TILING! BRILLIANT!

We have all come across some type of tiling in one way or another. We know about tiled floors, tiled showers, pool floors and mosaics!

In Geometry however tiling is more about placing different types of polygons together to make a pattern. There are three different types of tilings in Geometry. First is regular tiling which is using one type regular polygon, placing them evenly together to make 360° at the vertex. Now remember that regular means that all sides and angles are the same or congruent. There are only three polygons that can make regular tilings and they are a square, an equiangular triangle or a hexagon.

Here are some examples of regular tilings made by each of the polygons above.

The next type of tiling is semi-regular tiling which is using more than one type of regular polygon, placing them evenly together to make 360° at the vertex. Now you can use the same shapes as above but you can also use any other type of a regular polygon like a pentagon, octagon or decagon.

Here are some examples of semi-regular tilings that include some of these polygons.

You can see that every shape is some type of regular polygon but there is just more than one type of regular polygon that helps make the pattern.

The final type of tiling is called irregular tiling which is tiling a plane with non-regular polygons. Non-regular polygons do not have congruent angles or side lengths. Here are some examples of non-regular polygons.

When you put these types of polygons together you can make many types of irregular tilings. Here are some different types of examples that include both regular and irregular polygons.

Tilings can also be known as tesselations, which is the formal wording. There really are a ton of different tilings you can make and I would encourage to try some for yourself. Here are a couple that I made through the program Paint.

And remember, you can use your knowledge of tesselations in everyday life like if you ever want to tile your kitchen floor or your shower walls. You need to know the equation used to find if a certain polygon pattern will fit. Say you want to created a pattern using just regular hexagons. You plug your information into the equation (n-2) x 180. The letter n represents the number of side on the polygon and we know that a hexagon has six sides. Once you receive the answer to (6-2) x 180, you take the answer and divide it by n (in our case "6"). If the answer is divisible by 360° then you know that you can use that polygon to create a regular tessellation. If it is not divisble by 360° then you cannot create a regular tessellation but you can figure out what the reamining degree is and put a shape in that will fit properly.
                      Hexagon:                                                           Pentagon:
                     (6-2) x 180                                                        (5-2) x 180
          4 x 180 = 720 / 6 = 120°                                     3 x 180 = 540 / 5 = 108°
                    360 / 120 = 3                                             108 + 108 + 108 = 324°
                                                                                             360 - 324 = 36°
We now know that if you want to create a tessellation that uses regular pentagons we can only fit three pentagons together so we would need a shape that has an angle of 36° to make it semi-regular. Here is an example:

Good luck to you and your work with tessellations!