The name fudgeflake has been applied to the directly self-similar cyclic trihextal. I extend the name to include the corresponding inversely self-similar tile, distinguishing them as the cis-fudgeflake and the trans-fudgeflake.

The elements of the above figures can be considered as centred on 3 points
of a hexagonal grid (e.g. at the Eisenstein integers `e _{00}`,

I conjecture that these two series of tiles have a countably infinite number of members, as there is no obvious reason why the construction should stop working at any point.

These fractals share the following properties

- each figure contains
`3n`elements^{2} - each figure tiles the plane
- each figure has a similarity dimension of
`2` - each figure has
`c`-symmetry_{3} - 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 are contracted by a factor of
`√(3n`^{2}) - the elements are rotated by 30°
- the boundaries have dimension
`log(2n)/log(√(3n`^{2}))

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

These obviously tile the plane with one copy in the unit cell. With the
above orientations and placement of elements on a unit grid the tiling vectors
are `e _{n0} + e_{nn}` and

**Source:**

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

The fudgeflake can be traced back to C. Davis & D. Knuth, Number representations and dragon curves. J. of Recreational Mathematics 3: 66–81 & 133–149 (1970), many years before I found it on a systematic investigation of IFSs based on rings of elements. I must have seen it in Benoit Mandelbrot's The Fractal Geometry of Nature several years earlier as well.

**References:**

- Famous
Fractals: Fudgeflake at ThinkQuest: geometric construction of
boundary of
*cis-fudgeflake*- could be implemented as 1st order IFS or L-System. - Turtle Program for Fudgeflake, inter alia from G. Edgar:
- Fraktaltechnik
(Fudgeflake): recursive geometric construction for
*cis-fudgeflake*. - various constructions for the fudgeflake from Larry Riddle
- "India":
*cis-fudgeflake*, by William Gosper - "continuum":
*trans-fudgeflake*, by William Gosper

© 2000, 2017, 2018 Stewart R. Hinsley