Which shapes make floor tilings?


 Maximilian Greer
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1 Which shapes make floor tilings? Suppose you are trying to tile your bathroom floor. You are allowed to pick only one shape and size of tile. The tile has to be a regular polygon (meaning all the same sides and all the same interior angles, such as a square or an equilateral triangle). Our goal is to figure out which shapes will work. Vertices from one polygon (the corners where edges meet) have to touch vertices from other polygons. For a tiling to work, it means you would be able to cover the floor with only samesize copies of that tile without having gaps. Don t worry about what happens at the walls or with fixtures such as sinks or bathtubs we ll get a tile cutter later. 1. Sketch several regular polygons: an equilateral triangle, a square, a pentagon, a hexagon, an octagon. Can you draw a seven sided regular polygon? Make a guess as to which of these shapes will work to make a tiling. Are there reasons for your guesses? 2. Make cutouts using the separate sheet. There are regular triangles, squares, pentagons, hexagons, and octagons. From your experimentation, which ones work as tiles and which ones don t work? Do you think that tilings might be possible with other regular polygons that you don t have on the next sheet, such as regular heptagons (sevensided polygons) or regular decagons (tensided polygons)? Why or why not? 3. As mentioned before, if you can make a tiling, vertices must touch other vertices at certain points. Around those points, there are a number of angles from each of the polygons sharing that vertex. What is the sum of the degrees in all of those angles? Why is that the sum? 1
2 4. In item 3, you should have found that all the tiles meeting at a point have to have angles adding up to 360. We can use this to see which regular polygons can possibly be used in a tiling pattern. The key is to find out the size of each angle in a regular polygon. We would like to be able to answer questions like this one: what is the size of each angle in a regular heptagon? A regular three sided polygon is also called an equilateral triangle. How large is each angle? Why? A regular 4 sided polygon is a square. How large is each angle of the square? A regular hexagon can be divided into 6 equilateral triangles, as pictured at right. Using the angles in the equilateral triangles, determine the size of the angle marked in the first hexagon. 5. Use the results you have found so far to fill in the table below. Leave the second column blank for 5 and 7 sided polygons. Number of Sides Size of Each Angle (in degrees) Do you see a pattern in the table? Can you guess the angle size for regular pentagons and heptagons? 2
3 6. The dissection method used for the hexagon can be applied to other polygons. Consider the figure at right of a regular octagon, subdivided into triangles. One of the eight identical triangles is shown beside the octagon. Work in your group to determine the three angles of this triangle. Here are two hints: (i) the top angle is one of 8 identical angles that fit together around the center point of the octagon; (ii) the two base angles of the triangle are equal. Find the three angles below, showing all your work and explaining your logic. Based on your answers, how large are the angles in the original octagon? Why? 7. Repeat the steps of item 6 to figure out the size of the angles in a regular pentagon, as pictured at right. 8. Based on the prior two exercises, can you find a formula for the size of each angle in a regular polygon with n sides? If so, do that now, and then skip step 9. [Hints: Imagine that the polygon is divided into n triangles. One angle of the triangle is at the center of the polygon; how big is that angle? How big are the other two angles in the triangle?]. If you are not sure how to do this step, go on to step 9. 3
4 9. Repeat the steps of item 6 to figure out the size of the angles in a regular pentagon, as pictured at right. Then try step 8 again. 10. The correct answer to step 8 is given as follows: The size of each angle in a regular polygon with n sides is 180. Verify that this gives the correct answers for the polygons you have already considered: triangle, square, hexagon, octagon, pentagon. 360 n 4
5 11. Now that you know how to determine the size of angles in regular polygons, decide which polygons can tile a floor. Your goal is a complete list of the possible tiles, and an explanation of why no other tiles can work. (Hint: you need at least 3 tiles meeting at each corner point of the tiling why?) 5
6 Discussion. If you make a tiling pattern using regular polygons, there must be at least tiles at each corner. With exactly 3 tiles, each one must have an angle of 120 to completely fill up the space where the tiles fit together. We know that the hexagon has a 120 and in fact hexagons do make a tiling. What about 4 tiles at each corner? For that to work, each tile has to have an angle of 90 to fill up the space where the 4 tiles meet. We know that each angle of a square is 90 and in fact squares do make a tiling. Could 5 tiles meet at each corner? If so, each angle would have to be 360/5 = 72. That is not possible because no regular polygon has a 72 angle. To see why this is certain, notice that the angle of a square (the 4 sided regular polygon) is 90, which is too big, and the angle of an equilateral triangle (the 3 sided regular polygon) is 60, which is too small. Since there is no polygon in between these two cases, no regular polygon can have an angle of 72. So pentagons cannot make a tiling pattern. Could 6 tiles meet at each corner? That would require 360/6 = 60, which is the angle in an equilateral triangle. And we saw earlier that you can make a tiling pattern with equilateral triangles. Could 7 or more tiles meet at each corner? For seven tiles the angle would have to be 360/7 = That is smaller than the angles in an equilateral triangle. But notice that in our formula , as we increase the number of sides that also increases the angle. In fact, if you increase n the number of sides, n, that results in a smaller center angle 360/n. Then, we subtract a smaller amount from 180, and that produces a larger result. So if we want an angle that is smaller than 60 we have to decrease the number of sides below 3. But that is not possible. The same logic works for any number of tiles greater than 7: if there are t tiles then the angle of each tile would have to be 360/t, and with t > 6 that gives an angle less than 60. But as we just saw, all regular polygons have angles greater than 60. To summarize, the above remarks show that there are just three regular polygon tiling patterns. With hexagons you can fit three together at each corner, with squares you can fit 4 together, and with equilateral triangles you can fit 6 together. No other arrangement is possible. 6
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