JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING RESEARCH VOL. 1, NO. 1, 2021 Applying Multidimensional Geometry to Basic Data Centre Designs Pedro Juan Roig1,2,*, Salvador Alcaraz1, Katja Gilly1, Carlos Juiz2 1Department of Computing Engineering, Miguel Hernández University, Spain, Email: [email protected], [email protected], [email protected] 2Department of Mathematics and Computer Science, University of the Balearic Islands, Spain, Email: [email protected] * Corresponding author Abstract: Fog computing deployments are catching up by the Therefore, that leads to utilize a smaller amount of switches day due to their advantages on latency and bandwidth compared so as to link together all those hosts, thus allowing for the use to cloud implementations. Furthermore, the number of required of easier interconnection topologies among those switches [5]. hosts is usually far smaller, and so are the amount of switches needed to make the interconnections among them. In this paper, Hence, Fog deployments permit the implementation of an approach based on multidimensional geometry is proposed basic Data Centre designs, which may be modelled by means for building up basic switching architectures for Data Centres, of elements of multidimensional geometry, specifically, by in a way that the most common convex regular N-polytopes are using different types of N-polytopes. first introduced, where N is treated in an incremental manner in The organization of the rest of the paper goes as follows: order to reach a generic high-dimensional N, and in turn, those first, Section 2 introduces multidimensional geometry, next, resulting shapes are associated with their corresponding switching topologies. This way, N-simplex is related to a full Section 3 talks about platonic solids, Section 4 explains the mesh pattern, N-orthoplex is linked to a quasi full mesh Schläfli notation, afterwards, Section 5 exposes regular N- structure and N-hypercube is referred to as a certain type of polytopes in Euclidean spaces, later on, Section 6 presents the partial mesh layout. In each of those three contexts, a model is N-simplex, in turn, Section 7 demonstrates the N-orthoplex, to be built up, where switches are first identified, afterwards, then, Section 8 exhibits the N-hypercube, and finally, Section their downlink ports leading to the end hosts are exposed, along 9 wraps it all up by drawing some final conclusions. 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 II. MULTIDIMENSIONAL GEOMETRY any given port of a switch is to be forwarded through the proper It may be said that the concept of dimension in a outgoing port on its way to the destination host by using the mathematical object is related to the minimum number of appropriate arithmetic expressions in each particular case. Therefore, all those algorithmic models represent how their necessary coordinates to specify any of its given points [6]. corresponding switches may work when dealing with user data This way, an independent point may be considered to be an traffic within a Data Centre, guiding it towards its destination. object with no dimensions (0D), a line may be a 1-dimension object (1D), a plane may be a 2-dimension object (2D) and a Keywords: Data Centre, N-hypercube, N-orthoplex, N- space may be a 3-dimension object (3D). As a matter of fact, simplex, networking. our physical world is regarded as tridimensional, as any point within may be completely described by quoting its length, I. INTRODUCTION width and height. Fog Computing paradigm is an extension of Cloud Back in the ancient Greece, Euclid established the basis to Computing, where the computing assets are moved to the define the classical geometry within the three-dimensional edge of the network, thus improving some key performance space. Also in that time, Pythagoras came up with his indicators such as latency, jitter and bandwidth [1]. Pythagorean theorem, which permitted to measure distances Fog deployments are ever growing thanks to the rise of within the Euclidean geometry, bringing about related new wireless communication standards allowing for faster concepts such as length, angle, orthogonality or parallelism, transmissions, such as cellular 5G or Wi-Fi 6 [2]. which are the core of geometry nowadays. Since then, different types of Non-Euclidean geometries Moreover, other technologies may take advantage of those have been defined, either by applying some sort of constant improvements, such as Mobile Edge Computing or Mist curvature, such as in hyperbolic and elliptic geometries, or Computing, facilitating a change of paradigm in online otherwise, non-constant curvature, such as in Riemannian services and hyperconnectivity in the coming years [3]. geometry, even though the infinitesimal structure of any space However, it is to be noted that Fog environments are always behaves as Euclidean. related to limited geographical areas compared to Cloud ones, Therefore, focusing on the Euclidean space, where N=3, it as the former are localized in a certain area whereas the latter may be extended to any number of dimensions, where such a usually have a global scope. This implies that Fog domains number may be any natural number N. This leads to consider may have far less users and workload, resulting in the need of multidimensional geometries to those Euclidean spaces with much less hosts in their Data Centres [4]. N>3, which may not correspond to our physical world, but just an extension of it. Additionally, it may also apply when N<3. 1 JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING RESEARCH VOL. 1, NO. 1, 2021 III. PLATONIC SOLIDS IV. SCHLÄFLI NOTATION Back in the ancient Greece, Plato studied and classified the Moving on to the XIX century, a swiss mathematician so called platonic solids [7]. Each one was a regular called Schläfli came up with a simple notation to describe any polyhedron, made up by the same type of regular polygons, regular geometrical shape regardless its dimension, thus hence, 3D convex regular structures being composed of just making possible its use for any regular N-polytope. This is one type of 2D convex regular shape. Only 5 solids meet these known as Schläfli notation, or alternatively, Schläfli symbol. specifications, shown in Fig. 1. Such a notation provides a summary of all the underlying k- facets of any regular polytope of dimension N, and it takes the form {p,q,r,…,x,y,z}, having as many variables as N-1 [8]. Starting with this expression, by taking out variables on the right-hand side in an incremental way, it is revealed the different sorts of k-facets forming the overall shape. This way, Fig. 1. Platonic solids. {p,q,r,…,x,y} determines the type of its (N-1)-facets, whilst In this context, Plato associated them with the five classical {p,q,r,…,x} denotes the kind of its (N-2)-facets, and all the basic elements (fire, earth, air, water and ether), such as the way down to just {p} giving the sort of its faces, also known fire to the tetrahedron, the earth to the cube, the octahedron to as 2-facets. Therefore, it may be concluded that Schläfli the air, the icosahedron to the water, and eventually, the symbols gives details about the different kinds of k-facets dodecahedron to the constellations in heaven, even though within any regular N-polytope. Aristotle later considered that heaven was made of ether. Additionally, it is to be said that edges, being 1-facets, Additionally, Table I exposes the numbers of faces, edges would be represented by {}, because they are one-dimensional and nodes related to each of those platonic solids. Those data shapes and this notation grants them zero variables, which clearly reveal that there a two couple of shapes being dual of may seem obvious as edges only have one possible shape, each other, such as the cube and the octahedron on the one regardless of its length. Furthermore, this notation may assign hand, and the dodecahedron and the icosahedron on the other a negative number of variables to nodes, being 0-facets, and to hand, as each pair of items has the same number of edges, and empty set, being the (-1)-facet, hence it does not apply to them. their number of faces and nodes are just switched around. On the other hand, a peak figure might be defined as the This basically means that the nodes of one item may be elements being incident on a given k-facet. Following with the located right in the middle of the faces of its dual item, where Schläfli notation, by taking out variables on the left-hand side the inner one is inscribed and the other one is circumscribed. in an incremental way, it is revealed the different sorts of peak Besides, the tetrahedron is autodual, as the number of figures being incident on each kind of k-facet. This way, faces and nodes match, which makes that a tetrahedron may {q,r,…,x,y,z}stands for the vertex figure, that being an (N-1)- be inscribed or circumscribed to another tetrahedron. facet, {r,…,x,y,z} indicates the edge figure (the vertex figure of a vertex figure), that being an (N-2)-facet, and all the way TABLE I. K-FACETS ON THE PLATONIC SOLIDS down to just {z} gives the (N-3)-peak figure, that being a 2- Name of the Regular Polygons Faces Edges Nodes facet. Additionally, it is to be mentioned that {} would Polyhedron as its faces indicate the (N-2)-peak figure, that being a 1-facet. Tetrahedron Triangle 4 6 4 Cube Square 6 12 8 Alternatively, mixing both concepts together, such as k- Octahedron Triangle 8 12 6 facets and peak figures, it may be said that there is a number Dodecahedron Pentagon 12 30 20 of z (N-1)-facets of type {p,q,…,x,y}around each (N-3)-facet.