[b]Goal:  [/b]To enable Maths educators to use GeoGebra to understand some of the mathematics that supports the construction of regular plane tessellations. [br][b]Relevant Maths Keywords [/b][b]and Concepts: [/b]Tessellation or Tiling, Euclidean Plane, Regular shape,  Irregular shape, Regular Polygon, Interior angles, Exterior angles, Congruent shapes, Regular Tessellation, Irregular Tessellation[br][br][b]1.   [/b][b]Pre-Knowledge for exploring basic plane tessellations:[/b][b]Polygons[/b] are 2-dimensional shapes. They are made of straight lines (edges), and the shape is "closed" (all the lines connect up in vertices).[b]Regular Polygons[/b] are polygons that are equiangular (all angles are equal in measure) and equilateral (all sides have the same length).[b]You should also know that:[br][/b]·       a whole turn around any point on a surface is 360°[br]·  the sum of the internal angles of any triangle = 180°[br]·      the sum of the internal angles of any quadrilateral = 360°[br]·      the sum of the external angles of any polygon=360° (one whole turn)[br]·  the sum of the interior angles of a n-sided regular polygon = (n -2) × 180°[br]·      how to calculate or measure the interior angles of regular polygons

[b]Example:[/b]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 is the property that a figure coincides with itself under an isometry, where an [b]isometry [/b]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.[br][b]Types of Symmetry:[br][/b](a) [b]Reflectional [/b][b]symmetry.  [/b]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. [br](b) [b]Rotational symmetry. [/b]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°.[br](c) [b]Translational symmetry. [/b]An object has translational symmetry if you move it along a straight path without turning it to produce the same image.

A [b][size=100]tessellation[/size][/b] 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). [b]   [br] [br][/b][b] Regular Tessellations of the Plane [br][/b]Tessellations in which one regular polygon is used repeatedly are called [b]regular tessellations.[br][br][/b][b]Two key questions to consider[/b] – Which regular polygons will tessellate (or tile) the plane and why?How many different tessellations are possible in each case?

[justify]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[br]non-regular tessellation (or tiling)  of the plane.[/justify][justify]A [b]vertex [/b]is a common point where sides (edges) of polygons meet.  The [b]configuration [/b][b]of a vertex[/b] is the sequence of polygon orders that exist around it. Normally these orders are given in a sequence starting with the lowest order.[br]The [b]vertex configuration [/b]of each vertex in the tiling shown in Figure 1. is [b]3.3.4.3.4[/b] as each vertex is surrounded by two equilateral triangles, a square, another equilateral triangle and finally  a square.[/justify][justify]Clearly the vertex configuration of each vertex of a regular tessellations of the plane will be identical. [/justify]

[size=150][b]Only Three Regular Edge-Edge Plane Tessellations Exist[/b][/size]

[size=150][size=200][b]No other regular polygon will tile the plane as their inner angles are not a factor of 360°.[br][br][/b][/size][/size]See for instance a Pentagon: