Origami database

About origamis

Origamis – also called square-tiled surfaces – are translation surfaces which are obtained from gluing finitely many copies of the Euclidean unit squares along their edges by translations. They play a crucial role in the theory of translation surfaces because they lie dense in the moduli spaces of finite translation surfaces. Moreover they can be described in a simple way as pair of two permutations in the symmetric group $S_d$ (where $d$ is the number of squares the origami consists of). By their combinatorial nature many invariants as for example the Veech group - which in case of origamis is a subgroup of $\mathrm{SL}(2,\mathbb{Z})$ of finite index - can be explicitly calculated. [HL06, Sch05, Zor06]

How it works

This is a database of origamis. It contains all origamis up to degree 9 (up to isomorphisms by renumbering the squares) as well as other orgiamis of interest, e.g., normal origamis (up to degree 500). Currently the database contains up to 14640 orbits of origamis, of which at least 8679 are normal.

You can search the origamis by degree or by other properties. Note that most properties are shared between origamis in the same orbit. In order to reduce the size of the search result, it is advised to search for orbit representatives instead of actual origamis. If you want to do more advanced computations with origamis, you can download the database and the Origami GAP package.

Search for origamis:

Find specific origami

from permutations

Type in the (1-based) permutations of the origami in cycle notation (leave empty for $\text{id}$). We will then find the corresponding origami, provided it exists in the database. Note that the origami will be in a certain normal form, that is, its permutations will correspond to your typed ones only up to renaming.

By ID

If you already know the ID, you can go straight to the origami.

Feedback

If you have feedback regarding this website, the data in the database, the functionality of our origami GAP package or any other topic that our working group is working on please feel welcome to contact us:
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References

[HL06] Hubert, Pascal and Lelièvre, Samuel, Prime arithmetic Teichmüller discs in $\mathcal{H}(2)$, 2006
[Sch05] Schmithüsen, Gabriela, Veech Groups of Origamis, University Karlsruhe, 2005
[Zor06] Zorich, Anton, Flat surfaces, Frontiers in number theory, physics, and geometry. 2006
[We2015] Weitze-Schmithüsen, Gabriela, The deficiency of being a congruence group for Veech groups of origamis. International Mathematics Research Notices, No. 6, 2015 (1613–1637)