When tiling a floor, one uses the same identical shape repeatedly over and over again to cover the plane. There are only three regular polygons that have the capability to be repeated in this pattern (a regular polygon has all the same side lengths and angles) and they are the equilateral triangle, the square, and the regular hexagon. Although there is a limited amount of regular shapes that can be repeated in this pattern, there is an infinite amount of irregular shapes that have this capability. For example, any triangle can do the trick, as well as any 4-sided figure. In 1962, Solomon W. Golomb began researching replicating figures, which he later called “rep-tiles” and these shapes laid the foundation for the general theory of polygon replication. Golomb’s terminology for his rep-tiles is as follows: a replicating polygon of order k is one that can be divided into k replicas congruent to one another and similar to the original (Gardner 47). The replicating order of shapes is designated by “rep-k” For example, the trapezoid in figure 1 is a rep-4 polygon because there are four smaller, similar trapezoids that create the larger one. The parallelogram is a rep-2 polygon because it is only divided into two smaller shapes.
There are only two known rep-2 polygons and they are the isosceles right triangle and what is known as the “golden rectangle” which has a side ratio of 1 to 2; the unique property of the rep-2 rectangle is the reason it is used for the shape of printer paper, playing cards, and pages in books. The golden rectangle belongs to a family of parallelograms that have a special property where if the side ratios are 1 to k then the replication of that parallelogram is k. This parallelogram family is special because it is the only family of figures that exhibits all replicating orders. In fact, this family holds the only known examples of replicating orders that are prime numbers and greater than 3.
There are many types of rep-4 figures known, including every triangle and every parallelogram. There is only one known rep-4 pentagon and that is the sphinx-shaped figure below. There are three known varieties of rep-4 hexagons and these can be created by dividing a rectangle into 4 quadrants and then throwing one quadrant away. Two of the three known rep-4 hexagons are shown below.
There are no other standard polygons that are known with order 4 replications. There are non-standard “stellated” rep-4 polygons however that are known. Stellated indicates that two or more polygons are joined at a single point, like a corner.
Another interesting feature about standard rep-4 polygons is that each of them is also standard rep-9 and vice versa.
Because stellated polygons are not considered standard polygons, stellated rep-9 polygons are not rep-4.
A final theorem explained in this chapter refers to multiplicity and is goes as follows: Consider a figure P that can be divided into multiple congruent figures (not necessarily similar to figure P) and call these congruent figures Q. The number of Q’s that there are in figure P is the multiplicity of figure P. The multiplicity of figure Q is the number of P’s that fit in Q. If you multiply the multiplicities of both figures P and Q you get a replicating order of both figures.
This chapter was rather confusing at first and difficult to understand because of the technical terms that were being tossed around. After a while I understood the rep-k values and what k meant. I am still confused as to why the chapter started talking about tiling, because most of the chapter did not talk very much about tiling a floor, it talked more about the theorems and ideas that went along with dividing shapes with congruent figures. Also, the chapter was specific in the beginning saying that you must take smaller figures and piece them together to make a larger one, but throughout the rest of the chapter I believe they broke down big figures as much as they built up small ones. Overall, I understood the concept of breaking down shapes into smaller shapes using similar shapes, but I do not understand the purpose of this concept.
Your explanation of the format for repk and how k represents the number of figures that the original shape can be divided into helped me understand this chapter more. What I am confused about is the theorem at the end of the chapter with P and Q as shapes and s, t, and u as the multiplicity. The entire chapter all the pictures were divided into similar shapes as the original, but then the theorem said it did not have to be that way. I do not quite understand if the theorem is talking about the same topic as the rest of the chapter. Another part of this chapter that confused me was the part about stellated repk polygons. I think you explained what they are well, but neither you or the book described their significance to the chapter idea very well. I don’t quite understand their significance. I am as confused as you on the purpose of these repk polygons; are they only significant in tiling floors? Other than that I can’t think of what they could be used for. Overall, I thought this chapter was pretty difficult to understand but you did simplify the major points pretty well.
ReplyDeleteI found this chapter very interesting but at the same time very hard to understand. You did help me understand the repK a lot better than the book did. The most interesting thing I found was how many polygons they have discovered that can be replicated. It shows how much time they have put into this theorem/topic, especially with the polygons that have a unique shape like the Sphinx. But I still cannot figure out the P, Q, s, t, and u theorem. I wish they would have a figure breaking it down step by step. Overall, the chapter was easy to understand at the beginning, but the very end confused me a lot because they said the theorem didn't have to be like how they first stated it. I don't know what they meant by this at all. As a whole, the chapter was a good read and I enjoyed it, but the end was hard to understand. Also, like Charlotte above, I don't understand why these repK polygons are significant. I tried looking it up but could not find anything worth noting.
ReplyDeleteI also found this chapter to be quite difficult to understand at first. I thought you did a good job of explaining exactly what the different values of k a replication order can take on. I also liked how you included the diagrams of the different replications it made the descriptions a little easier to understand. What I found difficult, like Frank, was the theorems on p, q, s, t, and u. I found it difficult to connect them to actual “real-life” examples like how the book connected the replications on the sphinx and floor tiles. I found a connection between our chapters. In my chapter (50) they gave an example of mapping and color coordinating. Replication could be used as a part of the four-color theorem as well. Overall the chapter was interesting but also hard to follow and understand. I agree that they should have used the example of tiling more throughout the chapter it could have probably made it easier to understand. Towards the end, Gardner started talking about rep-k squared or cubed tiles and I don’t think that was necessarily needed just because he gave examples of different shapes inside of a square or a triangle per say. I think including that just made the chapter a little more difficult to understand.
ReplyDeleteWho knew tiling floors could be so confusing!? Anyways, I also found the specific terminology and to be complicated in understanding. I used your post to guide me through this chapter to help clarify the ideas conveyed for "rep-tiles." It was nice to get a second view on the explanations of rep-polygons, rep-k, and the P and Q theorems. Maybe an additional explanation of the P and Q multiplicity theorem at the end would have been nice, rather than just stating it entirely. However, you explained majority of the terms very well in a way us readers could understand when compared to the book. I really enjoyed the illustrations as well to show me a visual representation of different rep-k polygons and how the breaking down took place. I understand why the tile part came to be quite confusing. It does feel like the author of the chapter changes between breaking down shapes and building tiling without a solid proper transition. If you were to split the chapter, I would say that the theorems part is the more analytical advanced mathematics part, while the tiling part would be the partly recreational side to the math. Though, after trying to read through the chapter and understand the concepts, I feel like you did a good job focusing on the main math points of the chapter.
ReplyDeleteThis post pulled me into reading this chapter because of the real world application to tiling floors. I have family in the tiling floor business and never realized how much thought and mathematics went into doing the job correctly. Once I was reading the chapter I was quite confused on where " rep k" came into place, but I believe Molly did a great job explaining that to the readers. I also found that Molly talked about all main topics throughout the chapter such as the rep k, these rep polygons, and the theorems(P and Q). Some of these shapes and descriptions were very interesting in my opinion. It was interesting that some simple shapes can come together to create a beautiful new shape such as "The Ampersand" on page 52. A name that came up a lot throughout the chapter was Golomb. I found Golomb's research with all of these shapes and tiles simply amazing. It is amazing how much one person could find by just looking at shapes. Although the chapter was a bit difficult to understand at first, I did find the concept of reptiles quite intriguing. In addition, I believe Molly did a great job of making these concepts easier to understand for the reader.
ReplyDeleteThis chapter started out relatively straight forward and simple but as I progressed through the reading I found myself more and more confused. While I understood the books explanation of Rep-k, yours helped me to fully grasp the concept of rep-k. Another part of this chapter that I believe you did a good job of explaining is the "golden rectangle". It was interesting to me how much this particular type of rectangle is used in everyday life such as in printer paper, playing cards and the pages of a book. One thing that I did not understand after reading the chapter was the theorem that refers to multiplicity. While your explanation of it does help clarify it slightly it is still hard for me to understand and may be something that I have to research on my own to fully grasp the concept of it. All in all this chapter seemed, at first, like it would be a relatively easy chapter to understand. However when as I started getting farther and farther into the chapter it proved to get more and more confusing. I feel that you did a great job on pulling out all of the key points of this chapter and simplifying them so they were easier to understand.
ReplyDeleteThis chapter started out relatively straight forward and simple but as I progressed through the reading I found myself more and more confused. While I understood the books explanation of Rep-k, yours helped me to fully grasp the concept of rep-k. Another part of this chapter that I believe you did a good job of explaining is the "golden rectangle". It was interesting to me how much this particular type of rectangle is used in everyday life such as in printer paper, playing cards and the pages of a book. One thing that I did not understand after reading the chapter was the theorem that refers to multiplicity. While your explanation of it does help clarify it slightly it is still hard for me to understand and may be something that I have to research on my own to fully grasp the concept of it. All in all this chapter seemed, at first, like it would be a relatively easy chapter to understand. However when as I started getting farther and farther into the chapter it proved to get more and more confusing. I feel that you did a great job on pulling out all of the key points of this chapter and simplifying them so they were easier to understand.
ReplyDelete