Perfect Tilings

The tilings shown in this Demonstration each use only one shape, but various sizes of this shape.
Such a tiling is called an irregular tiling. If all the tiles have different sizes, the tiling is called perfect.
The area to be tiled can be a square, a regular triangle, or a larger version of the shape itself.


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This Demonstration shows nine perfect tilings where the tiles have the same shape as the area to be filled, and eight other perfect tilings where the area has a different shape.
For any area/tile combination, only the tilings with the least number of tiles are shown in this Demonstration.
In the case of "squaring the square", for example, many solutions are known that use more tiles than the example shown in this Demonstration.
A solved problem relating to Euclidean plane:
In 2005 F. V. Henle and J. M. Henle showed that the plane can be perfectly tiled using only one square of each integral size.
Open problem:
Is there a perfect tiling of a square with 1×3 tiles?
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