Quadrilateral Tiling with Textures

This Demonstration tiles the plane with textures defined from portions of an image. Controls are provided to select the location and magnification within the image for use in the tiling. The shape of the tiling can be changed by clicking inside the image area. It may be more convenient to work with a small portion of the full tiling (uncheck "tile image area") and to then adjust the image. Clicking the "tile image area" button retiles the image area.
The construction begins with an arbitrary quadrilateral with vertices , , , and . These four points can be changed by clicking (and dragging) in the image area. Clicking close to one of the vertices causes that vertex to relocate to the mouse position; dragging the vertex causes the quadrilateral to continuously resize.

The tiling begins by rotating the quadrilateral 180 degrees around the midpoint of each side. For example, the starting position in the image area shows a misshaped rectangle. This shape is rotated 180 degrees around the midpoint of the line segment connecting and , once around the midpoint of the side connecting and , once around the midpoint of the line connecting and , and once around the midpoint of the line connecting and . Now the original shape can be translated by and by to fill in the gaps. The construction can be repeated to fill up the whole plane, and it works for any quadrilateral, even those that are nonconvex.

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