Cell Complexes: The Structure of Space and the Mathematics of Modularity

Brendan Lane


The modeling of growing multicellular structures is of fundamental importance in investigating plant development. A distinctive feature of plants is that, with rare exceptions, cells do not move with respect to each other; the only differences in tissue topology are due to cell divisions. Corresponding features are found in other application domains, such as geometric modeling. Models from these diverse domains, taken together, constitute the field of developmental modeling. This dissertation is concerned with devising a mathematical and computational formalism for such modeling.

Examining simple examples of one-dimensional models shows that the mathematical structure of the cell complex is ideal for developmental modeling. The cell complex consists of mathematical cells of different dimensions, letting physical quantities of different inherent dimension sit in their proper place in the structure. A cell complex can be represented in an index-free manner, and topological operations on it, including the important case of cell division, can be effected in a local manner. Finally, the cell complex is built on neighbourhood relations which let developmental rules easily access values in neighbouring cells.

A novel data structure, the flip, records a single adjacency between cells in a cell complex; a flip table, the collection of all flips in the complex, is in turn sufficient to represent the complex itself. This representation is used to build the Cell Complex Framework, a C++ API which can be used for computational modeling of development in any number of dimensions. The framework provides basic operations such as iterating over a cell complex, adding, removing, dividing, and merging cells, and computing geometric information such as orientation, measure, and centroid. Some developmental models from the existing literature, in both two and three dimensions, are reproduced using the Cell Complex Framework, demonstrating its workability and expressiveness; these include both geometric models as well as models of biological systems. New models are also shown, including a model of turtle geometry on the surface of a 2D mesh and a three-dimensional model of the apex of the moss Physcomitrella patens.


Brendan Lane. "Cell Complexes: The Structure of Space and the Mathematics of Modularity". Ph.D. thesis, University of Calgary, September 2015.

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