Fog computing deployments are catching up by the day due to their advantages on latency and bandwidth compared to cloud implementations. Furthermore, the number of required hosts is usually far smaller, and so are the amount of switches needed to make the interconnections among them. In this paper, an approach based on multidimensional geometry is proposed for building up basic switching architectures for Data Centres, in a way that the most common convex regular N-polytopes are first introduced, where N is treated in an incremental manner in order to reach a generic high-dimensional N, and in turn, those resulting shapes are associated with their corresponding switching topologies. This way, N-simplex is related to a full mesh pattern, N-orthoplex is linked to a quasi full mesh structure and N-hypercube is referred to as a certain type of partial mesh layout. In each of those three contexts, a model is to be built up, where switches are first identified, afterwards, their downlink ports leading to the end hosts are exposed, along with those host identifiers, as well as their uplink ports leading to their neighboring switches, and eventually, a pseudocode algorithm is designed, exposing how a packet coming in from any given port of a switch is to be forwarded through the proper outgoing port on its way to the destination host by using the appropriate arithmetic expressions in each particular case. Therefore, all those algorithmic models represent how their corresponding switches may work when dealing with user data traffic within a Data Centre, guiding it towards its destination.
Applying Multidimensional Geometry to Basic Data Centre Designs