Download Kajol Parwani December 13, 2012 Math Investigation #6: Regular

Kajol Parwani December 13, 2012 Math Investigation #6: Regular Polygons and Tessellations This mathematical investigation is all about polygons and tessellation. This is related to our current unit of study, which is geometry. From looking at a variety of polygons, there are reasons why some tessellate and why some don’t. Tessellation is created when polygons are repeated over and over again covering a plane without any overlaps or gaps between the shapes. After figuring out why some polygons tessellate and why some polygons can’t tessellate, we have to try and figure out how this can be applied in real life situations. There is a reason why a regular hexagon can tessellate. First of all, take a look at what a hexagon tessellation would look like: In a hexagon, there are six sides and six angles in total. A hexagon with all equal sides and angles can tessellate. This is because this polygon’s angles add up to 720°. In order to figure out what one of those angles would be, I would need to divide 720 by 6. 720 ÷ 6 = 120 From this calculation, I know that in a regular hexagon, each of the interior angles is 120°. So this question is, why do these regular hexagons tessellate? When these hexagons tessellate, angles come together at the vertex (point at which adjacent sides of polygons meet), and the sum of the angles need to add up 360°. The reason for this is because it forms a reflex angle, so this is why they need to add up to 360°. From this, I know that in order for a regular hexagon (or for any polygon for that matter), to tessellate, the interior angles must be a divisor of 360°, otherwise they won’t tessellate. Now that I know why a regular hexagon can tessellate, I need to figure out why some other polygons, such as pentagons, can’t tessellate. Since a regular hexagon can tessellate because the interior angles are divisors of 360°, pentagons cannot tessellate because their interior angles are not divisors of 360°. A pentagon is a five-­‐sided polygon, and the sum of the interior angles is 540°. In order to figure out what one of the interior angles are, I would need to calculate: 540 ÷ 5 = 108 From this calculation, I can deduce that a single interior angle in a regular pentagon is 108°. This proves that regular pentagons cannot tessellate because 108 is not a divisor of 360. If somebody were to try and tessellate pentagons, they wouldn’t be able to because there would either be a gap, or it will overlap. Here is what it would look like: The diagram above shows what it would look like if somebody tried to tessellate pentagons. When they meet at the vertex, the angles that come together form a reflex angle (which is 360°). Since 108 + 108 + 108 = 324, the gap would have to be 360-­‐324, which is 36°. This shows that since the gap is 36°, and not 108°, regular pentagons cannot tessellate. There are a couple other polygons that can tessellate as well, besides hexagons. In fact, any polygon with more than six sides are not able to tessellate. Polygons with a sides of three, four, and six are able to tessellate. Let’s say x represents the number of sides of a regular polygons meeting at the vertices of the tessellation: If x = 2, interior angles of regular polygon are 360 ÷ 2 = 180 (no such regular polygon) If x = 3, interior angles of regular polygon are 360 ÷ 3 = 120 (regular hexagon) If x = 4, interior angles of regular polygon are 360 ÷ 4 = 90 (square) If x = 5, interior angles of regular polygon are 360 ÷ 5 = 72 (no such regular polygon) If x = 6, interior angles of regular polygon are 360 ÷ 6 = 60 (equilateral triangle) If x>6, interior angles of regular polygon are less than 60 (no such regular polygon) This shows that the only possibilities for polygons that can tessellate are the ones that have 3, 4, or 6 sides. One reason why an equilateral triangle will be able to tessellate is because a hexagon is made up of at least 6 equilateral triangles. So this proves that if a hexagon can tessellate, so can an equilateral triangle, and it can be linked back to why polygons tessellate. These six triangles can continuously make hexagons, so this will be a repeatable pattern, which results in a tessellation. Here is a table showing the same information: Regular Interior Angle Size 360° ÷ Interior Angle Tessellates? Polygon (°) Triangle 60 360 ÷ 60 = 6 Yes Square 90 360 ÷ 90 = 4 Yes Pentagon 108 360 ÷ 108 = 3.3 No Hexagon 120 360 ÷ 120 = 3 Yes Heptagon 128.57 360 ÷ 128.57 = 2.8 No Octagon 135 360 ÷ 135 =2.7 No Nonagon 140 360 ÷ 140 = 2.6 No Decagon 144 360 ÷ 144 = 2.5 No From this, I can deduce that some other regular polygons that will tessellate are equilateral triangles, regular squares, and regular hexagons, since the values 6, 4, and 3 are integers, whereas all of the other numbers include decimal values, so those can’t tessellate. Here is an image showing the interior angles of squares, equilateral triangles, and hexagons, and how they tessellate: Again, squares, equilateral triangles, and hexagons all can tessellate because an interior angle from each of them is a divisor of 360. When they meet at the vertex, it equals up to 360°). Now that we have covered the different types of regular polygons that will tessellate and the basis of why some regular polygons tessellate and why some don’t, it’s time to think outside of the box. Where can this concept of tessellation be seen outside of mathematics (in the real world)? There are a couple of examples in real life showing tessellation. The first real life application is in tiling. Tiling can be found in buildings, bathroom, floors, and walls. The most important thing to consider in tiling is that the shape of the tiles should cover the floor or walls without any gaps or overlaps. Most of the Norman Foster Buildings have tessellation either on their windows, walls, or bathroom floors. Norman Foster Builds are an architectural phenomenon as the founder of some of the most financially successful pieces of architecture. Here is an example of a Norman Foster Building in Denmark, showing tessellation (elephant house): This tessellation is a square tessellation. Tessellation is an advantage in architecture because the closer fitted the objects are together, the less waste material is produced. Since tessellations are shapes that are perfectly fitted, no waste material would need to be produced if the template includes a tessellation. This way, the architects are saving money, and using the concept of tessellations is advantageous for them. Another advantage for the architects while using tessellation is that even the simplest repeating pattern is aesthetically appealing. For many years, tessellation has been used in Islamic, Indian, and Istanbul architecture. Here is an example of tessellating Islamic structure: The designs in these countries are delicate and beautiful, and they are two-­‐
dimensional. Three-­‐dimensional is much more intricate and complicated, but once they are completed they are generally elegant and eye-­‐catching. Three-­‐
dimensional is a different situation. A good example would be a soccer ball. Since the surfaces aren’t flat, it is three-­‐dimensional. Soccer balls consist of pentagons and hexagons, so this creates a semi-­‐regular tessellation. From this real life application, I can conclude that tessellation is can be found in not only mathematics, but it can be found anywhere in the world if we just look around. Here is an image showing an example of tessellation in architecture: This image was taken in Louvre, Paris and it projects a two dimensional tessellation. It is two-­‐dimensional because the surfaces are flat. A rhombus is used as the two-­‐dimensional shape. This can tessellate because these rhombuses are squares and when all the angles meet at the vertex, the angle sum is 360°. Another real life application that can be used to express the concept of tessellation is through nature. A beehive (or honeycomb is a classic example of tessellation because it involves regular hexagonal shapes and it is a habitat for bees. This proves that tessellation doesn’t necessarily occur from man-­‐made objects, but it also occurs in nature. Another way that tessellation can be applied into the real world (through nature), is by the shells of turtles. Turtles have a protective covering, which is the shell, and these shells have a tessellation as well. It is a hexagonal tessellation, just like a honeycomb. I can tell that a honeycomb and a turtleback shell can tessellate because there is a consistent pattern of repeating hexagons. These hexagons can continue to tessellate because there are no overlapping hexagons, and there are no gaps between these hexagons. Here are images showing examples of tessellation in nature: The first picture projects a honeycomb. A honeycomb is a structure of hexagonal cells of wax, made by bees in order to store their honey and eggs. It’s noticeable through the image that these hexagons tessellate, so this is one real life application. The second picture shows the shell of a turtle, which also displays hexagonal tessellation. Even though it’s nature and everything is done naturally, there is a reason why bees produce hexagons in their honeycombs. First of all, bees want to enclose the largest possible area with the least amount of wax, so that they can save this. This real life application can also connect to the architecture real life application because tessellation is used to save materials, such as metal or wax. A circle for a honeycomb could work as well, but bees don’t use this because hexagons are the shape with the most sides that can tessellate. There are no other shapes more than six sides that can be next to each other without any spaces or overlapping occurring. The tessellation can sometimes be inaccurate because the wax might not be perfectly placed inside the honeycomb, and this can cause a change in the angles of the hexagons, so honeycombs are not necessarily accurate. Bees are a very interesting species, because a mathematician named Pappus found out that bees are like nature’s mathematicians. Each of the six hexagons is exactly the same width and they meet at an angle of precisely 120 degrees, which produces a regular hexagon. The efficient pattern for the storage of honey is remarkable to the wonders of nature and to mathematics as well. Tessellation can even be incorporated in art. In fact, tessellation has been used in art for hundreds of years now. A man named M.C. Escher was considered to be the father of modern tessellations because of the tessellation patterns he produced in his art. In 1925, Escher had produced his first art piece which included the concept of tessellation. It was a block print of lions which interlocked and covered the plane without any overlapping or gaps. He had thought that it was beautiful to have tessellation in his art, so after he created his lion block print, he produced tessellations of camels, squirrels, and birds. Escher had an obsession of tessellation and by the time he died, he produced 137. Escher had no idea that this tessellation had mathematics behind it, but his work was a great help to many other mathematicians and scientists. If it weren’t for Escher, not many of the tessellation paintings now would be there because his work of incorporating tessellations into art really influenced many artists. Sometimes, Escher had used tessellations in art to express his feelings, and this is still going on with art. Using tessellation in art displays great realism, while at the same time it shows the impossible perspective, and the art can also trick the eye. These patterns are used in art a lot because it is considered to be aesthetically appealing, and tessellation in art can be used to express the artist’s feelings and emotions. Here is an image showing an example of tessellation in art: This image on the left shows Escher’s lion piece which displays tessellation because there are no gaps or overlapping lions. Even though polygons were not used in his art, these lions are still a form of tessellation. It is still used today in art because it is physically appealing and very eye-­‐catching as well. There are so many ways that tessellation is shown in our real world, it’s just sometimes we don’t notice it. In the pineapples we eat, or the fish scales we see, or playing with a soccer ball, all of these include examples of tessellation. The ones I have explained are some basic examples where tessellation is incorporated in tiling’s, honeycombs, turtle shells, and finally in art. Tessellation can be accurate or inaccurate based on where it is found. If the shapes are made by something digital, such as a computer, then it is accurate because all of the sides and angles will be the same to one another. However, in real life this is not always the case. For example, tessellation can be inaccurate in tiling. There can be times when a tile chips off, this causing there to be a gap and the angle changes as well. When the angle changes, the shapes don’t tessellate. For example, if my bathroom floor consists of a tessellation of tiling and one of the tiles chips off, the interior angles of the tiles will change, and this will result in the vertex not equaling up to 360°. Once this happens, these specific tiles are not tessellating, so this is one way tessellation can be inaccurate in real life. Another way tessellation can be inaccurate is in origami. Accuracy is a vital importance when folding the patterns. Since the majority of the creases are being created in the tessellation as part of the grid, it’s important for the lines to match up with each other and for the pleats to be of equal width. It is very rare to have a case where a person folds perfectly along the lines, because usually people are a bit off at first. The closer the lines are to each other, the more accurate the tessellation is in the origami. However, it is very unlikely to have an origami piece that has a 100% accurate tessellation because this would mean that the lines would need to be perfectly folded along the lines. Here is an image showing equilateral triangles tessellating in origami (therefore creating hexagons): This is another way tessellation can be inaccurate. My results for this mathematical investigation are accurate because I found them by using rules and equations of geometry. From these rules, I could easily figure out whether or not some types of polygons tessellate. Another reason why my results are accurate is because of logic. By looking at my answers I know that there aren’t any outliers or anything outrageous said through my answers. Also, I already know that equilateral triangles, squares and hexagons can tessellate because in my own house I have two windows with triangles and squares tessellating, and I have also seen hexagonal tessellation before by looking at turtle shells. In addition, when I did the calculations to figure out which polygons tessellated and which didn’t, I made sure to double check my answers by re-­‐
calculating them again. Overall, my results make sense because I have used geometric rules and theorems, and I have also checked my calculations, so this proves that my results are accurate. There are a couple of ways of improvements and expanding this tessellation investigation. The first way this task can be expanded is by making an actual model. This whole investigation was done digitally on the computer, but by making a model, it can give a clearer understanding of tessellation because it will be a hands-­‐on experience. Cutting six equilateral triangles, squares, pentagons, hexagons, heptagons, and octagons will be the first step of creating the model. Now that we already know that only regular triangles, squares, and hexagons can tessellate, we can actually experiment with these polygons by moving them around. Also, we know that any polygon that has over six sides cannot tessellate because the interior angles get larger as the number of sides in a polygon increases, so the hexagon is the largest polygon (side-­‐wise) that can tessellate. After cutting out these various polygons, they can be arranged in a way to show the tessellation. Again, this tessellation probably won’t be perfectly accurate because one shape might be thicker than the other due to some inaccurate cutting. The last improvement that can be made for expanding this task is by exploring more on different types of tessellations with semi-­‐regular polygons. This investigation was focused on regular polygons, which means that all of the angles and the sides are congruent. Two or more types of regular polygons form semi-­‐
regular polygons, each with the same length of sides. Also, each vertex has the same pattern of polygons around it. Regular polygons are easy to work with because it only consists of one polygon tessellating, but semi-­‐regular polygons consists of two or more polygons tessellating. Now that we know what are semi-­‐
regular polygons, this task could be expanded by also creating a model of two types of semi-­‐regular polygons tessellating, and comparing these models to the previous models of regular polygons, and seeing how they are similar to each other and how they differ. In conclusion, tessellation surrounds us everywhere we go whether we see it though windows, or art, or even in nature. Tessellation is most likely to be even the slightest bit inaccurate if it is formed through nature, but on the computer, it is accurate because the sides and angles are congruent. Even if bees don’t know it yet, the structure of their habitat is based on mathematics. References Lion tessellation by M.C. Escher [Photograph]. (n.d.). Retrieved from http://www.tessellations.org/tesspics/DONT-­‐STEAL-­‐BANDWIDTH-­‐NOR-­‐ THIS-­‐COPYRIGHTED-­‐ART-­‐lions.jpg [Regular polygon tessellation] [Photograph]. (n.d.). Retrieved from http://www. coolmath4kids.com/tesspag1.html# [Tessellation in architecture] [Photograph]. (n.d.). Retrieved from http://www2. gvsu.edu/oxfordj/angie.html [Tessellation in honeycomb] [Photograph]. (n.d.). Retrieved from http://www. gscdn.org/library/cms/35/4335.jpg [Tessellation in turtle shell] [Photograph]. (n.d.). Retrieved from http://flextiles. files.wordpress.com/2011/11/tortoise-­‐shell.jpg