Polyforms follow more or less a common priciple. They are derived from a few rules from very simple basic forms. This is where the name Polyform comes from. You find a comprehensive collection at [PolyPages]. Once the rules for a polyform are defined, the next step is to determine how many different tiles can be made. The resulting tile set is the basis of the related puzzle. In the simplest case each tile occurs only once, but the are useful puzzles with multiple tiles too (e.g. [Logelium] DiDom).
It's usually assumed, that the pieces tile the area of a given form. But there are other kinds as we will see. I found a hint in [Vicher] Puzzle Pages showing a puzzle named Rounded Polyominoes. To me the name is a bit misleading. The tiles resemble of course Pentomino, but their behavior is completely different. The area of the form cannot be covered totally, but has small holes (!ups!).
Some time ago I experimented with tiles of squares, which are only connected by a vertex or edge. This results in partial overlapping of the tiles. I wasn't exited about the result, because the number of possible solutions increased extremely. This kind of tiles is known by the name Pseudo Polyforms, but there is not much attention to it (see [Esser] Polyform Pages).
So it seems natural to prohibit overlaps by an extra rule. This way you get a new and different class of puzzles which I named Bridged Polyforms.
There are 22 different bridged Tetromino tiles. Four squares are connected at least with an edge or vertex. The vertex connections require a bridge.
All tiles can be placed into an 8*11 unit rectangle. Because the Logelium allows various presentation styles of the solutions, an interesting series follows:
This style is nearest to the version of Miroslav Vicher. You can still observe the little holes which are left between the tiles.
The total number of the solutions is (still) unknown.
Here the bridges between the main cells of the figures are clearly visible. The decomposition into octagons and squares for bridges models the overlap restriction rule.
All three styles are based on the same solution.
This presentation style is not at all a tiling of the area, but transforms the puzzle into nodes and connectors. It looks pretty similar to stick puzzle types.
You can find more of these forms at [Logelium] Bridged Tetrominoes.
What makes me really exited about the bridged tiles, is that the principle can be applied to many other basic forms. This way you get a whole family of bridged polyforms.
We quickly start with equilateral triangles. With the bridge connections applied, we get eleven different tiles. Except of the mouse holes they fill an area of 33 units. The nice diamond below has exactly the shown two (times six) different solutions.
Many more forms with this elegant tiles you can find at [Logelium] Bridged Triamonds and [Logelium] Onesided Bridged Triamonds.
The number of Bridged Tetramonds jumps immediately to 82, which is already
a large number for puzzle solving.
The form on the left is using only a fraction of the Bridged Tetramonds, but is nice anyway.
There is no symmetric form with the ten bridged DiTans which has any solutions. Five tiles with odd parity can never add up to zero, which is the parity of any symmetric form. We find that the 15 figures of the onesided version can build a rectangle because of the uneven number.
I was a bit more successful experimenting with TriTans. With a lot of computing hours and some luck I managed to bring all 72 bridged TriTans into a rectangle form. Each tile consists of three tan shapes, which are connected by edges or vertices. The holes in the resulting arrangement are of various shapes, where each one reflects another type of overlap restriction.
The rectangle below consists of 6x9 squares of size √2 and is rotated by 45°.
You can find more forms with solutions at [Logelium] Bridged TriTan.
A subset of the bridged TetraTan are all tiles made of one square and two tans. This results in 94 different bridged DiTan+Mino. The picture shows a rectangle form of all these tiles.
An even more restricted subset consists of all tiles with a square and two butterfly like wings connected at least with one vertex point. The 50 tiles would fill 10 by 10 units, but the tile set is of uneven parity too. The picture below shows consequently an imperfect square.
The bridging principle is not exhausted with these puzzles, but it's getting more complicated to find more use cases.