James Lunde
December 8, 1999
Natural Patterns
Penrose Tiling
For this project, I am first going to explain the idea of tiling, then, I am going to explain Penrose tiling I an easily understandable way. To understand Penrose tiling, I did exercises with them. I also visited different web sites to get other information about the Penrose tiling.
The idea of tiling is very evident in everyday life. There are objects that tile planes everywhere that you look. From floors to walls, tiling is everywhere. The definition of tiling is a surface cover with shapes that are fitted together to cover an entire surface, without gaps and without overlapping. These surfaces for this tiling may be flat or curved. Take a soccer ball for example. A soccer ball is an example of tiling with hexagons. They are perfectly aligned all over the ball without gaps or overlapping. The object that is being repeated in a tiling pattern is called the protile. In the soccer ball example, the protile was a hexagon. However, there does not have to be only one protile. There may be many. For example, a tiling pattern may be made with hexagons and squares, both of those shapes being protiles.
There are many different ways one could choose to tile a plain. You could translate the object. This means that you would take the same object, and simply slide it one object to the left, right, top, and bottom, and continue to do that until you have a periodic tile. Another option would be to rotate the object around one of its edges midpoints. To accomplish a periodic tile this way, all you need to do is spin the shape around a fixed midpoint of one of the edges. Lastly, you could follow the rules of Penrose tiling.
Penrose tiling is a tile pattern that can consist of two different shapes, assembled in certain ways, which makes an aperiodic pattern. The two shapes are kites and darts. A kite is a convex shape. That means that it does not have any dips or bumps around the edges of the shape. A dart is a concave shape, the edges go into itself and a line can be drawn between two points in the shape, and part of the line will be outside the shape. The aperiodic pattern can also be established by using two rhombi, one with angles of 36 and 144 degrees, and one with angles of 72 and 108 degrees.
These shapes are then arranged in using a set of simple rules, which makes an aperiodic pattern. They can cover an infinite plane in an infinite number of ways. When the objects are expanded over the plane, there are certain local symmetries, but no specific patch is ever repeated.
A University of Oxford mathematics professor, Sir Roger Penrose, first found these Penrose
tiles. Sir Penrose is also famous for his work in three other scientific fields: gravitational theory,
the scientific basis of human consciousness, and tiling theory. Sir Penrose grew up around
mathematics, his mother was a doctor and his father was a medical geneticist, who used math in
his work as well as in his recreation. Sir Penrose came across his discoveries more or less on
accident. He wanted to cover a plane with different shaped tiles. By playing around with
different shapes, and after many years of work, he came across the Penrose Tiling method.
Although there are many more rules that are complicated, that one must follow to tile a plane
using Penrose tiling. There are very specific rules that you need to follow, or else your results
will not be what you want them to be.
By doing this project, I learned many new things about tiles and especially about Penrose tilings. I did further research on the internet for this paper, as well as following the research packet that I already had. I even found a java applet that let me make my own Penrose tiling, it was very interesting. Here is a listing of the websites that I used: