Metatrapezia

The demihexagon is an isosceles trapezium (trapezoid in American usage) in which three of the sides have the same length. The IFS which generates is deduced from a L-system that I saw somewhere on the web. Any trapezium in which the ratio of the parallel sides is 2:1 is the self-affine attractor of an IFS.

The demihexagon is the triamond.

The meta-figure technique can be used to create additional attractors which retain the properties of having a similarity dimension of 2, and being self-similar. The demihexagon has a 4-component IFS, which would normally give 14 different 1st level meta-figures.

The meta-figures are labelled according to the affine transformations replaced according to the meta-figure technique, the transform corresponding to the base (long edge) of the demihexagon being labelled A, that corresponding to the top B, and those corresponding to the sides C and D.

D-metatrapezia are mirror images of C-metatrapezia; A,D-metatrapezia are mirror images of A,C-metatrapezia; B,D-metatrapezia are mirror images of B,C-metatrapezia; and A,B,D-metatrapezia are mirror images of A,B,C-metatrapezia. Thus the number of dissimilar meta-figures is reduced to 10.

The demihexagon is degenerate - each of the 4 components of the demihexagon has a single axis of symmetry, so replacing a transform with the an equivalent transform with a reflection in that access generates the same attractor. However, applying the meta-figure technique to the different sets of transforms does not necessarily generate the same result. There are 16 possible base IFSs which generate the demihexagon, so there could be up to 160 dissimilar metatrapezia. However the effect of symmetry reduces this to 24. There are 2 C-metatrapezia, 4 A,C-metatrapezia, 4 B,C-metatrapezia and 8 A,B,C-metatrapezia. The additional 14 attractors are called isometatrapezia, and are labelled by the components which are reflected in their symmetry axis, e.g. C-iso-C-metatrapezium.

Of the 24 metatrapezia, 20 are singly connected; the exceptions are the A-, A,C-, A,C-iso-A,C- and A-iso-A,B,C-metatrapezia.

I have allocated trivial names to some of the resulting metatrapezia.

• demi-ring metatrapezium (A-metatrapezium)
• peaked metatrapezium (B-metatrapezium)
• eagle metatrapezium (A,B-metatrapezium)
• triangle metatrapezia (B,C-metatrapezia)
• anvil metatrapezium (C,D-metatrapezium)
• star metatrapezium (B,C,D-metatrapezium)

There are 4 different component layouts which generate the triangle metatrapezium, which is the 30° right-angled triangle. There are several other component layouts generating the 30° right-angled triangle which do not arise from the application of the meta-figure technique to the demihexagon.

The star metatrapezium has 5 pairs of components with hexagonal symmetry (D₆), and hence there are 12⁵ different IFSs which generate the same attractor, but with different component orientations.