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
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
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:
(reveal)
References
[HL06] Hubert, Pascal and Lelièvre, Samuel, Prime arithmetic Teichmüller discs in
[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)
