We all have fun in producing and looking at these symmetrical patterns.In some way symmetry is regarded as a measure of beauty.
But symmetry is not only central to Arts but also in Science: if there is a royal way to science, this must be symmetry.To understand this, just think about the fact that many objects built by man are symmetrical not only because symmetry is pleasing but also because they are more easy to create. So you can guess that whenever you see symmetry in Nature there is a simple process or algorithm that creates it.
Not a surprise that symmetry is a crucial concept in mathematics:mathematics being the basis of science.
Using the words of the great Feynman:
Why is nature so nearly symmetrical? No one has any idea why. The only thing we might suggest is somethibg like this: There is a gate in Japan, a gate in Nikko, which is sometimes called by the Japanese the most beautiful gate in all Japan; it was built in a time when there was great influence from Chinese art. The gate is very elaborate, with lots of gables and beautiful carvings and lots of columns and dragon heads and princes carved into the pillars, and so on. But when one looks closely he sees that in the elaborate and complex design along on of the pillars, one of the small design elements is carved upside down; otherwise the thing is completely symmetrical. If one ask why this is, the story is that it was carved upside down so that the gods will not be jealous of the perfection of man. So they purposely put the error in there, so that the gods would not be jealous and get angry with human beings. We might like to turn the idea around and think that the true explanation of the near symmetry of nature is this: that God made the laws only nearly symmetrical so that we should not be jealous of His perfection! The Feynman Lectures on Physics Symmetry in Physical Law Volume I Chapter 52, Section 52-9 (p. 52-12) Addison-Wesley Publishing Company, Reading. 1964.