Katherine E. Stange

Schmidt Arrangements

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

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

Gaussian Schmidt arrangement

Root -2 Schmidt arrangement

Root -7 Schmidt arrangement

Root -11 Schmidt arrangement

Root -6 Schmidt arrangement

Root -15 Schmidt arrangement

Gaussian Schmidt arrangement with Apollonian packing shown

Original Algorithm — A Selection of Fields

Gaussian Schmidt Arrangement

Schmidt arrangement of Q(sqrt(-19))

Schmidt arrangement and coset, for Q(sqrt(-15))

Schmidt arrangement of Q(sqrt(-11))

Schmidt arrangement of Q(sqrt(-7))

Schmidt arrangement of Q(sqrt(-163))

Schmidt arrangement of Q(sqrt(-67))

Schmidt arrangement of Q(sqrt(-43))

Schmidt arrangement of Q(sqrt(-10))

Schmidt arrangement of Q(sqrt(-6))

Schmidt arrangement of Q(sqrt(-5))

Schmidt arrangement of Q(sqrt(-2))

Schmidt arrangement of Q(sqrt(-3))

Miscellaneous Images

Gaussian Mandalas