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
Showing circles up to curvature 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-19))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement and coset, for 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.
Schmidt arrangement of Q(sqrt(-11))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-7))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-163))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-67))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-43))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-10))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-6))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-5))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-2))
Curvatures up to 30, coloured by curvature modulo 2.
Schmidt arrangement of Q(sqrt(-3))
Curvatures up to 30, coloured by curvature modulo 2. In this case circles overlap because of the roots of unity in the field.
Miscellaneous Images
Gaussian Mandalas
