Tiling the Plane with the Pinwheel Pattern
Click on the Triangle button to put a copy of the Pinwheel Triangle into the workspace.
It is instructive to create a super-tile for the pinwheel tiling yourself by combining five copies of the triangle. Triangles can be grouped by clicking and dragging to draw a rectangle around the group. Any group selected can be duplicated by clicking the Clone button, after which the whole cloned group can then be colored. Ungroup by holding down the Shift key while dragging a rectangle around the group.
A more efficient tiling process is available. With a triangle selected in the workspace, click the Expand button to create a super-tile consisting of five triangles, with the original in the interior, in the “T-arrangement.” Clicking the Collapse button undoes the super-tile construction, reducing an arrangement to the single interior triangle arrangement. Repeatedly clicking the Expand button builds larger and larger T-arrangement super-tiles as part of the pinwheel tiling for the entire plane. At any stage, it is possible to create a copy of the whole selected T-arrangement using Clone.
It is always possible to tile the plane with copies of any single triangle. For our triangle, two copies fit together to form a rectangle that can obviously be repeated endlessly to tile the whole plane.
The remarkable thing about pinwheel tiling is that there is only one way (duplicating the entire identical tiling) to match small pieces in such a way that the larger tilings line up. Experiment with this idea by starting with a single triangle. Color it red and Expand two or three times. Then Clone the whole super-tile to create a black super-tile and a red super-tile. Visually pick out a piece (one or a set of several triangles) of the red tile, and find a corresponding piece of the black tile. Then try to line up the matching red and black pieces. Are you convinced that there is no way to line up the matching pieces that extends to larger and larger parts of the tiling without matching the whole super-tiles? The operations of dragging, rotating, and flipping (often called translations, rotations, and reflections) are precisely the rigid symmetries of the plane.