Regular polygons and tessellations Criterion D: Reflection

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1 Regular polygons and tessellations Criterion D: Reflection Aika Kim 12/13/2012 Math 9+ Assignment task: Our task in this project is to identify what a tessellation is, and what shapes do and don t. But most importantly, we must connect tessellation to real life implications, reflect on where tessellation might be used, and think more than just regular polygons. 1) Why does a regular hexagon tessellate? A hexagon has 6 sides, which means that 4 triangles can be made within the shape. If there are 4 triangles, and each have a total of 180, then it would have a total angle of 720 (180 6=720). The equation to calculate what the angle would be for each side of the hexagon would be because there is a total of 720 degrees, and there are 6 sides in a hexagon. Therefore, each side should have a total angle of 120 degrees. As it shows in the diagram, the hexagons are tessellated by 3 sides. This is because 120 is a factor of 360, and 120 3=360. There are no left over angles, and it creates a whole circle (in the points of each they meet center part o the 3 shapes tessellated). If the angles on the sides are a factor of 360, then it tessellates. 2) Use the diagram on the right to explain why a regular pentagon will not tessellate A pentagon can create 3 triangles. If one triangle has a total angle of 180, then would be 540. And since a pentagon has 5 sides, each side should have an angle of 108 (540 5=108). A regular pentagon does not tessellate because 108 is not a factor of 360. For a shape to tessellate, all the angles on the sides must equal to 360 when the sides are put together. It needs to equal to 360 because if the sides are all put together to tessellate, then the angles should form a full circle. If they all connect, and if all the

2 angles of the sides add up to 360, then it would tessellate. For a pentagon, it doesn t because 108 is not a sum of 360. If you combine 3 pentagons, then it would add up to 324 degrees, leaving 36 degrees an open space. 3) Which other regular polygons will tessellate? Justify your answer Shape Sides Sum of interior angles Shape Each angle (sides) Triangle Quadrilateral Pentagon Hexagon Heptagon/Septagon

3 Octagon The only shapes that tessellate triangle, quadrilateral, and hexagon. This is because each side has an angle is a factor of ) Write a reflection, addressing areas such as real life applications, accuracy and improvements. Many things around us tessellate, but we never acknowledge it. Most things that are built need to tessellate in order to stay in place or balance. For example, a brick we see on roofs of houses are tessellating with usually rectangles. Kitchen walls also use tiling pattern. It uses glass/clay material because it has the best resistance in steam and heat, and it is also easy to take care of. Tessellation is very important because it is first of all neat, but also, if things weren t tiled and all put together, then it would create wholes and spaces between which can cause trouble. What if the roof didn t tessellate? If the bricks weren t all tiled and lined up in order, then there would be a whole in the roof. Architecture has a lot to do with tessellation, and people are usually used to looking at rectangles tessellating. Many of the buildings that we see have a tessellating pattern, and the designs and shapes used are tessellated as well. Tessellation has a design but it is used also because spaces can be reduced from it. By tessellating a shape (if it is possible), there is no left over space. For situations like rooftops, you wouldn t want left over space. Therefore, tessellation is very important in architecture.

4 An obvious tessellation that we see and are all familiar with is beehives. It is famous for being a tessellation of a hexagon. Then why are they shaped like that? According to the research I did, having cells in a hexagonal pattern is the most efficient use of space, therefore, giving the highest number of cells for a given area using the least amount of wax. Hexagon has 6 sides, meaning that 6 other hexagons can be attached to tessellate. Bees need a lot of cells because of its huge amount of eggs. Therefore, it adapted to be able to build the most efficient beehives. Another example that everyone is familiar with is a soccer ball. Soccer balls that we know are made out of regular polygons. This is because this shape is able to close. What I mean by this is that it can make a shape that doesn t have a corner/end. Off course, if you try, that is possible, but not all shapes can do so especially in a ball. All the examples that I gave previously are all made out of the same shapes. But do they always need to be so? Does a tessellated shape have to be built up by the same shapes only? They do not. A tessellation can be seen in quilts and origami as well, which are made up of different kinds of shapes and sizes. Quilt is a well known example of a tessellation. Although this example is a design tessellated with the same shapes, not all of them have to be. If different shapes match up and form one quilt, then it means it is tessellated. An object is tessellated if different patterns or shapes fill in an area without making any space in- between. If you look at it closely, this is made out of rectangles lined up in different angles, but in the end, forming a point where the 4 shapes add up 360. Therefore, this example is just made out of many different triangles or rectangles, combined into different shapes and sizes to form a pattern/tessellation.

5 Although this is not the greatest example, origami is a type of tessellation as well. However, as I mentioned, it is not made out of the same shapes. When you fold an origami, the shapes that form are all from the sides that are left over. In other words, when you fold a shape, it creates another, and in the end, every fold you do forms a shape, causing the one sheet to form different shapes all forming from one another. This diagram is not something that we are familiar noticing, but the relationship the shapes have is the same relationship tessellating shapes (that we all notice) have. This is made out of quadrilaterals, triangles, and rectangles, but all of these are outlined to form shapes such as hexagons and trapezoids. A regular tessellation is when a tilting pattern is formed. But this example isn t a tiling pattern. Therefore, it might not be a tessellating diagram. However, it has something mutual to tessellating shapes, which is the fact that all the sides and corners that meet with one another all add up to 360 degrees. However, in real life, we do not see many things that are tessellating. Most of the things that are, are made with a machine or a computer. It needs precise measurement to form an exact tessellation. If you are folding an origami, for example, the folding you are doing may be more precise than measuring in a ruler because you are matching up the corners so that it can bisect. Besides these unusual situations, many things that we draw or make by hand aren t perfect. Therefore, it is hard to form a tessellation using regular shapes in real life (such as beehives, for example). If we extend different angles of shapes so that there wouldn t be any open space (for example, the picture above), then you wouldn t call it a tessellation. It needs to form a pattern (a tiling pattern), that shows that same order can fill one area of space without leaving any open space, and so that all sides of the shape are connected with another.

6 Then why do shapes tessellate? The explanation is very similar to the meaning of tessellation. There are tessellations, and regular tessellations. Tessellations can be made up by different kinds of shape with not really a tile like pattern, as long as they don t overlap, or form a gap. This is an example of a tessellating shape. There isn t a pattern or a specific shape, but all the shapes do not create space nor do they overlap. However, a regular tessellation is different. A regular tessellation must be in a tile pattern that goes on forever without gaps of overlaps, using the same pattern or shape. The shapes must be regular polygons, and the vertex must be the same as well. If you have 2 shapes (for example, a hexagon and a triangle), the triangle has to have the same vertex, and the hexagon must have the same as well. Therefore, the two may be a different size, but each of the same shapes have the same angles, as well as the vertex. All the interior angles of each equilateral is the same 60 degrees. 6 of the sides add up to 360. This diagram shows that one point is met by 6 triangles. If you look at the diagram, 6 of the triangles form a hexagon. Therefore, a hexagon should form a regular tessellation as well: If 6 triangles form a hexagon, one side of the triangle is one side of the hexagon. If one side of the triangle is 60 degrees, in order to form a side in the hexagon, it requires two triangles. This is why one side of a hexagon has an angle of 120, and there are only 3 that is needed to tessellate, and form a 360 combined with the shapes. If you cut the hexagon in half, then one side would have an angle of 60. When 3 (the amount that is needed to form a 360) is multiplied by 2 (since it is divided by half), it would be 6, which is like I mentioned before, the amount needed to tessellate for a triangle. All of the regular polygons that tessellate has a pattern like this, except that the other regular polygon

7 have more sides, which means there would be less amount of shapes needed to tessellate, and more triangles within the shape. Although our focus for this assignment is to find proof and relationships of regular polygons, it could be extended by investigating a pyramid tessellation in three- dimensions. The reason why investigating pyramids in three- dimensions is a little different from looking at two- dimensional polygons, as well as some three- dimensional shape is because a pyramid does not have all the same sides. For a cube, for example, there are six sides of the same shape/square. However, in a pyramid, there is a square base, with 3 triangles, meeting at the top point. Therefore, the tessellation would be a little different from others. So how would this three- dimensional pyramid tessellate? Three- dimensional pyramid does tessellate. This is because it is not much of a different to normal two- dimensional triangles. I wasn t able to find a 3D shape that would help me show it visually, so I would have to explain it: If the two bases of the pyramid went back to back, then it would create a diamond. A diamond could tessellate: A diamond would tessellate because it is the same thing as two triangles put together which is what I did to demonstrate that the pyramid would tessellate. Therefore, three- dimensional pyramid tessellation would essentially look like this:

8 A pyramid would tessellate because all sides are 60 degrees, and as I said before, if a side has 60 degrees, it would tessellate because it is a sum of 360 which is what would match all the points together without leaving any space/gap. Also, a triangular prism would tessellate because it can be thought as an extended version of a triangle. 3 pyramids together (tessellated) would form a triangular prism because all the sides have the same area, as well as the angle, which means that it would tessellate without leaving any space (left over angles). Therefore, a pyramid would tessellate. Then would tessellation work on other regular polygons that are three- dimensional? The answer to this is yes. If it tessellates when it is two- dimensional, then it would for three- dimensional as well. This is because if the angles it has in 2D tessellate, the 3D would be the same kind of relationship (just the matter of how thick it may be, etc.). The angles will be the same, therefore, it will tessellate (if all of the shapes are three- dimensional of a regular polygon). To conclude, not all regular polygons tessellate, and there is a relationship/ difference in angles between shapes that tessellate, and don t. Three- dimensional have a lot in common to two- dimensional, and tessellation are seen in our daily lives as well.