In the case of fudgeflakes composition of appropriately scaled IFSs is a productive means of producing new tiles.

There are potentially 16 order 9 tiles generated from combinations of first
order fudgeflakes (involving the cis- and trans- fudgeflakes and their mirror
tiles). However a composition of an IFS with itself does not produce a new
tile, but instead produces a higher order dissection of the original tile, and
adding mirror tiles to the mix doesn't produce additional tiles (other than
mirror tiles), instead generating the other tiles in the same and different
orientations. Thus there are two novel order 9 tiles. If the cis-flowsnakes are
denoted **F _{n}** and the trans-flowsnakes

Further tiles can be produced by composition of fudgeflakes of different orders, and by composition of composite fudgeflakes with fudgeflakes or other composite fudgeflakes. I have presumed that adding mirror tiles to the mix still doesn't generate further tiles, and haven't investigated their involvement further. Apart from this I conjecture that all cases generate tiles, and the only judgement necessary is to eliminate those which are higher order dissections of lower order tiles.

In addition to the 2 order 9 (3^{2}) tiles there are 6 order 27
(3^{3}) tiles, 8 order 36 (3×12) tiles, 20 order 81 (3^{4} and
3×27) tiles, 24 order 108 (3^{2}×12) tiles, 10 order 144
(12^{2} and 3×48) tiles, 8 order 225 (3×75) tiles, 30 order 243
(3^{5}) tiles, 40 order 324 (12×27 and 3^{3}×12) tiles, and
so on.

The order 27 tiles are **F _{1}.F_{1}.F_{1}**,

The order 36 tiles are **F _{1}.F_{2}**,

The order 81 tiles are **F _{1}.F_{1}.F_{1}.F_{1}**,

The order 108 tiles are **F _{1}.F_{1}.F_{2}**,

The order 144 tiles are **F _{2}.F_{2}**,

The order 225 tiles are **F _{1}.F_{5}**,

The order 243 tiles are all combinations of 1st order fudgeflakes other than
**F _{1}.F_{1}.F_{1}.F_{1}.F_{1}.F_{1}**
and

It is possible to calculate the number of tiles of particular orders based
on 1st order fudgeflakes alone. When `n` 1st order
fudgeflakes are combined there the resulting tiles have `3 ^{n}` elements. The number of IFSs is

compositional order | order (# of elements) of tile | number of tiles |
---|---|---|

2 | 3^{2}
= 9 |
2
(2^{2}-2) |

3 | 3^{3}
= 27 |
6
(2^{3}-2) |

4 | 3^{4}
= 81 |
12
(2^{4}-2^{2}) |

5 | 3^{5}
= 243 |
30
(2^{5}-2) |

6 | 3^{6}
= 729 |
54
(2^{6}-2^{3}-2^{2}+2) |

7 | 3^{7}
= 2187 |
126
(2^{7}-2) |

8 | 3^{8}
= 6561 |
240
(2^{8}-2^{4}) |

9 | 3^{9}
= 19683 |
504
(2^{9}-2^{3}) |

10 | 3^{10}
= 59049 |
970
(2^{10}-2^{5}-2^{2}+2) |

11 | 3^{11}
= 177147 |
2046
(2^{11}-2) |

12 | 3^{12}
= 531441 |
4028
(2^{12}-2^{6}-2^{4}+2^{2}) |

13 | 3^{13}
= 1594323 |
8190
(2^{13}-2) |

14 | 3^{14}
= 4782969 |
16254
(2^{14}-2^{7}-2^{2}+2) |

15 | 3^{15}
= 14348907 |
32730
(2^{15}-2^{5}-2^{3}+2) |

There are 8 tiles based of compositional order 2 for each pair of fudgeflake
orders. For tiles of compositional order 3 there are three cases, depending on
whether 1, 2 or 3 fudgeflake orders are involved. The first is considered
above. The second has 24 tiles (3 permutations of fudgeflake order x
2^{3} selection of cis- versus trans- fudgeflakes), and the 3rd 48
tiles (6 permutations of fudgeflake order x 2^{3} selection of cis-
versus trans- fudgeflakes). Similar calculations can be performed for higher
compositional orders, but in some cases one has to account for the generation
of lower order tiles (e.g.
**F _{1}.F_{2}.F_{1}.F_{2}** does not
generate a new tile).

© 2000, 2017 Stewart R. Hinsley