Fudgeflakes

The cis- and trans-z-30-cyclotrimers are self-similar fractal figures with a similarity dimension of 2, and a fractal boundary. They are composed of 3 parts arranged in a triangle, each part being rotated by 30° with respect of the figure, and either directly (cis-) or inversely (trans-) similar to the figure.

The dimension of the boundary of the fudgeflake is given at ThinkQuest as log(2)/log(3^½).

These are the first members of an uncountably infinite set of fractals. As I have encountered the term fudgeflake used to describe (the boundary of) the cis-z-30-cyclotrimer, I appropriate this term to cover all members of this set, the above figures becoming the first order fudgeflakes.

The transz-30-cyclotrimer is a special case of Gosper's continuum frac-3-tile.

The parts of the above figures can be considered as centred on the centers of an array of 3 hexagons. Additional fractals are found when additional parts are added corresponding to the centres of additional rings of hexagons surrounding these 3 hexagons. This is equivalent to placing the parts on the vertices of n,n+1-sided hexagonal array.

This results in 2 countably infinite series of fractals – the cis- and trans- nth order fudgeflakes.

These fractals share the following properties

• each figure tiles the plane
• each figure has a similarity dimension of 2
• each figure has c3-symmetry
• each figure is topologically equivalent to a circle, i.e. it is connected and there are no voids
• there are cis- and trans- variants for each n
• the elements rotated by 30°
• the boundaries have dimension log(2n)/log(sqr(3n²))
• each figure contains 3n² elements

As n tends to infinity the attractor approaches a regular hexagon.

In some cases additional fractals can be generated by 'replacing' each element of a fractal with a 'copy' of a second fractal. (This can always be done with a 'copy' of the first fractal, but in this case the result is the original fractal, as is to be expected from the self-similarity property.) Some care, or experimentation, is required to get the scale, position and orientation correct. However if the case of the fudgeflakes this technique is particularly easy and productive. The scale factor is found to be 1/n for the nth order fudgeflakes.

Denoting the 1st order cis-fudgeflakeas F, and the 1st order trans-fudgeflake as f, we can recursively combine these to form a uncountably infinite set. (This cardinality can be demonstrated by a diagonal argument analogous to that that proves the uncountability of R.)

Combining 2 units gives 2 new fractals, corresponding to F.f and f.F (F.F and f.f are the same as F and f).

Combining 3 units gives 6 new fractals (F.F.F and f.f.f are the same as F and f).

The number of fractals increases with increasing number of units.

Fudgeflakes of different orders can be combined in the same fashion. If we denote the nth order cis-fudgeflake as F_n, and the nth order trans-fudgeflake as f_n then we can have such fractals as F₁.F₃ and F₃.f₂. As n approaches infinity that attractor for F_n.F₁ approaches a regular terhexagon. Similarly the attractor for F_n.F₂ approaches a dodecterhexagon.

Source: All are independently discovered, although I have since found copies of the 1st order cis- and trans-fudgeflakes on the web.

References:

1. Famous Fractals: Fudgeflake at ThinkQuest: geometric construction of boundary of cis-fudgeflake - could be implemented as 1st order IFS or L-System.
2. Turtle Program for Fudgeflake, inter alia from G. Edgar:
3. Fraktaltechnik (Fudgeflake): recursive geometric construction for cis-fudgeflake.
4. "India": cis-fudgeflake
5. "continuum": trans-fudgeflake.

© 2000 Stewart R. Hinsley