Problem of the Month
(January 2008)
The problem of tiling equal polyominoes in squares has been well-studied. This month we investigate tiling equal polyominoes in frames, squares missing from the centers of squares. We also investigate tiling equal polyominoes in circles, a totally symmetric set of lattice points whose distance from the center is smaller than a constant. Can you find examples of the smallest frames or circles made from various polyominoes? What other shapes can be tiled well with equal polyominoes?
ANSWERS
Frame Tilings:
Small Polyominoes
Pentominoes
Hexominoes
Heptominoes
Octominoes
Nonominoes
(George Sicherman)
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(George Sicherman)
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Decominoes
(George Sicherman)
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(George Sicherman)
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Undecominoes
(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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Mike Reid gave these non-trivial frame tilings for some larger L polyominoes:
Off-Center Frame Tilings:
Small Polyominoes
Pentominoes
Hexominoes
(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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Heptominoes
(George Sicherman)
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(George Sicherman)
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Octominoes
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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Nonominoes
(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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George Sicherman investigated triangular frames made from polyiamonds...
...and hexagonal frames made from polyhexes:
Circular Tilings:
Small Polyominoes
Pentominoes
Hexominoes
Heptominoes
Octominoes
Nonominoes
(George Sicherman)
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Decominoes
Undecominoes
I also investigated tilings of donuts, circles missing from the centers of circles. I even allowed disconnected regions. My results are here.
Claudio Baiocchi suggested that we look for full symmetry configurations. We call these Baiocchi figures.
Baiocchi Figures:
Small Polyominoes
Pentominoes
(George
Sicherman)
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(George
Sicherman)
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(George
Sicherman)
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Hexominoes
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(George
Sicherman)
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(George
Sicherman)
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(George
Sicherman)
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(George
Sicherman)
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(George
Sicherman)
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(George
Sicherman)
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(Corey Plover)
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(George
Sicherman)
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(George
Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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George Sicherman also investigated Baiocchi figures made from other polyforms:
Polyiamonds
The 9-iamonds are here.
Polypents
(Károly Hajba)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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The pentapents can be found here.
Polyhexes
The hexahexes can be found here.
Polyhepts
The pentahepts can be found here.
Polyocts
(Károly Hajba)
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(Károly Hajba)
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(Károly Hajba)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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(George Sicherman)
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The pentaocts can be found here.
The Baiocchi figures for polycubes can be found here.
If you can extend any of these results, please
e-mail me.
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