My investigation into the Space Allocation Problem has led me to the concept of Block Algebra (BA).

BA is a method of explaining relationships between two rectangles, using the 13 atomic relationships (Figure 1) of Interval Algebra (IA) (Allen, 1983). BA is the combination of two IA relationships – one for each Cartesian axis – giving a potential 169 atomic relationships (Balbiani, Condotta, & Del Cerro, 1998) which are shown in Figure 2.

IA 13 Atomic Relationships

BA 169 Atomic Relationships

As an example let’s consider the relationship (o, m), this indicates that the edges of the two rectangles overlap (o) in one axis and meet (m) in another axis. So if we make the leap that these two rectangles are rooms we have mathematically determined the spaces are adjacent to one another.