Tessellation Maker

A tessellation asks for less than it seems to. Cover a flat surface with copies of a shape, leave no gaps, allow no overlaps, and never stop. That is the whole demand. It is harder to meet than it looks.

A Cornell outreach course states it with a mathematician’s care: a tiling is a collection of tiles that cover the plane without gaps or overlaps, each tile a single connected piece with no holes. A kitchen floor passes. So does a honeycomb. A regular pentagon does not, because its corners measure 108 degrees, three of them leave a wedge too narrow for a fourth, and no shuffling removes the wedge. What separates the hexagon that works from the pentagon that cannot is arithmetic, and the arithmetic is short enough to do at a table.

The sum at every corner

Wherever tiles meet at a point, their angles must add to a full turn, 360 degrees. A regular polygon with n sides has corners of (n - 2) x 180 / n degrees: 60 for the triangle, 90 for the square, 120 for the hexagon. If one kind of regular polygon is to do the whole job, its corner must divide 360 evenly. Six triangles work, four squares work, three hexagons work. Past six sides the corner is wider than 120 degrees and narrower than 180, so it divides nothing. Three regular tilings exist, and the list is complete.

Mix the polygons and the same sum opens more doors. Insist that every corner of the pattern show the same polygons in the same order, and exactly eight more appear, called semi-regular or Archimedean. Each is named by reading round one corner. In 4.8.8 a square and two octagons meet, and 90 + 135 + 135 = 360. With the three regular ones that makes 11 tilings, first treated systematically by Johannes Kepler in 1619.

The sum is necessary and not sufficient, which is where the subject stops being a homework exercise. Seventeen combinations of regular polygons close up round a single point, in twenty-one different orders, and only eleven of those can be repeated at every corner. Two pentagons and a decagon meet perfectly, 108 + 108 + 144, and still cannot be carried on to fill the plane. That is why the tool on this page does not draw the eleven from a stored picture. It grows each one outward from its corner rule, polygon by polygon, and the pattern either completes or contradicts itself.

Borrowed edges

Regular polygons are a narrow diet. Any triangle at all tiles the plane. So does any quadrilateral, convex or dented, however lopsided.

The reason is a half turn. Spin a quadrilateral 180 degrees about the midpoint of one side and the copy lands against the original, sharing that side exactly. Repeat at the other sides and the four corners of the shape gather round each meeting point, where they add to 360 degrees and close. Now bend a side. If the bend on one half is repeated, turned over, on the other half, the half turn still carries the side onto itself and the neighbor still fits. The straight line was never the point. The side only had to be its own image under the motion that brings the neighbor in.

That idea is the rule behind every interlocking figure: each stretch of a tile’s outline is a copy of another stretch, carried there by a motion that also carries the tile onto its neighbor. Draw a square, replace the top edge with any curve, and slide a copy of that curve to the bottom. Do the same from left to right. What one side gives away the opposite side takes back, and the figure tiles by sliding alone. A textbook from Saint Louis University gives exactly these steps and calls the result a sort of inkblot test: only afterwards do you decide whether you have drawn a bird or a man.

Try it with numbers. Choose the square with sliding edges in the tool above and load the bird. Its outline has swollen to 1.26 times the width of the square it came from, yet with the tile set to 50 mm wide the readout gives an area of 15.75 square centimeters. A plain square at the same scale has a side of 50 / 1.26 = 39.68 mm, and 39.68 squared is 1,575 square millimeters: the same 15.75. Drag any point as far as you like and the figure stays put. The area belongs to the hidden square, not to the bird.

M. C. Escher, born in 1898, made this his game for life. He first saw the tiled walls of the Alhambra in 1922 and went back in 1936 for three days of careful color sketches. The Moorish tiles were geometric. His had to be recognizable as creatures, and he later wrote that he “spent a large part of my time puzzling with animal shapes.”

How systematic he became is less well known. In 1937 his half-brother, a geologist, sent him a list of crystallography papers, and one of them, by G. Pólya, showed a sample pattern for each of the seventeen symmetry groups a repeating plane pattern can have. From 1937 to 1941 Escher worked out for himself which shapes can surround themselves the same way everywhere and how their edges must be related, and he wrote the results into a notebook. He allowed only slides, turns and glide reflections. A plain mirror would force an edge to be a straight segment, which is no use for a lizard. He left 134 numbered symmetry drawings.

A mathematician reached the same ground separately. Heinrich Heesch studied such tiles in the 1930s and in 1963, with Otto Kienzle, published a proof that asymmetric tiles of this kind come in exactly 28 types. He named each by lettering its edges: T for an edge matched by translation, G for a glide, C with a number for a turn. Escher, by then very ill, had found all but one of them more than twenty years earlier.

Slides, turns and glides

Each motion leaves its signature on the pattern. A slide keeps every copy the same way up. A tile that fits by sliding alone is always a bent parallelogram or a bent hexagon with opposite sides matched: two pairs of edges or three, never more.

Turns come in few sizes. The half turn about the middle of an edge is what makes every triangle and quadrilateral work, and it is the heart of a test named for John Conway. A shape passes if its outline splits into one pair of edges matched by a slide and up to four more that are each their own half turn. It then tiles by half turns alone. A quarter turn about a corner ties two neighboring sides of a square together, and tiles pinwheel round that corner. A third of a turn does the same for a hexagon, and Escher built his lizards on it.

The glide is the odd one. It is a mirror flip followed by a slide along the mirror line. It turns left-handed copies into right-handed ones, so rows of figures face one way and then the other, as footprints do. Because the flip never happens without the slide, no edge has to lie on a mirror and every edge stays free to curve.

Between the points a visitor places, the tool draws a centripetal Catmull-Rom spline. An analysis first published in 2009 proved that, in its family of curves, this is the only version that never forms a cusp or a loop inside a segment. That matters for an edge about to be copied a thousand times.

Color, then plastic

A tiling is also a map, and a map wants coloring so that neighbors differ. Escher found that when four tiles meet at every corner two colors are enough, checkerboard fashion, and that patterns with three tiles at a corner need three. No flat map needs more than four. That is the four-color theorem, conjectured in 1852 and first proved, with a computer, by Appel and Haken in 1976. The tool searches for the fewest colors that keep neighbors apart and lets you spend up to four.

Turning a tile into an object adds a problem that geometry does not have: real edges have width. Two printed tiles drawn to touch will not go together, so each outline is moved inward by half the wanted gap. Ask for 0.2 mm of clearance and every side retreats 0.1 mm; the gear-shaped tile set to 50 mm comes out 49.8 mm wide. How much gap is enough depends on the machine. The University of Florida’s makerspace allows 0.20 to 0.25 mm between separately printed parts that must slide on its two printers, and says such figures are learned by printing a test. So 0.2 mm is a first guess to test with two tiles. It is not a promise.

Print the tile lying flat. The first layer is squeezed against the heated bed and comes out a little wider than drawn. Cura’s definition file calls the fault elephant’s foot and carries a setting against it, Initial Layer Horizontal Expansion: a negative value there pulls the first layer in to make up for the squash. The tile can also leave room for the flare. A bevel sets the rim of one face back from the outline, so the beveled face of the tile belongs face down.

The method has limits of its own. It makes only patterns in which one shape repeats in a regular way, and the tool offers nine of Heesch’s types, none with mirror symmetry. It will not find the convex pentagons that tile, a list that grew to fifteen kinds and was shown complete only by a computer-assisted proof announced in 2017. It cannot produce the hat, a single shape described in 2023 that tiles the plane but never periodically. And when a dragged edge makes the outline cross itself there is no tile at all, which the tool reports instead of drawing one.

What survives all of this is the first idea. A shape fits its copies when its outline is a set of borrowed edges and every loan is repaid somewhere else on the same tile. Pull one edge and watch the debt move.