Polysquares in the Cubic Grid


2024/12/06
Connecting squares edge by edge at 90° or 180° you get pieces, where all corners are points of a cubic grid. They are called Polyominoids, and the number of different pieces is given at OEIS sequence A056846 . If only connections at 180° are allowed, all corners of the pieces are in a single plane and we have the polyominoes. If all corners of the pieces are in two neighboring planes and the squares are orthogonal to these planes we have pieces equivalent to polysticks.

Given a special set of pieces we can look for figures, which can be constructed with these sets. The following table shows some examples . Click the numbers of pieces to see the constructions.

Number of Squares Number of Pieces Total Number of Squares
3 11 33
1,2 or 3 14 38
4 80 320
3 or 4 91 355
5 780 3900

What about physical pieces? Since physical squares have a thickness greater than 0 the unit length of the cubic grid must be at least the edge length of the squares plus the thickness of the squares. The connection between two squares can be made by triangular or quadratic prisms. Printing those pieces with or without support is easily possible; the obj-files are here. But if you use such pieces in a construction with vertical squares, you need a frame or connections to other pieces to prevent them from falling apart. Squares with holes at their edges might be a solution. The holes can be connected by little hooks, but these hooks don't stick perfectly, so that the connection isn't sturdy enough. Furthermore the hooks can't be applied when the pieces are already in the correct position.

Squares with slots at the centers of their edges are another possibility. Four kinds of clamps are needed to connect the slots of different pieces. These clamps can also be used to create the pieces. In this case they should be glued to the squares. Now printing different pieces isn't neccessary, you only need one square shape.

At last I added truncated pyramids with slopes of 45° to both sides of the squares. This way connections with U-clamps between the added parts are possible and we get angles of 90° between the squares. Slots for the clamps seem to be useful, and if there are slots, screws with nuts are also possible.

The slots are not neccessary, if the printed clamps are bigger or if fold back clips are used instead. The fold back clips are easier to apply or to remove in a construction.

Here are two constructions with different physical pieces made of three squares.

Even with a construction plan it isn't easy to make such figures.

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