2013/09/17

A Pseudo-Polyomino or Polyplet is a pattern formed by several equal-sized squares connected by common edges or joined at right angles by common corners. Since the connections of squares at corners are rather virtual it would be nice to make a physical bridge between those squares. On the other hand you need some space for the bridges while constructing figures with the pieces and you can get it, if you round off the other corners.
If all corner connected squares are bridged, we get a lot of holes in the pieces, which can't be filled. Therefore a valid rounded Polyomino shouldn't have unnecessary bridges.
From the Pseudo-Pentomino in the picture we can derive two Rounded Pentominoes and three one-sided Rounded Pentominoes.

The chosen bridges must produce a spanning tree with ordinary polyominoes as vertices. In the example above the piece consists of two monominoes and one tromino with three possible bridges one of them unnecessary.
I learned about the concept from Miroslav Vicher' site and the Logelium site. Then I wrote a computer program to create such pieces and constructed symmetric and other pattern with them.
At last I made pieces from different materials like acrylic or magnetic foil.

For n=1..6 the number of Rounded Polyominoes is given by A056840 in OEIS. I tried to expand the series to n=10 and got:

 Number of Squares 1 2 3 4 5 6 7 8 9 10 Number of Pieces 1 2 5 22 99 580 3557 2395 155437 1057516

If you start with Polyplets you can separate the holey pieces and count how many different spanning trees can be obtained from these pieces. Eric Harshbarger created the pieces up to n=10 and got the same numbers. Unfortunately there was a difference for n=11.

Beside the ordinary rounded sets I also explored the one-sided sets and some polarized set, which are equivalent to rounded polyrectangles or rounded polyrhombs. Here are some constructions for:

I attached little magnets to the 177 one-sided Rounded Pentominoes cut from red acrylic to fix them on iron bars.

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