Tiling submitted by Andrew Hudson.

The scaling factor is $\frac{\sin(\frac{3\pi}{n})}{\sin(\frac{\pi}{n})}+\frac{\sin(\frac{2\pi}{n})}{\sin(\frac{\pi}{n})}$.

All tiles appear in one chirality only, so markings were omitted.

8-fold cyclotomic aperiodic tiling with dense tile orientations (CAST DTO).

A polygonal version of Kenyon 2. The edges are generated by the morphism: a->b, b->c, c->d, d-> b’a’ (where x’ is the inverse of x).

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings.

Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

In order to generalize Danzer’s 7-fold tiling to n-fold symmetry, where n>5 is odd, L. Danzer and D. Frettlöh introduced trapezoidal tiles, each one the union of two triangles with edge lengths of the form $\sin(k \frac{\pi}{n})$. It needs some further effort, including the introduction of three …

One of several substitution tilings found by L. Andritz using similar right-angled quadrilaterals.

This is the dual partner of Conch, which has more details. The scaaling factor of this rule is either of the (complex conjugate) expanding roots of $x^4 - x^3 + 1 = 0$.

A substitution rule shown on R. Kenyon’s homepage: http://www.math.brown.edu/~rkenyon/gallery/gallery.html with inflation factor that satisfies: $x^4+x+1 = 0$.

A volume hierarchic version of Conch.

This tiling and Nautilus are dual tilings generated by non-PV morphisms. As such they are the first step in a generalisation of the work of G. Rauzy, P. Arnoux, S. Ito and others for PV substitution rules. The work that developed out of G. Rauzy’s seminal paper [Rau82] .

The inflation factor for …

One manifestation of the famous Penrose tilings. In fact, this is the first manifestation found by Penrose, the Penrose rhomb, the Penrose kite-dart and the Robinson triangle tilings are refinements of this one. (You may also click ‘Penrose’ below ‘MLD-class’ above to see the others.) Their …