A “Circle Limit III” Backbone Arc Formula

Year: 2008 Authors: Douglas Dunham; Luns Tee2

Core claim

For any (p,q,r) pattern in the Circle Limit III family, the backbone equidistant curve makes an angle ω with the bounding circle given by a single hyperbolic-trigonometric expression, which generalizes Coxeter’s Circle Limit III formula and is antisymmetric in p and q.

Topics

Circle Limit III, hyperbolic geometry, Poincaré disk model, equidistant curves, pattern families, mathematical art

Domains

hyperbolic geometry, hyperbolic trigonometry, non-Euclidean geometry, geometry of curves, trigonometric formulas, mathematical art, Escher-inspired pattern design, tessellation

Methods

geometric interpretation in the Poincaré disk model, hyperbolic trigonometric derivation, algebraic manipulation of angle formulas, comparison with special cases

Media

computer rendition, fish pattern imagery, Poincaré disk model, bounding circle

Source status

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