The origin of the term drafter comes from the wooden or plastic triangles used for technical drafting (see [Wolfram] Drafter). The times of drafting that way are vanished completely into history. A drafter is a symmetrically cut equilateral triangle, also known by the name 90-60-30 triangle. Naturally drafters live in triangle grids. Not really surprising that you can construct polyforms with the drafter shapes. A short survey can be found at [MathPuzzle] Eternity by Ed Pegg, solutions of convex forms with TriDrafter at [Vicher] TriDrafter by Miroslav Vicher.
The number of DiDrafter built from two basic elements is six at first sight, and if we have a closer look there are seven more. These are classified “rejected” (unusable) on the MathPuzzle page above. The seven more are not at all useless, but only have the property to break the triangle grid, where the other ones are grid conform. The grid jumps make the resulting puzzle on the contrary more interesting and complex. The addition “extended” is used to distinguish the tile set from the standard version.
This picture shows the seven additional and non grid conform DiDrafter tiles with related triangle grid elements:
The box shaped tile is a special case. This tile can be viewed as grid conform or as dragon tile depending on how you look at it. The ring structure created by the box tiles are marked hatched in the picture further below. This reflects the ambivalent behaviour of the tiles.
The other tiles conform to the grid.
It looks unlikely that forms can be filled with so many dragon tiles, each one creating a grid jump, but actally many solutions exist. The DiDrafter tiles have a parity problem, so the following example uses a doubled tile set.
There are seven dragon tiles, the naming of tiles breaking the grid structure. This effect leads to kite rings (see also [Logelium] DiDom Kite Rings ). The difference to the DiDom kites rings is that there is no grid rotation, but only a grid shift by half a unit. Nevertheless there are a lot of similarities.
To make the structure of the grid jumps more transparent, the following picture shows the same solution as above with different colours of the grid sections. You can see six kites rings, two of them are cascaded.
Different from DiDom dragon rings there is only a finite number of grid orientations. The base grid can enclose three other grid orientations, these are marked light yellow, red, green and blue. More are stacked inside the rings and have no direct connection to the others. All in all there are twelve different possible grid orientations. Please note that each ring consists of one or more stripes and is therefore directed. This phaenomenon is much better visible in the [Logelium] TriDrafter example.
Surprisingly grid shifts can occur without any dragon tiles. The solutions shown at [Vicher] TriDrafter contain grid shifts, although only grid conform TriDrafters have been used. This shows that the PolyDrafter extension with the dragon tiles is quite natural, because you can't escape the grid shifts anyway. Another approach of creating the extended tiles is cutting areas out of the pictures below that contain both yellow and green triangles. The extension of the PolyDrafter tile set is a closure which makes the set complete.
Having a closer look on drafters it becomes visible that the grid shift are not exceptions but rather the regular case. (see also [Logelium] TriDrafter).
 |
The tiles of the famous Eternity puzzle consist partially of drafter shapes and therefore can create dragon rings as well. This effect was discussed in the Eternity community by the phrase "against the grain", but this is a bit sloppy, because it sounds like only two possible orientations. The rings are more rare in this case, because the tiles are made mainly of triangles and some drafter. None of the Eternity tiles cross a grid border. If we consider dragon rings when finding solutions, the search methods get considerably more complex. On the other hand we will possibly loose solutions by restricting ourselves to only grid conform ones. |
More examples of Eternity dragon rings are shown at [Vicher] Eternity.