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Dynamic visualization of planar hyperbolic symmetry via the Klein model. (English) Zbl 1274.51017

Summary: Planar hyperbolic geometry results from replacing Euclid’s fifth postulate by its negation, with the consequence that there are infinitely many lines through a given point that are parallel to a given line not containing the given point. In the Klein model, hyperbolic points are represented by the points interior to a fixed horizon circle, and hyperbolic lines are represented by the open chords of the horizon. The Klein model is a projective model that distorts both distance and angle measurements. Each pair of distinct lines in a hyperbolic plane will be intersecting, sensed parallel, or ultraparallel. These three cases give rise to circles, horocycles, and equidistant curves, respectively. Basic isometries of a hyperbolic plane are reflections, rotations, parallel displacements, and translations.
We examine some examples of discrete hyperbolic symmetry through rosette groups, horosette groups, frieze groups, and groups generated by reflections in the sides of a triangle, as well as regular and semiregular tilings. We can visualize these isometries and resulting discrete symmetry patterns within the Klein model. The projective format provides a view as if hovering in a helicopter in hyperbolic space over a hyperbolic plane filled with these symmetry patterns.

MSC:

51M09 Elementary problems in hyperbolic and elliptic geometries
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
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