Magic tile shape grows organically, never creating a pattern

This is the most complex shape known that can fit together organically. It was just discovered by mathematicians in 2023. It is easy to cookie-cut these out of clay (notice the cutter I made at the top). Placing the tiles is tricky because it is only logical to seek a pattern, but that does not work. Starting with a center tile and moving outward in a spiral around it seems to be the best way. Mathematicians are seeking to prove that placement can grow infinitely without ever repeating a pattern. Making the cookie cutter in Fusion 360 was easier than expected because the shape is built from the pie-slices that result from cutting a regular hexagon into six pieces midway across the straight sides. Because of the complexity of the shape I have found that it is best to print multiple cutters (I can do eight at a time), and stamp the shapes without using stretch wrap (letting them dry overnight in the cutters). The randomness seems confirmed in that when I piece together a few dozen tiles it is very difficult to do a count (because they are not in rows). In addition, to piece together 28 tiles requires turning eleven of them over - if there was a pattern I would expect to turn over exactly half of them. One issue: To create a setting with straight sides it appears I will need a dozen shapes.

Related Pictures

Making complex ceramic tile shapes by 3D printing your own cookie cutters

This was done on an affordable RepRap printer. The red plastic templates were drawn in Fusion 360 and sliced and printed using Simplify3D. A wooden block was used to press these cookie cutters into the clay. The plastic wrap made sticking a non-issue (and rounded the corners nicely). Commercial bottled glazes were applied to this low fire talc body by brushing (in three coats) after bisque - the rounded corners make brushing easier. The tiles were fired at cone 03. This is an old classic design that I discovered when researching Damascus tile. The toughest obstacle was learning how to use Fusion 360. It turns out that cookie cutters are a starter project for many 3D software packages, there are lots of videos on making them.

Make multiple cookie cutters for the best result

Using the "create pattern" function of my CAD software, Fusion 360, I created a grid of duplicate cookie cutters. That enables 3D-printing up to 25 at a time. While it is impossible to remove them while the clay is soft I can leave them in the cutters overnight, the next day they just drop out. Creating the rounded top edges is easily done first using a little stretch wrap as shown, then it is just a matter of pushing the cutter the rest of the way through and setting them aside to dry. This method creates a high-quality edge finish. Of course, the embossed design was first stamped into these Polar Ice porcelain tiles using a letterpress plate.

3D printed cookie cutters used to make this tile design

3D-printed cookie cutters to make fish tiles

This was made for our "AmacoGlazeLovers" Facebook group page (notice one fish is swimming the other way). Only three tile shapes are needed. The fish cutters were 3D printed to both cut and stamp at the same time. Multiple sizes of the triangle and trapezoid were made to accommodate irregularities and keep joints tighter. The clay is M340 and the glazes are Amaco Celadons and Potter's Choice (for brushing). I found the best way to paint them was to glue them down to a plaster slab with a few drops of glaze (it was easy to scrape off when the three coats had dried).

Links

URLs	https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile Mathematicians have discovered an elusive ‘einstein’ tile Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats.
URLs	https://en.wikipedia.org/wiki/Einstein_problem Einstein Problem on Wikipedia The einstein problem seeks the existence of a single tile shape that can tessellate only in a nonperiodic way
Projects	Cookie Cutting clay with 3D printed cutters We are finding more and more applications for this simple process of cookie-cutting shapes in ceramics. You won't believe whats possible and how easy it is to get started.
Glossary	3D Design 3D Design software is used to create dimensionally accurate objects by sketching 2D geometry and transforming it using tools to rotate, extrude, sweep, etc. The software generates the polygon surface.

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