An approach to structure in architectural and urban design. 2. Algebraic representation and local structure
In this second paper it is shown that any simplicial complex <I>K</I>(<I>X</I>), with a finite vertex set <I>X</I>, possesses a natural algebraic representation in a specified exterior algebra. This also leads to a representation of patterns by polynomials in that algebra. The idea of the complementary complex, with antivertices, also finds an algebraic representation in an extended exterior algebra. The algebraic representation is used to illustrate the role of the obstruction vector <I>Q</I>(<I>K</I>), introduced in the first of these papers. Furthermore the local geometry of the complex is explored by introducing the idea of shomotopy and the associated group structure {<I>S<sub>q</sub></I> *} whose generators are linked to the ideas of a <I>q</I>-hole and <I>q</I>-object.
