One of our art teachers said this is the idea that fills in the missing link for him with 3d printing--the link between hand-made artwork and digital manufacturing. Other teachers responded similarly, feeling little affinity for a machine that prints digitally designed objects until they could see how hand craft can be a vital part of its use.
I'm working with a high school art teacher and her ceramics class. I made a Processing sketch they can use to generate patterns from several different shapes. Check it out, it's here: http://openblackboard.com/processing/shapetile/.
Conceptually the code of the program is very similar to the one Josh presents above for Turtle Art. Each shape object is made up of a shape method, called by a row method that draws the shape across the window (and a second offset row if you set offsetEvenRows to true), called by a makeRows method that repeats the rows top to bottom. The amount of horizontal and vertical overlap is determined by the slider numbers you give the shape, as well as the size of the shape.After configuring the shapes they way you want click the save button and download the image. You will want to tweak it before you get your final pattern.
The next step is to crop the image down to what you want just for the tile, so open it in Preview, or some image editor on a PC, and crop down to the desired area. Since you probably want the tile to repeat horizontally and possibly vertically as well this may be tricky, especially if you are aiming to have a square tile.
Save the cropped image and open it in Illustrator. Now you will trace it to prepare it for conversion to a vector graphic. Click Object > Live Trace > Tracing Options. Check Preview and spend some time trying the different Presets. When you have one you like, click the Trace button. At first I tried Detailed Illustration on this one, but when I completed the next step, importing to Tinkercad, I saw that the preset did something very different than I expected, so I retraced it with the Lettering preset and that worked much better.
Click Save as, and choose SVG.
Now in Tinkercad use the section in the upper right to import the SVG file into a new design. It will come in bigger than you want it, so scale it down, holding shift while you drag a corner to keep the aspect ratio.
The height will have decreased as well so raise it back up to about 6mm. Then add a 2mm layer to tie it together.
And now some results! I can't wait to see them glazed.