360°. (c)

**Translational symmetry.**An object has translational symmetry if you move it along a straight path without turning it to produce the same image.

## Defining a Tessellation and a Regular Tessellation of a plane

**tessellation**can be defined as the covering of a plane with a repeating unit consisting of one or more shapes (regular or irregular) in such a way that: • there are no open spaces between and no overlapping of the shapes that are used;• the covering process has the potential to continue indefinitely (for a surface of infinite dimensions– Cartesian Plane).

**Regular Tessellations of the Plane**Tessellations in which one regular polygon is used repeatedly are called

**regular tessellations.**

**Two key questions to consider**– Which regular polygons will tessellate (or tile) the plane and why?How many different tessellations are possible in each case?

## 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.

**Only Three Regular Edge-Edge Plane Tessellations Exist**

## Equilateral Triangle 3.3.3.3.3.3 Tiling

## Square 4.4.4.4 Tiling

## Hexagon 6.6.6 Tiling

**No other regular polygon will tile the plane as their inner angles are not a factor of 360°.**See for instance a Pentagon: