A symmetrical 5 element rendition of the Rauzy tile can be created.
As this is c2-symmetric, additional fractals can be generated by rotating elements about its centre. As the construction used for the Rauzy tile does not place the centre at the origin, we have to discover and calculate the coordinates of the centre.
The centre is the midpoint of the line connecting the first "triple junctions", i.e. red/blue/gold and green/magenta/gold. Each triple junction is the limit of a spiral. These are defined by the recursions
where T1 consists of contraction by 0.54389012692½, rotation by 235.311°, and translation by the vector (-1,0).
The former converges near the point (-0.59574393,0.25442587) and the latter on the point (0,0). [ Errors in the numerical evaluations of the contraction and rotation will result in errors in the calculation of the convergence of these series. These are not important at screen resolutions. ]
Rotating elements about the centre creates 12 distinct new fractals, as show below. Each occupies half the area of the base Rauzy tile, and in each case 2 copies can be combined to recreate the original Rauzy tile.
Related Fractals: Rauzy tile
Source: Derived mechanically from the Rauzy tile.
© 2001 Stewart R. Hinsley