R. William Gosper presents a continously varying set of fractal rep-3 tiles with a single degree of freedom. This set is characterised by each element being unrotated with respect to the overall figure, but reflected in a line with respect to the overall axis. Thus the transforms for the members of the set are composed of a 3½-fold contractions towards an arbitrary point, reflection in an arbitrary axis, and one of 3 translations. It is necessary to identify the vectors corresponding to the translations.
We may arbitrarily select the point to which the contraction is performed as the origin, and the reflection axis as the Y-axis. The points corresponding to the 3 vectors must correspond to a triangle, and we can arbitrarily fix the length of one size of the triangle as 1. Conjecturing that we can place one vertex at the origin, we can then arbitrarily fix the 2nd point at (2-½, 2-½). Further, conjecturing that the set contains a c3-symmetric member, we can deduce from geometrical considerations that the 3rd point and corresponding vector is at (2-½-cos15°, cos15°). The validity of these conjectures is confirmed by experimental plotting of the attractor of the resultant IFS.
Rotating this figure by 105° we can see that this figure (on the left) is identical to the trans-fudgeflake (on the right).
Because of the cyclic symmetry, the same attractor is generated when the elements a rotation by 120° or 240° degrees is added to the transforms. Rotation by 60°, 180° or 300° generates a mirror image, as does using vectors (0,0), (-2-½, 2-½) and (cos15°-2-½, cos15°).
The attractor can be stretched perpendicular to the axis of reflection, while retaining the property of self-similarity. If the axis is the Y-axis, then the vectors become (0,0), (2-½.x, 2-½) and ((2-½-cos15°).x, cos15°), with non-zero x. This procedure only results in a rep-tile if the elements are unrotated, or rotated by 180°.
Equivalently, the attractor can be stretched parallel to the axis of reflection, while retaining the property of self-similarity. If the axis is the Y-axis, then the vectors become (0,0), (2-½, 2-½.y) and (2-½-cos15°), y.cos15°), with non-zero y.
The rep-9 implementation of the trans-fudgeflake has 2 degrees of freedom. However this does not necessarily imply that the figure has a 2nd degree of freedom: the rep-3² parallelogram also has 2 degrees of freedom, but the rep-3 parallelogram only one. However I do not have a proof that there is only a single degree of freedom.
Source: Reverse engineered from images provided by R. William Gosper.
References:
© 2001 Stewart R. Hinsley