Cyclomer Teardrops

Introduction

Teardrop fractals are a subset of grouped element fractals, independently discovered by myself and R. William Gosper. Teardrop fractals are derivable from cyclically symmetric fractals with a central element. This page covers those teardrop fractals derived from the cyclomer-monomers. All teardrop fractals tile the plane. All teardrop fractals derived from the cyclomer-monomers are rep-tiles.

The homeolineotrimers are also cyclodimer-monomers, and hence have corresponding teardrops. The homeolineomers are two-fold degenerate, as the same figure can be obtained by 180°, thus giving potentially 2 teardrops for each homeolineoner. In practice the doesn't result in a 2^nd trans-homeolineotrimer (rep-3 parallelogram) teardrop, as this, like the trans-homeolineotrimer is continuously variable with a single degree of freedom, and both figures below to the same continously variable set. The z₀-homeolineotrimer has a greater degree of degeneracy, but the derived teardrops are equally degenerate.

Image	Unit Cell	Base Fractal	Min. # in Unit Cell
		trans-homeolineotrimer	2
		z0-homeolineotrimer	2
		z1-homeolineodimer	2
		z2-homeolineodimer	2
		z2-homeolineodimer	2

Each of the cyclotrimer-monomers are 6-fold degenerate, as they have 3-fold rotational symmetry plus reflectional symmetry. The resulting figures also contain show a considerable degree of degeneracy, and there are a total of 10 distinct simply connected teardrops, plus 2 disconnected figures.

Image	Unit Cell	Base Fractal	Min. # in Unit Cell
		0-cyclotrimer-0-monomer	3
		0-cyclotrimer-120'-monomer	3
		0-cyclotrimer-60-monomer	3
		0-cyclotrimer-60'-monomer	3
		0-cyclotrimer-180-monomer	3
		60-cyclotrimer-60-monomer	3
		60-cyclotrimer-60'-monomer	3
		60-cyclotrimer-180-monomer	3
		60-cyclotrimer-120'-monomer	6
		60-cyclotrimer-0-monomer	6

The 2 cyclotetramermonomers are 4-fold degenerate. There are 6 teardrops, the other two figures not being simply connected.

Image	Unit Cell	Base Fractal	Min. # in Unit Cell
		cis-0-cyclotetramermonomer	4
		trans-0-cyclotetramermonomer	4
		trans-90-cyclotetramermonomer	4
		trans-180-cyclotetramermonomer	4
		cis-270-cyclotetramermonomer	4
		trans-270-cyclotetramermonomer	4

The cyclohexamermonomers are 6-fold degenerate. However of the 12 possible figures only 4 are simply connected teardrops.

Image	Unit Cell	Base Fractal	Min. # in Unit Cell
		cis-0-cyclohexamermonomer	6
		trans-0-cyclohexamermonomer	6
		cis-300-cyclohexamermonomer	6
		rans-300-cyclohexamermonomer	6

Related fractals - homeolineomers, cyclomers

Sources: All of these were independently invented. However a considerable propertion have also been obtained by R. William Gosper.

data file teardrop.xil; generator gef.pl; images generated at 6x5 per screen at 1280x1024

References:

• R. William Gosper's rep-3 countablegon (trans-homeolineomer teardrop)
• R. William Gosper's teardrop frac-n tiles (cis-tetracyclomer-monomers)
• R. William Gosper's Frac-4-tiles from complex radices 2 and -2, digits 0, 1, 1^1/3, 1^2/3 (60-cyclotrimer-60-monomer)
• R. William Gosper's teardrop Frac-7-tile (cis-cyclohexamermonomer)