About the Tiling Project

This a simple method for generating radial tilings using dart-rhombus tile sets, with 3-fold or higher rotational symmetry. These tile sets will tessellate a plane with no gaps or overlaps. The method can generate tilings with 3-fold or greater symmetry. The tilings are non-periodic, but the tile sets are not aperiodic. The same tile sets could be used to create tilings with translational symmetry, though each example I show is a tiling with rotational symmetry. This is an edge-to-edge tessellation in that adjacent tiles share only full sides.

In these examples, I have superimposed an image on each tile, but I show diagrams below without the decoration. Sometimes the image is programmatically varied from tile to tile. In these cases the tile with its marking is no longer a true tiling in the mathematical sense, though the underlying structure is. The marking or decoration of the tiles, other than by systematic coloring, often obscures the structure. Infinitely many interesting tilings are possible without this obfuscation, but I'm less interested in the math than I am in inventing a quasi natural image.

The method I use to generate the structure is a programmed version of the following. Starting from a center point, divide 360 degrees into three or more equal angles using congruent rhombi tiles. Each tile will share a side starting at the center point and terminating at a vertex. At each vertex divide the angle between adjacent rhombi into six equal angles using five lines. The first and last of each set of lines will intersect forming one dart tile adjacent to each rhombi. Use four of these dart tiles to fill each of the four remaining angles between the darts created by intersecting lines. Four dart points each at the rhombi vertices will exactly fill the space between the darts adjacent to each rhombi.

Add an outer row of rhombi tile to each dart side away from the center point. Add a dart to each outer rhombi extending away from the center point. The angles between each of these darts will fit exactly one or two dart tiles. Fill each of these angles between the darts with dart tiles. Continue adding rows of rhombi and darts in this same manner.

The method works for any order of symmetry, 3-fold or higher. The tilings generated have four different vertices, regardless of the symmetry order. Though each tiling is non-periodic, lacking translational symmetry, they contain isolated regions that exhibit translational symmetry.

The interior angles for the dart and rhombus for each of the symmetry orders can be easily calculated. For 5-fold symmetry, the rhombus angles are 72 and 108; the dart angles are 24 and 288.

Similar tilings using the same tile sets can be produced by starting with darts in the center. When a tiling is created with darts in the center from the tile set with rhombi in the center, then the order of symmetry is three times that of the rhombi-centric tilings. These tilings have three different vertices, regardless of the symmetry order.

These dart-centric tilings can be simplified from a dart-rhombus to a single triangle tiling by removing the rhombi and converting each dart to a triangle. Likewise, a dart-rhombus-dart can be converted to a rhombus.