Erich Friedman's Math Magic for December 2009 posed this problem, considering pentominoes and smaller polyominoes as well. He posted dissections by himself, Berend van der Zwaag, Mike Reid, Gavin Theobald, and me. He later posted improved dissections by Livio Zucca and Helmut Postl.
Here I show only dissections of the monomino, domino, trominoes, and tetrominoes. For dissections of pentominoes, see Globular 3-4-5 Pentomino Dissections.
Edo Timmermans has also investigated dissections in which the polyomino at scale 3 is undivided. For want of a prettier term, I call such dissections subglobular.
The first table below shows both unconditional dissections and globular ones. The unconditional dissections are taken from Math Magic.
The second table shows unconditional and subglobular dissections. If you find a solution of either kind with fewer pieces, please write.
In these tables, where Edo and I found dissections with the same
number of pieces, I give Edo's solution.
Such cases are tagged ET (GS).
Because no pieces are rotated, the dissection can be stretched horizontally and vertically to form a rectangle with any dimensions. In particular, it provides a globular dissection of any straight polyomino. If the straight polyomino has 3 or more cells, this globular dissection is minimal.
1O | 2I | 3I | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
EF | 4 | GS | 4 | EF | 4 | GS | 4 | EF | 4 | GS | 5 |
3L | 4I | 4L | |||||||
---|---|---|---|---|---|---|---|---|---|
BZ | 4 | EF | 4 | GS | 5 | BZ | 4 | ET (GS) | 5 |
4N | 4T | ||||||
---|---|---|---|---|---|---|---|
GS | 5 | ET (GS) | 7 | GS | 5 | GS | 6 |
3L | 4L | ||||||
---|---|---|---|---|---|---|---|
BZ | 4 | GS | 4 | BZ | 4 | ET | 5 |
4N | 4T | ||||||
---|---|---|---|---|---|---|---|
GS | 5 | ET | 6 | GS | 5 | ET (GS) | 6 |
Last revised 2024-07-17.