When you really want to tile more than one layer but triple tiling is just too much of a good thing, surely the happy medium is double tiling.
- How may a mosaic of more than 900 sections be double tiled with congruent triangles along the 6 guidelines listed below?
Here are two examples of double tiling with congruent triangles. The first example demonstrates most guidelines of this puzzle while the second follows the toughest guidelines as well.
In the first example, eight overlapping 26.6°- 63.4°- 90° triangles double tile a square mosaic of 15 sections where:
“Double tiling” is taken to mean that every section of a mosaic is completely covered by portions of exactly two tiles and that all tiles lay completely within that mosaic.
The tiles are congruent triangles.
Each tile is uniquely oriented.
The mosaic is edge-contiguous in that all sections may be visited along a single unbroken path that stays within the mosaic while crossing tile edges from section to section without touching any vertex.
In the second example, four overlapping congruent 30°- 60°- 90° triangles double tile a triangular mosaic of 4 sections where, moreover:
Every angle is a whole number of degrees.
No distinct lines are parallel. (Parallel tile edges may, however, lie along a single continuous line.)
Bounty challenges, achievability unknown
Double tile a mosaic other than the second example above that follows all 6 guidelines and has no holes.
Double tile a mosaic that follows all 6 guidelines and whose outline is not bilaterally symmetric.
(All interesting double tilings, including those with fewer than 901 sections and/or those that disregard some of the above guidelines, deserve votes of approval.)