A taxonomy of crystallographic sphere packings

A geometric study of circle packings and ideal class groups

“…We call A superintegral if the orbit under its supergroup is integral. Superintegrality is used in [5] and [13] to help classify crystallographic circle packings, and again in [12] to help classify Kleinian circle packings and bugs.…”

Domains of Convergence for Polyhedral Packings

“…Property (1) follows from σ T j being a reflection, fixing P and exchanging the half-spaces cut out by P . For property (2), let d be some circle on N perpendicular to c, and call either of their points of intersection t. Consider the line tangent to N at t and also tangent to c. As d and c are perpendicular, is also perpendicular to d. Now, consider the unique right cone tangent to N with base d and call the apex of this cone a. The segment at is perpendicular to d and by construction, it is also tangent to N .…”