Falt surfaces examples

Installation

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

$ sage -combinat install

The latter 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.

To get introduced to Sage, look at the documentation on 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 invariant (rk: it is not yet clear which property should be attached to the teichmueller curve and which one 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