Perfect tiling

A perfect tiling is the tessellation of a portion of the Euclidean plane in which all tiles are similar to each other and to the entire portion, but no tile is congruent with any other.

Triangle tiling

Every right triangle, but not an isosceles triangle, can be perfectly tiled with two tiles (Figure 1). Other triangles can also be perfectly tiled. An example with six tiles is shown in Figure 2.

Hexagon tiling and the golden ratio

There are only a few types of polygons with at most six sides that allow for perfect tiling. The only polygons of this type known to date, besides the aforementioned right-angled non-isosceles triangles, are special hexagons subdivided into two tiles each, in which the aspect ratio is defined by $s=\sqrt{\mathrm{\Phi }}$, i.e., $s^2=\mathrm{\Phi }$. The aspect ratios $\overline{BC}:\overline{HK}$, $\overline{KB}:\overline{DC}$, $\overline{FE}:\overline{AK}$, $\overline{AF}:\overline{EG}$, $\overline{AB}:\overline{FE}$ and $\overline{AF}:\overline{BC}$ form the golden ratio (Figure 3).

For this reason, and because of their special shape, these hexagons are also called the "golden b".

Geometric illustrations

• Figure 1
  Figure 1
• Figure 2
  Figure 2
• Figure 3
  Figure 3

References

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