the final final...

A lot of math goes into a lot of things that we see or do every day without us having any idea. I was surprised when I learned about the flower petals being in Fibonacci patters and that music is mathematically composed. However I was intrigued when I realized there is a mathematical sequence behind tiling. The tiles I walk on and see everywhere on a daily basis are arranged according to mathematical law. It seems so simple and it makes sense after all they are composed of geometrical shapes which normally would trigger the math light bulb in my head. For whatever reason it seems tiling escaped that thought process.
Tiling by definition is the covering of an entire plane with non overlapping figures. The key to tiling, as laid out by St. Thomas Aquinas, is symmetry. All that really means is that the proportions of the images you are looking at should be equal or mirrored. This component makes the tiling pleasing to the eyes and allows things to fit together in a way that is orderly and makes sense. Ceramic is the most common material used to create tiles and has been being used by man to create decorative or practical tiles for 4000 years. Decorative tiling can be found in ruins and different buildings all around the world. They were first made by hand and laid out to bake in the sun and remained flat. They could then paint them to make them more appealing. After a while they were able to make a mold out of wood that would help make a consistent size, shape and patter on each tile. The manufacturing and decorating of the tiles was perfected during the Islamic period in Persia.
Now-a-days we have machines to do all the making and painting of the tiles that we use all over the place. The most common patter is monohedral which uses only one shape and size of tile. It’s common to find tiling like this in practical places like many kitchens and bathrooms. These tiles and many others are considered edge-to-edge because they share a side with the next tile as opposed to being a free shape. In the 1970’s Sir Roger Penrose developed the Penrose tiling which is non-periodic, meaning there is not repetition within the pattern. These types of tilings seem to remind me of a flower usually and often take up larger spaces. We see a lot of frieze patterns in the museums and buildings we visit in Italy. This type of tiling is not meant to fill a plane like a floor or a wall, but is instead used as borders to other planes and/or materials. My personal favorite type of tiling was developed by M.C. Escher as a result of his study on the Regular Division of the Plane. It is called lattice tiling and can contain pretty much any pattern you can imagine. When the individual tiles come together at the sides they fit together like a puzzle piece in order to complete their shape. I read in multiple articles that Escher was known for his eye for abstract beauty. My favorite was one that said he “likes to challenge the logic of seeing.” He would sit there and draw random animals and people interlocking in every way possible until he could come up with an infinite pattern with aesthetic value and after 40 years of practice he mastered his obsession.
The first example of complex semi-regular tiling that comes to mind is the tilings we saw in the Vatican museum on the floors. They are from all over the place, brought to the museum not just to cover the floor but to be on display. They were there and we saw them but I don’t know that we were able to appreciate their worldly value. They create a mosaic picture which in some cases which can tell a story just as much as the statues and paintings in the room around it. Not to mention the fantastic colors and patterns created even without a picture. Just about every church and museum we have visited has had a wide variety of tilings on display that I feel often get over looked because we are so focused on looking up. I think from now on I will take a second to look down at the patterns in the floor or on the walls or borders as well as the art in the vicinity.