Hole Dissections: Introduction

Author : Gavin Theobald

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[n] indicates the plane with a {n} hole, i.e. an n-sided polgon hole. Similarly [p/q] indicates the plane with a {p/q} hole i.e. a polygram hole.

With hole dissections we are attempting to change the shape of a hole. This can always be done with the same number of pieces as the equivalent dissection. For example a triangle can be dissected to a square in four pieces, and a triangular hole to a square hole also in four pieces. We don't count the plane itself:

If the dissections can be arranged to share a piece, then we save a piece. This may not always be possible, but is in this case:

What if we want to have the holes sharing a common centre and orientation? This can always be achieved with just one extra circular piece:

In fact, we can position the two holes exactly where we like with a circular piece. Since we can always position the holes in this way, the following dissections will be shown without this piece.

Hole dissections are “slideable” if the pieces can be rearranged by sliding each piece without any piece ever overlapping another.

The following tables allow the number of pieces required for a hole dissection to be calculated. They give the number of pieces for each of the two shapes. For a few dissections this figure can be improved. This is generally when a dissection can make use of tessellations. Also for a few dissections I've not been able to find a solution in so few pieces, for example the table predicts that [4] to [7] can be dissected in 5 pieces, but I've not been able to find anything better than 6 pieces.

The numbers in these tables are the smallest number of pieces required to convert a shape to a P strip element. Superscript figures require pieces to be turned over. Entries marked with an asterisk * have strips that are wider than a square, hence may require an extra piece.

3 4 5 6 7 8 9 10 11 12
2 1 3 2 4* 3 5 3 4

5/2 6/2 7/2 7/3 8/2 8/3 9/2 9/3 9/4 10/2 10/3 10/4 12/2 12/3 12/4 12/5
3* 3 6 5* 4 3 7 7 6  10 6 6 5 5 5 6 5*

1 2 2
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