The kites we talk about are diagonally symmetric Tetragons. You can find a fine presentation of their properties in [Math] Mathematische Basteleien (German). The kites are always based on a triangle, which is mirrored and combined with the original one. In German language kite and dragon is the same word (Drachen), so you will find some hints to dragon related stuff. A very simple kite is constructed from half a regular triangle, also known as 30°-60°-90° triangle through mirroring. The ratio of the sides of the triangle is 1:2:√3.
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The kite is called simple, because it is conform to the triangular base grid in any position. The vertex points of the kites always fit onto vertex or middle points of the grid, if you combine them by edges. The 15 different tiles made from five 30°-60°-90° triangle kites build an interesting puzzle. The picture above shows how the tiles fit onto the grid. |
More complex kites appear from rectangular triangles with the edge ratio 1:2:√5, if you connect the long edges. I found the idea first in [Puzzle] Zucca , where tiles made of such triangles are named PolyDoms. At the same place you find a description, how the vertex points of the kites leave the base grid and switch to another grid tilted by an λ = atan(¾) angle. This leads to various complications, which other puzzles don't have. Nevertheless you can fill interesting forms with PolyDoms, but the systematic calculation of all solutions requires more thought. The 13 DiDoms constructed of two of such triangles will be subject for study through the following pages. It is possible to fill a twelve vertex form with a double set of DiDoms. A first solution is shown in [2] and inspired me to have a closer look.
The two pictures show all possible areas for the tilted grid completely enclosed by the base grid. You can see that all areas are combinations of segments with five units. The segments1 to 6 or 7 to 10 are compatible with each other.
To find systematically all solutions of forms with Dom based tiles I use a stepwise approach, like a multistage rocket. The method works in principle for all puzzles with tiles inducing grid jumps.
| 1 | At first we have to find all possible by the angle λ or -λ tilted secondary grids, which can be embedded in the basic form. The smallest segments are of area 5, if completely enclosed. Which segments are useful at all depends of the type of figures of the puzzle. There can be many disconnected secondary grids, if the primary area is large enough. They can even lay at the border of the form and also be incomplete. In this case there are no rings anymore but chains. The end points of the chains match the intersection of the outer form with the grid borders. |
| 2 | If a secondary grid is large enough to contain another ring, we can take it as primary grid an repeat the first step. This process can be continued recursively and ends always, because the inner grids get smaller and smaller. This way we have a tree of kite ring structures, where each leaf is a starting point for the Logelium solver. Each kite ring or chain is tilted by the angle n * λ against the base grid, where n is a positive or negative integer. |
| 3 | The positioning of the kites is crucial for the solver. Kites can only live on the edges, which separate two different grids. The kites of a ring have furthermore always the same orientation, i.e. the acute vertices point altogether either clockwise or against. Each ring must be looked at separately. All tiles which contain no kites must live completely in one of the partial grids, only kites cross grid borders. |
| 4 | Now we decompose the form by removing the kites. Each partial grid, which is now separate, can be rotated to conform to the base grid. After this normalisation the problem will be solved with the standard solver. By recomposing all parts again we get the complete solutions of a dedicated ring constellation. |
On the page DiDom Kite Rings you find a variety of ring structures with DiDoms which illustrate the method.
The structure of the grids and rings is pretty simple in case of a dodecagon.
ad 1: The points of the tilted grid are marked blue for better understanding. If we recognize the eightfold
symmetry there are ten fitting segments. Segment 4 and 5 are not useful, because they cut off areas, which are impossible
to tile with DiDoms. Because the area of all DiDoms is two, we can only use even numbers of segments, i.e. segment 1+3 or
1+6 or 2+3. The segments 1+2+3+6 cut off an area with an odd number of cells.
The segement pairs 7+8, 7+9, 8+10 and 9+10 are symmetry redundant, so there is only one left. The combination 7+8+9+10 is
the only possible with four segments.
ad 2: There are 10 different kite positions with segments 1+3, the other segment pairs have 15.
For the following position of kites there are exactly two solutions which are mostly equal.
There are no solutions for all other 9 possible positions of kites with the same segments.
These kite positions with the segments 3+6 have 96 solutions, and all share the same inner part.
All other combinations of segments have no solutions. This completes the search for solutions with DiDom tiles for this form.
