Escherization and Ornamental Subdivisions

M.C. Escher

Escherization ``Escherization,'' by Craig S. Kaplan and David H. Salesin. SIGGRAPH 2000, the 27th International Conference on Computer Graphics and Interactive Techniques. New Orleans, Louisiana, USA, July Computer Graphics and Geometric Ornamental Design Craig Kaplan, University of Washington 2002

Escherization Problem statement Given a closed plane figure S (the “goal shape”), find a new closed figure T such that: –1. T is as close as possible to S ; and –2. copies of T fit together to form a tiling of the plane.

Escherization

Tessellations Geometric pattern, which is able to fill an infinite plane without any overlaps or gaps Individual tiles can undergo rigid body transformations

N-hedral Property N-hedralMonohedral

Trivial dihedral case N-hedral Property

Symmetry Symmetry groups

Measure of Closeness How to compare two shapes? Metric insensitive to scaling, rotation, and translation Polygon Turning Numbers Arkin, E.M., Chew, L.P., Huttenlocher, D.P., Kedem, K., and Mitchell, J.S.B. An Efficiently Computable Metric for Comparing Polygonal Shapes. PAMI(13), No. 3, March 1991, pp

Polygon Turning Numbers

Optimizing over Tiling Space function FINDOPTIMALTILING ( GOALSHAPE ; FAMILIES ) : INSTANCES  CREATEINSTANCES ( FAMILIES ) while || INSTANCES || > 1 do for each i in INSTANCES do –ANNEAL (i; GOALSHAPE ) end for INSTANCES  PRUNE ( INSTANCES ) end while return CONTENTS ( INSTANCES ) end function

Results of System Performs well on convex and “nearly convex” shapes

Results of System System can fail on an already repeatable tile System tends to fail on shapes with long, complicated tiling edges Vertices can be converted into control points to form curves User manipulation can improve results

Voronoi Diagrams ``Voronoi Diagrams and Ornamental Design,'' by Craig S. Kaplan. ISAMA '99, The first annual symposium of the International Society for the Arts, Mathematics, and Architecture. San Sebastián, Spain, June 1999, pp

Voronoi Diagrams Division of a plane based upon the proximity to a set of point or line generators Generators with weights

Voronoi Diagrams

Parquet Deformation

Applications to Other Works Parquet Deformation Circle and Square Limits