K-Tiling the Plane with a Minkowski Sum of Segments

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The Minkowski sum of two sets of points and in the plane is the set ; this extends to any number of sets , , , ... as .


Two distinct points and different from the origin define a lattice, the set of points .

The polygon can be translated by one of the lattice points. If the translates cover the plane without overlapping, they are said to tile the plane. If each point of the plane is covered exactly times, the polygons are said to -tile the plane.

In this Demonstration, drag the locators to define the endpoints of segments from the origin. The Minkowski sum of a set of segments is a convex polygon. Choose a tiling lattice; the polygon is translated by points of the lattice, centered at the origin. A consistent shading in the graphic indicates a -tiling.


Contributed by: Alexandru Mihai (February 2015)
Realized with the help of Mellisa Sherman-Bennett, Alexander Dunlap, and Dat Nguyen
Open content licensed under CC BY-NC-SA



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