The Mathematics

Each product in this store is represented as a mathematical object known as a graph. The notation of such objects are generally written as G = (V, E) where V is the set of vertices and E is the set of edges that connects the vertices. In each figure, each vertex represents either a Heading, Section, Definition, Subsection, Paragraph, or Subparagraph, and the edge that connects two of these vertices represents the hierarchical structure of the particular Act. That is, a line connecting a Heading to a Section indicates that the Heading is divided into subclasses known as Sections. Because of the hierarchical construction of all legislation, the graph, G  is in fact a mathematical object known as a tree (a subclass of graphs) due to the fact that G  is (i) simple in that there are no loops (that is, a Section’s Subsection is not the Section itself); (ii) it is connected in that there is a path between every pair of vertices (that is, a Subsection cannot exist without being connected to a Section); and (iii) there are no cycles, in that there is no sequence of vertices starting and ending at the same vertex (that is, a Subsection cannot lead to a Paragraph that leads to the Section under which the Subsection is found within).

Examples of some trees can be seen below.

The hierarchical levels seen in each product represents the hierarchy of G and not the hierarchical titles given to the vertices. For example, the fact that a vertex falls on the fourth hierarchical ring does not guarantee that it is a Subsection. This is due to the fact that not every Section is subdivided into Subsections. We can account for this inconsistency, however, by adding a third dimension to G. Consider any edge e ∈ E  which has the form e = {u, v}  for two vertices, u and v. We may define the function χ(e) which assigns to that edge a specific colour depending on which type of vertex is involved. The set of all colours is then χ(E).

The tree G is presented in Reingold-Tilford circular layout for the efficient and tidy arrangement of layered vertices.

See below for an infographic which attempts to visually explain the above mathematics.