This then leads on to longitudinal coherence as this basic idea can then lead to using this knowledge in order to explain more complicated concepts of area. Figure 9 shows an example of irregular tessellation, in which all the polygons are regular and congruent. The basic idea within tessellations is the shapes that are used and how the fit with one another as well as the reasons why. Irregular tessellations are those that are formed by irregular polygons, or by regular polygons but that do not meet the criterion that a node is a vertex of at least three polygons. Tessellations also relate to Liping Ma’s beliefs on profound understanding of fundamental mathematics. The square is used as it highlights the four corners of the Earth while the hexagon that is used represents heaven. A triangle is used a simple regular shape that represents harmony and a hum consciousness. A star is also often used in the Islam religion and can differ between a 6, 8, 10 or 12 point star. The most common shapes that are used for tessellations within this religion is an equilateral triangle, a square and a hexagon. Eschers Circle Limit III depicts a tessellation of the hyperbolic plane using the Poincaré disk model with congruent fish-like shapes. Tessellations can be seen within the Islamic religion. Regular shapes can create a tessellation as when all of the vertices touch one another then it all adds up to the sum of three hundred and sixty degrees. For example, if you rotate a circle then it is still a circle making it congruent. All of these shapes are congruent which means that they are all the same size and even if you were to rotate a shape then it would still be the same. The difference between both of these is that with irregular shapes you have to rotate them in order to make them fit together and this is not necessary for regular shapes. The regular shapes include squares, hexagons and equilateral triangles whereas other triangles and quadrilaterals are irregular shapes. There are two different types of shapes that can occur within a tessellation, regular and irregular. When all of the shapes are the exact same then a tessellation occurs. By being identical I mean that all the shapes should be the same and the same size. The most important thing about these shapes is that there should be no spaces or gaps within the pattern and that they are identical. Tessellation is the arrangement of one or more identical shapes that fit together perfectly to create a pattern. Simple tessellation patterns have a basic design using a geometric shape like a square or triangle they can also be more complex using irregular shapes. Like playing football? Well that ball is an example of tessellations using hexagons and pentagons. Like chocolate? Who doesn’t?! Well that also shows tessellation as well as the skin on a pineapple. Inside your kitchen? The tiles will show examples of tessellation as will the floor if it is wooden. Look at the tiles in your bathroom, they will all be the same. Believe it or not, tessellation is something surrounding us in our daily lives.
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