Application of Semitotalistic 2D Cellular Automata on a Triangulated 3D Surface

Abstract

This paper presents the application of semi-totalistic, also called outer totalistic cellular automata on any three-dimensional (3D) surface. Cellular automata (CA) can be applied in controlling the state of a freeform building envelope. Such an intelligent ‘skin of a building’ can have any shape and a certain ‘organic’ appearance which can be dynamically controlled in response to the changes of the external conditions or users’ requirements. Any 3D surface can be triangulated. This means that it can become a grid of topologically identical elements. With the exception of boundary conditions, applicable to the elements positioned at the edge of the surface or around holes, every triangle in the grid has exactly three neighboring triangles. As with a CA, every element of a triangulated surface can be individually assigned with characteristics such as color or transparency level, which is analogous to a CA ‘state’. Therefore, it is possible to control to some degree the state of the whole surface taking advantage of the emergent behavior of the CA. The concept of CA on a triangular tessellation is discussed followed by discussion about the entities of irregularity of a grid, called ‘holes’ and ‘edges’. The concept of an ‘organic’ pattern in the context of CA is also briefly discussed. Two-dimensional (2D) CA in the triangular tessellation is discussed and implemented. A brief study covers the entities of neighborhood, type of rules (semi-totalistic and general), rule encoding and search for rules that meet given criteria. An implementation is performed on a regular triangular grid, irregular triangular grid and an imported triangulated 3D mesh which is irregular and has holes. A selected 2D triangular CA is applied on an imported triangulated 3D model.