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!

TRANSFORMATIONS!

Transformations in Geometry are changes in the position of a shape on a coordinate plane. What that really means is that a shape is moving from one place to another. There are four basic movements: translations, rotations, reflections and glide-reflections.
Translations or slides are a motion in which all points are moved the same distance and in the same directions.
                      

Rotations or turns, are where one point stays the same and the remaining points are turned around the center of rotation.
                             

Reflections, flips, or mirror reflections are determined by a line in the plane called the "line of reflection" or "mirror line" and is described as a transformation in which the figure is the mirror image of the other.

                                     

Glide-reflections combine both a slide and a reflection. This is when a translation and a reflection are performed one after the other.
                  

You can practice these transformations by using a gridded piece of paper. Here is a video that uses music to help remember each of the transformations and what they are.(http://youtu.be/hFajxpVeobY).  Practicing is the best way to truly understand how to perform each task. Have fun and see you next week!

Shapes and Figures.... Both 2D and 3D

In this week’s post I am going to be talking about shapes and figures and what it takes to be considered a shape or figure.

                Do you know the difference between a polygon and a polyhedron? To be perfectly honest it is fairly simple. The formal definitions of each term are as followed. A polygon is a plane figure with at least three straight sides and angles, and typically five or more. Here are some examples:

                                          

                                         

A polyhedron is a solid figure with many plane faces, typically more than six. Another way of explaining them would be that a polygon is flat plane or figure (it is 2- dimensional) and a polyhedron is figure with many planes (usually made up of many polygons and is a 3- dimensional).  Here are some examples of some polyhedrons:

                                                 

Within these figures there are many more shapes that are different from one another. Some examples are pyramids, prisms, cones and cylinders:

               

       Pyramid                      Prism                    Cone                 Cylinder

Now, we remember from last week’s post that “right” means creating a 90° angle. This applies with polyhedrons as well. If you have a right pyramid, cone or cylinder that means that a straight line from the apex or center of the figure down to the base creates a 90° angle and if you have a right prism then all faces have to be rectangles. The word “oblique” means it’s neither parallel nor at a right angle to an implied line, basically meaning slanted. If you have an oblique pyramid, cone or cylinder than a straight line from the apex or center of the figure is slanted creating an angle other than 90° and an oblique prism is when the faces are not perpendicular to the plane of the base.  All of the examples above are right polygons, now here are some examples of oblique polygons:

                                     

Oblique Hexagonal   Oblique Pentagonal        Oblique                       Oblique

       Pyramid                    Prism                       Cone                         Cylinder

The reason some of the figures above, such as the pyramid and the prism, have words like hexagonal and pentagonal in their name is because the bases are a specific shape. You can see that the base of the pyramid has six sides which makes it a hexagon so adding “hexagonal” to the name helps add more detail to understanding what the shape looks like.