Chapter 7: Penrose Tiles

     There are two forms of tiling; periodic and non-periodic tiling. Periodic tiling is a form of tiling in which you can outline a section of the image and it will be the same when shifted but not rotated or reflected. Non-periodic tiling is a form of tiling that uses each individual tile that uses each individual tile to create a larger or smaller version of the tile (this is defined as tiling through inflation or deflation). Penrose tiling is a form of non-periodic tiling in which the patterns are created from the shapes and the shapes are cut in half and then "glued" back together to form the original shapes through inflation or deflation.
   
     Penrose tiles generally lack symmetry since they are non-periodic. However, they can be constructed with a high levels of symmetry. The problem is that we generally don't recognize it. The chapter uses the universe as an example; it contains a cryptic mixture of order and what seem to be deviations from the order when in the grand scheme of things, still comply to the order (or at least try to).

     In R. Penrose's "Pentaplextiy: A Class of Nonperiodic Tiling of the plane," Penrose goes on to explain all the different method's on how to create these grand penrose tiles. However, he also goes into detail as to how all of the math is involved into the proofs of how these work which weren't too easy to follow and understand.

     While going through C. Radin's "Miles of Tiles," he gives a wonderful explanation of exactly what penrose tiles are and gives a huge array of examples to help the reader follow along as he explains. For example, he uses the crystalline structures of elements to show the natural occurrence of periodic tiling. However, when he gets to a few other examples, he only uses coordinates and points with their restrictions which can allow the reader to stray away from the text out of utter confusion or complete boredom.

     With R. Berger's "The Undecidability of the Domino Problem," he uses an example of dominos creating periodic tiling that is straight-forward and easy to comprehend. This allows the reader to get a glimpse of tiling without getting flustered through all the problematic vocabulary and procedural aspects of periodic and nonperiodic tiling. However, like the other two sources, the problem comes when the math does. When they start explaining the math of it, the phrasing is very confusing and hard to follow.

     The biggest problem with the three sources was when it came to the math. With the math lacking examples to show exactly how the math itself works, all the explanations seemed to just be a cluster of confusion. However, this chapter really emphasized the beauty on penrose tiling rather than the math itself. It also showed how this beauty is found in nature. This chapter really accentuates that one should really appreciate the beauty behind these patterns that most people just find in coloring books.