The Golden Ratio, f=(1 + 5½)/2, is a number which turns up in a number of places in mathematics. The reciprocal of its square root is a root of the polynomial c + c²=1, and hence is a candidate for the contraction ratio of a two element tile. There is indeed such a tile, published by Karl Scherer as the golden bee. (I presume this to be a reference to the letter rather than the insect.)
The golden bee is constructed by as follows: the red component corresponds to contraction by c=1/f½, rotation by 3p/2, and translation by (0, c); the blue component corresponds to contraction by c²=1/f, reflection in the Y-axis, and translation by (1, 0).
The same c is also a root of c + c3 + c4=1 and of 2c2 + c3=1. The tile corresponding to the former is the golden rectangle, which I originally encountered investigating the former equation. However both are conveniently derived mechanically from the golden bee. A 3 part dissection of the golden bee also corresponds to the 3rd polynomial. (There is an alternative 3 part dissection of the golden bee, not shown, corresponding to the 2nd polynomial.)
The same mechanical derivation process can be applied recursively to the golden rectangle. because of the dihedral (d2) symmetry of the golden rectangle there are 192 possible IFSs, which give 10 distinct attractors, 8 of which are connected.
4 attractors correspond to the polynomial c + c3 + c5 + c8 + c9=1, i.e. are dissected into 5 copies, each of which is a different size.
4 attractors correspond to the polynomial c2 + 2c4 + c6 + c7=1.
Thera re two attractors corresponding to the polynomial c2 + c3 + 2c4 + c5=1.
3 Further 5 part attractors can be derived from the 3-part dissection of the golden bee, one corresponding to each of the 3 polynomials above. Of these one is a double golden rectangle, one is a 5-part dissection of the golden rectangle, and the 3rd is disconnected.
Tiling: The golden rectangle and double golden rectangle self-evidently tile the plane with a single copy in the unit cell. Similarly the mechanical 5-element derivatives of the golden rectangle tile the plane with 2 unequally sized copies in the unit cell. The golden bee tiles the plane. Any tiling uses copies of at least 2 sizes. I have not been able to show that there is a periodic tiling. Any periodic tiling of 2 unequal sizes contains at least 8 elements in the unit cell.
Source: The golden bee was (trivially) reverse engineered from an image given on Karl Scherer's pages. The golden rectangle was independently discovered. The remainder are mechanically derived from the preceeding.
References
© 2002 Stewart R. Hinsley