Theory
Tilings by regular polygons
A tessellation, or tiling, covers the plane with tiles that leave no gaps and do not overlap. Tessella deals with edge-to-edge tilings by regular polygons in which every vertex looks the same.
Such a tiling is named by its vertex configuration: the number of sides of each polygon around a vertex, in order. In 4.8.8 every vertex is surrounded by a square and two octagons.
The polygons around a vertex must add up to a full turn. A regular polygon with $n$ sides has an interior angle of
\[\frac{(n - 2) \cdot 180^\circ}{n}\]so a tiling with a single kind of tile needs that angle to divide 360°. Only three polygons qualify, and they give the three regular tilings:
| Polygon | Interior angle | Tiles at a vertex | Vertex configuration |
|---|---|---|---|
| Triangle | 60° | 6 | 3.3.3.3.3.3 |
| Square | 90° | 4 | 4.4.4.4 |
| Hexagon | 120° | 3 | 6.6.6 |
Allowing more than one kind of polygon, while still requiring every vertex to be the same, gives eight more: the semi-regular, or Archimedean, tilings.
Status
| Vertex configuration | Polygons | Status |
|---|---|---|
| 3.3.3.3.3.3 | triangles | Implemented |
| 4.4.4.4 | squares | Implemented |
| 6.6.6 | hexagons | Implemented |
| 4.8.8 | squares, octagons | Implemented, rectangular layout only |
| 3.12.12 | triangles, dodecagons | Planned |
| 4.6.12 | squares, hexagons, dodecagons | Planned |
| 3.6.3.6 | triangles, hexagons | Planned |
| 3.4.6.4 | triangles, squares, hexagons | Planned |
| 3.3.3.3.6 | triangles, hexagons | Planned |
| 3.3.3.4.4 | triangles, squares | Planned |
| 3.3.4.3.4 | triangles, squares | Planned |
Each planned tiling has a placeholder file under
src/semi_regular/TODO/. They hold a
header comment and no code, and tess.scad does not import them.
Derivations
- Hexagons: the spacing of the hexagon layouts and the tile count of the radial one.
- Octagons: the octagon dimensions behind the 4.8.8 spacing.
References
- https://jwilson.coe.uga.edu/EMT668/EMAT6680.2001/Hembree/tessellations/hingedtessellations.html
- https://e.math.cornell.edu/people/belk/projects/LaraMerling.pdf
- https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_regular_polygons#Archimedean,_uniform_or_semiregular_tilings