Schmidt Arrangements

Schmidt arrangements are circle/disk orbits under certain arithmetically interesting groups of Mobius transformations.  They also make great adult colouring images.

• Interactive Schmidt arrangement visualizer — explore the arrangement of any imaginary quadratic field in your browser: pan, zoom, colour, and inspect the exact Bianchi matrix of every circle
• Schmidt orbits — choose three $\mathbb{Q}(i)$-rational points; the circle through them and its whole $\mathrm{PSL}_2(\mathbb{Z}[i])$-orbit are drawn over the Gaussian arrangement
• Visualizing Imaginary Quadratic Fields —  a two-page expositional paper on Schmidt arrangements (pages 16-17)
• The Farey Structure of the Gaussian Integers — a four-page expositional paper on their relationship to Farey Fractions
• An Illustration in Number Theory — a three-page expositional paper on their relationship to Abelian sandpiles
• Visit my code page to make your own Schmidt arrangements with my software.

A Selection of Fields

Original Algorithm — A Selection of Fields

Miscellaneous Images

Gaussian Mandalas