Scripts

I maintain some computer scripts for computations relating to my research.  These are for the computer algebra systems Pari/GP and Sage.  You are welcome to download and use these scripts; they and the software to run them (Pari & Sage) are free.  Please let me know if you encounter errors.

Notes on usage: 1) To use a Pari/GP script, load the script by typing "\r filename" at the prompt, where filename includes the path, then see the script for descriptions of functions; 2) To use a Sage script, type "attach('filename')" at the prompt; it also works simply to cut and paste the text of the script file into the first cell of a notebook; 3) To use a Sage notebook, upload it to the notebook interface.

Some of this material is based upon work supported by the National Science Foundation under Grant No. 0802915 and the National Science and Engineering Research Council Postdoctoral Fellowship No. 373333. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation of the USA (NSF) or the National Science and Engineering Research Council of Canada (NSERC).

Images of Schmidt Arrangements


The Schmidt Arrangement of the Eisenstein integers.
For the field K = Q(sqrt(-3)) with ring of integer O_K, a K-Bianchi circle is the image of the real line under a Mobius transformation in PSL_2(O_K). This image shows the K-Bianchi circles of curvatures less than or equal to 20*sqrt(3) which intersect the fundamental domain (including boundary) of the Eisenstein integers. Curvatures are all multiples of sqrt(3); the colour indicates whether the multiple is even or odd. See my recent paper, Visualizing the arithmetic of quadratic imaginary fields. Created with Sage. Click on the image for a larger version. Please request permission to use this image elsewhere.


The Schmidt Arrangement of the Gaussian integers.
Circles up to curvature 30. Note: this image includes circles in PGL_2(Z[i]), hence it is actually two copies of the Schmidt arrangement at right angles. See my recent paper, Visualizing the arithmetic of quadratic imaginary fields. Created with Sage. Click on the image for a larger version. Please request permission to use this image elsewhere.


The Schmidt Arrangement of Q(sqrt(-2)).
Circles up to curvature 30*sqrt(2). See my recent paper, Visualizing the arithmetic of quadratic imaginary fields. Created with Sage. Click on the image for a larger version. Please request permission to use this image elsewhere.


The Schmidt Arrangement of Q(sqrt(-7)).
Circles up to curvature 30*sqrt(7). See my recent paper, Visualizing the arithmetic of quadratic imaginary fields. Created with Sage. Click on the image for a larger version. Please request permission to use this image elsewhere.


The Schmidt Arrangement of Q(sqrt(-11)).
Circles up to curvature 30*sqrt(11). See my recent paper, Visualizing the arithmetic of quadratic imaginary fields. Created with Sage. Click on the image for a larger version. Please request permission to use this image elsewhere.


The Schmidt Arrangement of Q(sqrt(-15)).
In blue are the K-Bianchi circles up to curvature 60*sqrt(15). Since Q(sqrt(-15)) has class number 2, the extended Bianchi group produces more circles (in orange). The Schmidt arrangement is disconnected because the field is non-Euclidean. See my recent paper, Visualizing the arithmetic of quadratic imaginary fields. Created with Sage. Click on the image for a larger version. Please request permission to use this image elsewhere.

For other Schmidt arrangements, visit the folder of images.

Sage Mathematics Software

I do a wee little bit of development for the mathematical software Sage.