The formal approach to shapes and their algebras, as it appears in shape grammar theory, has been reviewed. It starts with geometric elements and their partial algebras, continues to shapes, their algebras, and boundaries, as well as algebras that calculate with shapes and their boundaries. There is a number of new concepts introduced along the way. These include diagonal decompositions and their algebras which simplify calculations with shapes, b-paired diagonal decompositions which extend calculations with shapes and their boundaries from diagonal shapes only to all shapes, and m-order boundaries which extend the concept of shape boundaries and allow for calculations with multiple representations of shapes. It also shows that algebras of shapes are infinite direct sums of diagonal algebras.

This is the second paper in the series on shape decompositions and their use as shape approximations. This time we investigate hierarchical and topological shape decompositions or hierarchies and topologies. We showed earlier that bounded decompositions behave the same way as shapes do. The same holds for hierarchies and topologies, which are special kinds of bounded decompositions. They are distinguished by their algebraic structures, which have many important properties to facilitate their application as shape approximations. We provide an account of their properties with an emphasis on their application.

Shapes play an important role in many human activities, but are rarely seen in their natural form as raw and unanalyzed. Rather, shapes come analyzed, structured in terms of their certain parts, forming shape decompositions. Different kinds of shape decompositions are developed, the most interesting among which are the decompositions that could be used as shape approximations. Two kinds of such decompositions, discrete and bounded, are examined in greater detail. Computations with shapes conducted in the framework of shape grammars and related shape algebras have been standard for over 3 decades. Similar computations are possible with analyzed shapes or shape decompositions. Different algebras to compute with shape decompositions are developed and compared to the shape algebras. The measure of their agreement determines how well the shapes are approximated by their decompositions.