These are IFSs in which the elements are laid out in circles. Dimers could be considered as cyclomers, but are more conveniently considered as lineomers.
Excluding dimers, there are 11 known 2 dimensional figures (2 are non-fractal) in this category, some of which are well known fractals. There are 3 pure cyclomers, and 8 cyclomermonomers (in which a central element is added within the circle). Cyclomers are known with more than one ring, but none of these are 2 dimensional. No 2 dimensional figures have been identified in which the elements of a ring do not all have the same orientation.
All of these figures tile the plane.
| Image | Name(s) | Min. # in Unit Cell |
 |
(cis-)fudgeflake cis-(30-)cyclotrimer |
1 |
 |
trans-fudgeflake trans-30-cyclotrimer |
1 |
 |
pseudo-Sierpinski triangle 0-cyclotrimer-0-monomer |
1 |
 |
(z-)60-cyclotrimer-60-monomer | 1 |
 |
equilateral triangle rep-22 triangle 0-cyclotrimer-60-monomer |
2 |
 |
(w-)60-cyclotrimer-0-monomer | 2 |
 |
square rep-22parallelogram (0-)cyclotetramer |
1 |
 |
Mandelbrot quintet Minkowski sausage red cross dragon cis-cruciopentamer cis-(z-)(26.5-)cyclotetramer-(26.5-)monomer |
1 |
 |
trans-cruciopentamer trans-(z-)(26.5-)cyclotetramer-(26.5-)monomer |
1 |
 |
(cis-)flowsnake Peano-Gosper curve cis-(z-)(19.1-)cyclohexamer-(19.1-)monomer |
1 |
 |
trans-(z-)(19.1-)cyclohexamer-(19.1-)monomer | 1 |
As any 2 dimensional linear IFS is a rep-tile, and therefore has a periodic tiling of the plane, it can be deduced that any 2 dimensional cyclomer must have 2-, 3-, 4- or 6-fold symmetry, and therefore a multiple of 2, 3, 4 or 6 members in any ring. With more than 6 members in a ring the central void is too large to be filled by a single element, whilst if there are two rings with more than 1 member in the inner ring it is necessary to modify the outer ring so that its members lie on straight lines rather than arcs.
These constraints means that there are 8 possible layouts for the elements of a cyclomer.
There are 10 known dimers, 4 of which are homeomeric, and which can reasonably be considered as the n=2 members of the cyclo-n-mer series. There are 5 known homeolineotrimers, and several more c2-symmetric heterolineotrimers, which can reasonably be considered as the n=2 members of the cyclo-n-mer-monomer series.
Packing constraints mean that there are no cyclohexamers; the angles between lines joining the corresponding vertices must be 120°, but to fit 6 together at one point requires that these angles be 60°.
This leave 5 layouts. There are 2 cyclotrimers, 1 cyclotetramer, 4 cyclotrimermonomers, 2 cyclotetramermonormers and 2 cyclohexamermonomers, giving the 11 cyclomers displayed above.
There are two 2 dimensional cyclotrimers, the cis-z-30-cyclotrimer, and the trans-z-30-cyclotrimer. I have encountered the former (and its boundary) on the web, the boundary being known being labelled as the fudgeflake. I am generalising this term to cover two series of À0 2 dimensional fractals of which these are the first members.
The trans-z-30-cyclotrimer is a special case of Gosper's continuum frac-3-tile.
There are two 2 dimensional cyclotetramermonomers. These are the first members of two series of À0 2 dimensional fractals with an approximate cross shape, which converge on a cross as n tends to infinity. I introduce the term cruciomer for these series, with the alternative names of first order cross fractal and cruciopentamer for these fractals.
The exact value of the rotation of their elements is arctan(0.5).
There are two 2 dimensional cyclohexamermonomers, the cis- and trans-19.1-cyclohexamermonomers. The former is well known as the flowsnake or Peano-Gosper curve. These are the first members of two series of À0 2 dimensional fractals with an approximately hexagonal shape, which converge on a hexagon as n tends to infinity. I generalise the term flowsnake for these series
The exact value of the rotation of their elements is arctan(sin(60)/2.5), or 19.106...°.
Both cyclotrimers tile the plane with one copy per unit cell.
The pseudo-Sierpinksi triangle and z-60-cyclotrimer-60-monomer tile the plane, with one copy per unit cell.
The equilateral triangle obviously tiles the plane. The equilateral triangle is the special, symmetric, case of the 22triangle, which is the first of an infinite series of IFSs with n2elements, all of which share the same attractor. The unit cell consists of 2 copies, forming a rhombus, each with an inverse orientation with respect to the other.
The w-60-cyclotrimer-0-monomer also tiles the plane. The unit cell agains consists of two copies, each inverted with respect to the other. However in this case the 2 copies have a common centre, rather than a common edge as is the case for the equilateral triangle.
The 0-cyclotetramer or square obviously tiles the plane, with one copy per unit cell. This is the special, symmetric, case of the 22 parallelogram, which is the first of an infinite series of IFSs with n2 elements, all of which share the same attractor.
All the cyclotetramermonomersand cyclohexamermonomers tile the plane with one copy per unit cell.
Conjecture: that these are the complete set of 2 dimensional cyclomers and cyclomermonomers.
Having on several occasions discovered additional 2 dimensional fractals within a class I thought I had already exhaustively catalogued I am not confident that there are no more cyclomersand cyclomermononers.
Related fractals - homeodimers, hextals, fractominoes, fudgeflakes, cruciomers, flowsnakes
Sources: The cis-cyclotetramermonomer(aka cis-cruciopentameraka cross fractal aka Mandelbrot quintet) was reverse-engineered from an image and recursive construction found on the web. The trans-cruciopentamer immediately follows by application of the trans technique. The remainder, with the possible exception of the square, were independently discovered, except that the analytical value for the rotation of the flowsnake was calculated from a L-system construction published in Scientific American.
References:
© 2000 Stewart R. Hinsley