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,
25-27 July 2000.
• 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
• Monohedral
• N-hedral
N-hedral Property
• Trivial dihedral case
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. 209-216.
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, 7-11
June 1999, pp. 277-283.
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