Magic tile shape assembles organically, never creating a pattern

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This is the most complex shape known that can fit together organically, without a pattern (dubbed the Einstein tile, it was discovered by mathematicians in 2023). It has six sides of 1 unit length, six of 1/1.73205080757 and one side measuring twice the latter. Placing the tiles is tricky because it is only logical to seek a pattern, but there is none. One method is to start with a center tile and move outward in a spiral, being ready to backtrack and place them upside down when a piece cannot be fit. The tile shape is a product of connecting four identical irregular pentagons - each made by cutting a regular hexagon into three pieces. To achieve the maximum precision, 3D print multiple cookie cutters and let them stiffen and shrink in the cutters. To round the upper corners use stretch wrap) when stamping (print cutters in both orientations if doing this). Tiling a floor or wall will present issues with the number of edge tiles that need to be cut. However, cutting at least some custom edge shapes is practical because only 90 and 120 degree angles are needed.

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

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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.

3D-printed cookie cutters used to make this tile design

3D-printed cookie cutters to make fish tiles

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Only three tile shapes were needed. The fish cutters were 3D printed to both cut and stamp at the same time. Multiple slightly different 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). These are small and we found that a good 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).

Make multiple cookie cutters for the best result

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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.

Video: Create a cookie cutter/stamper in 3D software, print it and use it

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This cookie cutter can both cut and stamp the piece (notice the 3D render in the centre, the logo is 2mm lower than that cutter around the outside). We make them by rolling a slab to 3.2mm (1/8in) thick, applying stretch wrap over it and then pressing the cutter/stamper into it (using a wood block). Then just peel away the plastic and the outer waste clay and a perfect crest is left. This method enables using clay of almost any stiffness. We find that softer clay works best, just peel it up from the board, apply slip to the back, position it on the side of the leather hard ware and press it down (from the centre outwards). On the lower right is a crested mug that has just been glazed. Upper right is a crest that has been glazed and fired. About the cookie cutter: We create them with 0.8mm wall thickness (twice the width of the 0.4mm extruder on the 3D printer). We export the vector image (made in Illustrator) into Fusion 360 and then add elements to stabilize and hold the profile in place. This cutter is 8mm tall and the stamp lines are 5mm tall. The crest is 52mm (2 in) wide. This whole process may sound a little intimidating to you - but we are working on a step-by-step video.

Videos

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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