Tesselations
by Kurt Komoda(taken from my MySpace Blog)
So, here I am . I have arrived home from work too late to attend training tonight. Again. Perhaps, I shall do something constructive. Hmm, maybe I shall play more "Call of Duty 4" on the XBox 360. No. I'll do something constructive. Yes. Perhaps, I shall tell you of one of my artistic hobbies. I'm not saying that I have excelled at this thing, but only that it is something I have taken great interest in over the years. I know I am not terrible at it. I've seen the works of several others who have delved into this virtually intouched realm, and I can safely say that I have at least made it over the high wall and into the garden. Where I stand in the garden, I cannot say. Most of you probably didn't even know there was a wall or a garden.
 What I am talking about are Tessellations. It's what M.C. Escher referred to as "Regular Division of the Plane." Basically, it is filling a surface with the same exact shape or shapes. At it's simplest, a checkerboard is a tessellation. The earliest of complex tessellations can be found in the Alhambra. Unfortunately, the Moors, who built the palace, were forbidden to depict living things in their works of art. Today, tessellation is mostly relegated to mathematicians and crystallographers. Escher took it further than anyone before or after. His creative mind, eye and hand were like no other, and I have always been in awe of his work.
In 1995, a man named Kevin D. Lee, of Sandpiper Software, created a program called "Tesselmania" that allowed even children or people who like Wolfsheim to create their own tesselations. As the story goes, Tesselmania was a product of "The Learning Company" and was later bought by a toy company and effectively buried forever. I have never been able to find a fully functional copy, although I have it for both Mac and PC. It's an extremely limited program. It runs in 256 color mode, has only one undo level, and draws like the earliest paint programs(no blending or mixing). The tesselations, themselves, can only be comprised of straight lines between a limited number of points. The demo version lacks a save function- so you'd have to take screen grabs and paste them into Photoshop or something. The mac demo version only stays on for 30 minutes at a time.
Nevertheless, it is an extremely useful program. I wish someone would update it. It would be perfect for Illustrator. I've seen other programs, including an Illustrator plug-in, but none are as useful as Tesselmania.
In the division of the plane, there are 3 main principles at work: glide reflection, translation, and axles(or rotation). A system of labelling the 28 known types of asymmetric tiling- using combinations of the 3 principles- was created by German mathematician, Heinrich Heesch, and is known as the Heesch Type. Tesselmania allows you to create using 15 of the 28 types. I won't drown you in the details of the types.
Below are some of my efforts. You'll immediately notice the aforementioned primitive graphic style of the images.I would say that Tesselmania is only a starting point for identifying ones tiles. Further work could be done in another program to create a finished product, but I have so far found such an effort unnecessary. When creating these tiles, I have tried to adhere to the 3 rules laid out by M.C. Escher, concerning the shape of the tiles:
- Tiles must have a closed form. That is, the entire object must be represented. A tile cannot be part of something that trails off, but is not shown.
- The outline of the object must be as recognizable as possible. Simply making some wobbly shape and cramming the image of, say, a dog all rolled up into it doesn't count. I've seen this rule broken again and again. Another way of breaking this rule is the most common travesty performed by aspiring tessellators: the filing of empty space within the tile, or the creation of space filing tiles around your tile- the purpose of which is to make the tiles "work." If we were to allow this, then any shape could be made into a tile, and then made to fit by accomodating tiles surrounding it. I have done this, here:
You see, I created this tile, but it didn't really work; I still had that little square space where the bottoms of the wings meet. So, I just left it there. Very lazy. Years ago, I patted myself on the back for this little creation. Today, I am disgusted by it.
- The outline must have no indentations and bulges that are too shallow or too deep. This is simply for the matter of distinguishing one tile from an adjacent tile. Long, spindly spikes, for example, and their complementary crevaces(for there must be equal-sized indentations for the tiles to fit together) become confusing to the eye.
Anyway, here are my experiments. The colors in some are just horrible, so just try to ignore that, for now.
A bit too line-y, I think. Escher said that the shapes of birds and fish seemed to be the easiest to create.
I think that this one could really be developed into something nice.
Again, this one needs tweaking. Did this one on the mac, and my 30 minutes was about to run out.
Eh. A bit forced.
This one I like. I hate the colors I've chosen here, but I like the design.
I almost like this one, but the designs really shouldn't be 3/4 views of any animal.
Okay, this one is just pure silliness, but it's what the shape looked like to me.
I was trying, for some reason, to make a Corgi puppy.
I think this one is my favorite.
Lopsided and stupid.
I was really trying to make more mythologically-based tessellations, but this is all I came up with.
You guys should download the program and try it out for yourself. In Windows, you have to right click on it, and in properties, set it to open in 256-color mode. For the Mac download, I don't think you have to do anything but put the appropriate tesselmania folder on your desktop, or wherever. You might have to do something so it opens in OS9(Classic). Remember, though- you only have 30 minutes at a time with the Mac version, and no save function on both. Take screen shots(command+4, on the Mac...PrntScrn on the PC).
Please post any questions you have about this tutorial in Agony a Forum! The theory is that maybe your problem is also someone else's. Posting your questions to the forum not only helps others, but also allows others to provide their own answers. Additionally, please feel free to post any of your own tesselations to the forum!
Click below to enter:
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