Two discrete forms of the Jordan curve theorem
American Mathematical Monthly
A local-global principle for vertex-isoperimetric problems
Discrete Mathematics - Kleitman and combinatorics: a celebration
Analytical Honeycomb Geometry for Raster and Volume Graphics
The Computer Journal
Polyominoes with minimum site-perimeter and full set achievement games
European Journal of Combinatorics
On minimal perimeter polyminoes
DGCI'06 Proceedings of the 13th international conference on Discrete Geometry for Computer Imagery
Isoperimetrically optimal polygons in the triangular grid
IWCIA'11 Proceedings of the 14th international conference on Combinatorial image analysis
Multi-agent Cooperative Cleaning of Expanding Domains
International Journal of Robotics Research
| Hi-index | 5.23 |
In the plane, the way to enclose the most area with a given perimeter and to use the shortest perimeter to enclose a given area, is always to use a circle. If we replace the plane by a regular tiling of it, and construct polyforms i.e. shapes as sets of tiles, things become more complicated. We need to redefine the area and perimeter measures, and study the consequences carefully. A spiral construction often provides, for every integer number of tiles (area), a shape that is most compact in terms of the perimeter or boundary measure; however it may not exhibit all optimal shapes. We characterize in this paper all shapes that have both shortest boundaries and maximal areas for three common planar discrete spaces.