A Modulo Krinkle tiling uses one tile. Its edges all have the same length, and its shape comes from three whole numbers, m, k and t, and an Offset switch. Miki Imura found the family in 2025. Copies of the tile fill the plane in a fan of congruent wedges that swirl round a centre. The page names a tiling by the triple (m, k, n), where n counts the directions that the edges take.
The tile. Draw two routes of unit steps between the same two points. Each route has k + 1 steps, and the routes bound the tile, which so has 2k + 2 edges. The directions run 0 to n − 1, 360° ÷ n apart. The lower route steps in directions 0, m, 2m and so on, counting in jumps of m on a clock with k hours, and ends with a step in direction k. The upper route takes the same steps with its first and last swapped. The tile figure in the options shows each edge's direction when k is 12 or less.
The wedge. Slide the tile along two fixed steps and the copies stack into a wedge, with one tile in its first row, two in its second, and so on. The staircase along each side of a wedge repeats the tile's lower route or its upper route. The wedge opens at exactly 360° ÷ n.
The fan. Turn the wedge by 360° ÷ n, over and over, and each new wedge slots against the last, because the staircase on one side of a wedge is the staircase on the other side of the next. Without offset, k wedges close up to the centre of a fan, and t copies of that fan turn to fill the circle, so the tiling has t-fold symmetry. With offset, half of the n wedges follow the same rule and the other half are those wedges given a half turn about the middle of one edge. The tiling then has 2-fold symmetry. The offset also changes n, so it changes the tile.
Grain boundary. A third arrangement extends one staircase between two neighbouring wedge directions in both directions. A half-plane of each lattice fills either side. The boundary repeats along its length, while the two sides use tile orientations one step apart.
Periodic too. The same tile also tiles the plane periodically. Choose Repeating to see it. That makes these tilings non-periodic rather than aperiodic, which is the difference from the Hat and the Spectre, whose tiles admit no periodic tiling at all.
Edges. Every edge is a step of one direction, and the two tiles that share it hold it as the same step. Single puts one bump on each edge, and Double puts an S-shaped pair. With offset, only Double fits. Height sets how far each edge swings, and the red stripes show where tiles collide. Wedge outlines draws the wedge boundaries, and thicker lines mark the seams between fans.
Construction. Play builds the fan wedge by wedge: the first wedge grows row by row, then each next wedge, and last the other copies of the fan together. Drag the slider to stop at any point. It needs the Fan arrangement.
Drag to move. Scroll, pinch or double-click to zoom. Turn with the turn buttons, a two-finger twist, a right-drag, Ctrl and drag, or Shift and the arrow keys. The address bar keeps your place, tile, colours and edges, so you can share a view.
This page needs WebGL 2, which this browser does not offer.