Flat surfaces examples

Installation

You need to install sage-combinat which is done with the following command 

$ sage -combinat install

This installs a modified version of Sage. For more on what this command does or how to reverse it, you can read this step by step tutorial.

For an introduction to Sage, check the documentation at http://sagemath.org/.

Using interval exchange transformations

Permutations of interval exchange transformations are created 

sage: p = iet.Permutation('a b c d','d c b a')
sage: p.connected_component()
H_hyp(2)

Square-tiled surfaces

Let us build the genus 2 origami with three squares 

sage: o = Origami('(1,2)', '(1,3)')
sage: print o
(1, 2)
(1, 3)

And now, we build its Teichmueller curve and compute some of its invariants (rk: it is not clear yet which properties should be attached to the Teichmueller curve and which should be attached to the Veech group) 

sage: t = o.teichmueller_curve()
sage: G = t.veech_group()
sage: G.index()
3
sage: G.nu2()  # elliptic points of order 2
1
sage: G.nu3()  # elliptic points of order3
1
sage: G.ncusps()  # number of cusps
2
sage: t.sum_of_lyapunov_exponents()
4/3