Quick start

One possibility to navigate the site is to use the search box on top of each page (right). Just type in a term, and a list of local sites containing this term shows up. Choose the one from the list which seems to fit.
E.g., if you look for the Ammann-Beenker tiling, you may type 'Ammann' in the search box. This yields a list with some links to tilings, one of them being the Ammann-Beenker tiling. You will also see a link to 'Ammann bars', leading to the explanation of this term in the glossary.

Navigation and content

Substitution tiling sites

Currently this site contains a collection of substitution tilings. To show a list of all present tilings, click 'Substitutions' on top of each page (right). Each substitution tiling has its own page. This page contains an image of the substitution rule, an image of a large patch, usually some additional information and references, as well as a classification by some properties of the present tiling (e.g., is the tiling flc, or is it a model set?). These properties are listed on top of the page. If you click on one of the properties (e.g., known matching rules) a list with all tilings sharing this property will show up.
The images of the patches are in some bitmap format, usually gif or png, optimized for being viewed in a browser. Sometimes there are also ps or eps files available. If so, this is indicated by a line 'Patch in vector format' below the patch image. Click this line to open or download the file. Then you should be able to scale it according to your preferences, without loss of quality.

Glossary

On top of the page you see also a link 'Glossary'. Following this link shows a collection of terms which are relevant in the mathematical theory of substitution tilings. Throughout the sites, a lot of terms are linked to the glossary, such that one can quickly access the explanation of each term. To give an example, here follows a link to the definition of repetitive.

Taxonomy

The present tilings are classified according to their properties. E.g., under known matching rules we list all the tilings of which we know matching rules to generate them.
For a list of all categories, see taxonomy.

About us

Nonperiodic substitution tilings turned out to be (or give rise to) interesting objects in many fields of mathematics, be it geometry, dynamical systems, topology, combinatorics, number theory, algebra, mathematical physics or statistical mechanics. Nevertheless, there is no comprehensive list of these tilings, and no common terminology. This is a pity. Many tilings are 'discovered' more than once, several equal concepts are known under different names, sometimes the same term has different meanings.
We - E. Harriss and D. Frettlöh - decided to help to fill this gap. For a start, we made several one- and two-dimensional substitution tilings available at this place in a consistent form, together with some relevant information about each one, and we will continuously add more. We apologize that we can not be as precise as a scientific journal, and that we can not list all known substitution tilings; the latter simply because there are infinitely many of them. Nevertheless, we welcome any suggestion.