Fundamental domains for congruence subgroups

Instructions

• drag red dots to change original polygon
• F button changes whether the polygon is drawn as filled in or just as outline
• 1,2,3,4 are some preset polygons; generally some fundamental domains
• Click on the + and - points to either add or remove a point to the polygon
• There are 5 preset polygons, which can be changed as described. The preset polygons are F₁, F₂, F₃, F₄, F₅; the first three are fundamental domains for PSL(2,Z); the fourth is somewhat random; the fifth is a domain for Gamma(0,2), and also used in papers of Caroline Series and Kulkarni (links below) in describing a geometric interpretation of the continued fraction algorithm.
• The matrices S, T, U, R are given by:
  $T=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],S=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],R=S{T}^{\mathrm{-1}}=\left[\begin{array}{cc}0& -1\\ 1& -1\end{array}\right],$
  $U=-S{T}^{\mathrm{-1}}S=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right],V=TS{T}^{\mathrm{-1}}=\left[\begin{array}{cc}1& -2\\ 1& -1\end{array}\right]$

These matrices are special because:
• T and S generate PSL_2(Z), and are generators corresponding to the domain F₁
• R and S generate PSL_2(Z), and are generators corresponding to the domain F₂
• T and R generate PSL_2(Z), and are generators corresponding to the domain F₃
• T has infinite order and fixes infinity
• S has (projective) order 2 and fixes i
• R has (projective) order 3 and fixes the 6th root of unity $(1+i\sqrt{3})/2$
• U is the conjugate of T^-1 by S, and has infinite order and fixes 0
• V has (projective) order 2, and fixes (1+i)/2. Together, T and V generate Gamma⁰(2), and domain F₅ is a fundamental domain corresponding to these generators.
Here, the generators "corresponding to a domain" mean that edges of that domain are identified by application of these matrices.
• When you click to change the matrix, this is applied to the image polygon; two polygons are displayed, one, F, and its image MF, ie., the result of appling the matrix M to F.

To Do

• Need to make it impossible to enter [0,0,0] for point
• Add in links and references, especially to paper by Caroline Series on Continued Fractions and Hyperbolic Geometry
• Also Kulkarni's paper
• Add instructions and description!
• Add premultiplication by T, S, Tinverse, as well as postmultiplication; possibly add other "standard" matrices
• Note that now p, q, q2 can be moved by mouse move, but the images can not do we want to change this?
• perhaps it's better to allow that p,q,q2 all have the option of being either complex or rational, rather than current restriction
• Note, currently M can be edited to have negative determinant, which may produce weird results, and generally is not allowed; should we allow this and edit code to cope with it or not bother?
• Also maybe deal with case q=q2?

• The mathematics:
  • Deciding what to include?
  • Showing how to write an element of PSL_2(Z) in terms of T and S; maybe in terms of other generators also;
  • ie. include implementing algorithm in Caroline Series paper
  • Eventually similar for other subgroups of SL_2(Z), giving generators, relations, etc
  • Use this to write numbers as continued fractions, and give corresponding graphical
  • Look into Desin d'enfant? etc etc...
  • Also include a corresponding "chopping up rectangles" continued fraction explanation and other methods
  • make disc model version... good for displaying infinity
  • draw farey symbols; make it possible to draw the farey symbols and convert to domains and vice versa
• The programming:
  • Problem: tried to switch to everything integers, but integers get too big, do doesn't work; using integers would be good for GCD; without this, may crash... so... need an overhaul at some point; look for warnings in js file to try and locate places to check whether integer
  • Bug: currently matrix is recomputed everytime mouse moves; need to stop this, and probably other unnecessary computations
  • Need to fix: don't have to redraw whole thing when mouse is moving, only when down/clicked
  • Some lines are drawn twice in tessellations; should be more careful and fix this
  • when a point is moved by mouse, we get a coordinate, as a decimal... try using continued fractions to get most reasonable fractional approximation?
  • make the canvas etc... Do we do this also as SVG? And what about latex versions,
  • and a download button, so people can use images and include them in papers they write?
  • screen should be scalable; both manual and auto scale options
• Goals?
  • Educational, learning about complex numbers, etc, this whole area of number theory;
  • Research tool: how much can this be used to explore domains, complex functions, modular forms?
  • Research aid: for drawing diagrams to include in papers...
• The cosmetics: Need a proper css and sort out the old fashioned html etc!
  • variable/responsive canvas width; variable by user?
  • make side panel moveable to convenient location
  • Use nice latexy stuff in displaying information nicely, e.g., the matrices, formulars, rationals, continued fractions, etc
• What else has been done? Find a list of links to other similar projects - hope none idential! Also list of references to online and off line material - papers, books
• Want to make a jointly editable version of this file, and generally files worked on, with back ups; maybe need to find location..