Poly-Arc-Squares

(by Peter F.Esser)


During our holidays in summer 2002 I saw this ancient Roman mosaic, which tiles the plane with squares whose straight sides are replaced by arcs. This is a basic cell to make polyforms a little bit different from the usual polyominoes.
Putting four or five basic cells together you get 7 or 21 distinct pieces respectively. With the seven pieces you can easily pack a 7x4 rectangle but to fill a 7x15 rectangle with the 21 penta-pieces will take you some time. Miro Vicher has this puzzle on his site, too. For larger constructions a computer program would be helpful.


The following table shows the number of pieces with n basic cells. Depending on whether the pieces can be flipped (two-sided) or not (one-sided), we get different counts. Subsets are given if the pieces are made up from arc-dominoes. Click on the numbers to see some constructions with the sets.

Number of Squares 1 2 3 4 5 6 7 8
Two-sided 1 1 3 7 21 60 208 704
Made up from Dominoes - 1 - 5 - 39 - 394
One-sided 1 1 4 11 35 110 392 1372
Made up from Dominoes - 1 - 9 - 72 - 777

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Tri-Arc-Squares

4 one-sided pieces, total area = 12.

Tetra-Arc-Squares

7 two-sided pieces, total area = 28.
5 made up from dominoes, total area = 20.

11 one-sided pieces, total area = 44.
9 made up from dominoes, total area = 36.

Using the last set you can ask for 3-fold replicas ot the tetrominoes. Unfortunately a construction for the L and skew-tetromino failed. For the T-tetromino it's impossible due to a checkering argument.

Two-Sided Penta-Arc-Squares

21 pieces, total area = 105.

Three rectangles 3x35, 5x21 and 7x15 can be packed with the pieces. Especially the 7x15 rectangle is a puzzle not too easy nor too hard to do it manually. Since 21=16+4+1 combined 1, 2 and 4-fold replicas of the pentominoes might be possible. Indeed all constructions except for the X-pentomino can be made. Two examples are shown.

One-Sided Penta-Arc-Squares

35 pieces, total area = 175.

You can see the 5x35 and 7x25 rectangles and a 5-fold replica of the X-pentomino with a hole for the scaled piece. It should be possible to get replicas of the other pentominoes as well. The five 5x7 rectangles cannot be used to make the larger rectangles, because they do not fit at the border.

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