Hyperbolic Prisms, Poicaré Disc, and Foams

Alberto Tufaile, Adriana P. B. TufaileSoft Matter Laboratory

EACH-University of São PauloBrazil

C’est souvent à partir d’une petite experience que la nature d’um phénomène se precise. Pierre-Gilles de Gennes

tufaile@usp.br

Hyperbolic Prism

• The hyperbolic prism is a transparent optical element with spherical surfaces.

Poincaré Disc• The points of the

geometry are in a n-dimensional disc, and the straight lines consist of all segments of circles contained within a disc that are orthogonal to the boundary of the disc.

Foams• We are talking about liquid foam or aqueous

foam. For example, pour a bottle of beer. A foam or froth is formed. The prism was inspired in such type of foam.

Geometrical Optics• Foams – Hyperbolic prisms – Kaleidoscopes;• Kaleidoscopes – Reflections – Möbius transformations• Möbius transformations – Hyperbolic geometry –

Poincaré disc model;• Poincaré disc – Fractal – Chaotic scattering of light;

Möbius and Stereographic Projections

-Inversion in a Sphere;-Preservation of circles;-Presevation of angles...

Iterations

Hyperbolic Kaleidoscope

Simulation Experiment

Some Results

Prism

Kaleidoscope

Exploringthe hyperbolic

prism

Thank You