Chapter 5 was about polygons and Replicating Figures. There are only three regular polygons. The equilateral triangle, the square, and the regular hexagon. These regular polygons can be used for making a floor that is very identical in shape and can be always repeated to cover the plane. But there are infinite numbers of irregular polygons that can provide this kind of tiling. Not many people knew about polygons about being made larger or smaller copies of themselves until 1962. In 1962 Solomon W. Golomb, who is a professor at the University of South California started working with replicating figures or as he called it "rep-tiles".
I believe there a two main concepts to this chapter that the author tried to communicate with us. The rep-2 figures and the rep-4 figures. The professor Golomb had some terminology.It was, replicating polygons of order k is one that can be divided into k replicas congruent to one another and similar to the original. Polygons of rep-k exist for any k, but they seem to be scarcest when k is a prime and to be the most abundant when k is a square number. There are only two types of rep-2 polygons. The isosceles right triangle and the parallelogram with sides in the ratio of 1 to the square root of 2.A parallelogram with sides of 1 and sq. root k is always rep-k proves that a rep-k polygon exists for any k. There is only one example, Golomb asserts, of a family of figures that exhibit all the replicating orders. When k is 7, a parallelogram of this family is the only known example. Rep-3 and rep-5 triangles also exist as well. Rep-4 figures, there are many of them. Every triangle is rep-4 and can be divided. Any type of parallelogram is a rep-4. There is only one type of rep-4 pentagon, which is call the sphinx-shaped figure. Golomb was the first to discover its rep-4 property. There are also three types of varieties of rep-4 hexagons. If the rectangle divided into four quadrants and one quadrant is thrown away, the re-maining figure is a rep-4 hexagon. There are no other known standard polygons with a rep-4 property.However, there are, "stellated" rep-4 polygons. A stellated polygon has two or more polygons joined at single points.One fact about the rep-4 polygon is that every known rep-4 polygon that is a standard type is also a rep-9 polygon. And it can also be the other way around every known standard rep-9 polygon is also a rep-4 polygon. Those are two main concepts I saw in the chapter. The rep-2 and rep-4 figures.
This chapter wasn't that easy for me to understand the first time. I didn't really know what replicating figures were until I read this chapter. I had to read this chapter twice to fully understand it. I also thought the examples in the chapter also helped me as well. This chapter didn't really surprise me because I was already expecting something new to learn and difficult to understand. I thought this chapter was also pretty interesting to read as well because I learned a few new things about replicating figures.And last but not least this chapter did raise many questions for me the first time I read this so I definitely recommend reading this twice.
Great summary of chapter 5! Even with many terms and definition tossed around in the chapter you conveyed them into a nice summary.
ReplyDeleteI also had to read this chapter twice to actually understand it. There were many puzzling concepts but your summary helped me picture them a lot better.
ReplyDeleteI like how professor Golomb called these rep-tiles. That's pretty clever. I agree that this topic was pretty perplexing throughout the chapter. A re-read was definitely helpful in understanding this concept and the terminology that was thrown into it.
ReplyDeleteThis was a good summary! the concepts were somewhat hard to follow but good job
ReplyDeleteYou did a good job with the summary. Some of the concepts were hard to understand, but your summary made it easier to understand.
ReplyDeleteThis is a great summary about the chapter, good job explaining it well. Like people have said, the concepts were difficult to understand but you did a pretty good job with them.
ReplyDeleteI am really impressed with your summary since I had trouble understanding this chapter. I thought it was going to be an easy concept to understand yet it was much more difficult then I had expected. I also felt like this chapter really reminded of Friday's class since since we had to repeat the drawings of triangles to construct parallel or perpendicular line.
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