9853

The Action of the Modular Group on the Fundamental Domain

Choose eight transformations from the set of controls on the left. Each transformation can be either an inversion or a translation. Choose the exponent of each of these transformations (-1, 0, or 1) from the set of controls on the right. This Demonstration shows the fundamental domain and its images under the consecutive compositions of the chosen transformations.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The modular group is the quotient of the group of integer matrices with by the subgroup , where denotes the identity matrix. This group acts on the upper half-plane by . It turns out that the modular group is generated by just two elements: inversion, given by the matrix , and translation, given by . The fundamental domain is the subset of the upper half-plane formed by all points such that and . The action of the modular group tiles the upper half-plane exactly with images of the fundamental domain. This tiling requires the use of the entire infinite fundamental domain, so it cannot be demonstrated by Mathematica. However, this tiling can be approximated by considering the images of a finite part of the fundamental domain under a finite number of compositions , where each and is either a translation by an integer or a reflection.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+