Geometric dual

It is significant that both V- and D-tessellations are unequivocally interrelated because they form geometrically dual graphs. For a brief expression of a relation between these two lattices let us designate a total number of facets in V- and D-lattices as/yE and E, and the total number of edges as BV1 and liV) z, respectively. For dual graphs the relation is [100,146,147]... 

The geometric dual G of a graph (7 is a graph obtained from G as follows ... 

This prescription for construction is iUostrated using G — fi s The vertices and edges of G2B are indicated by empty circles and full lines, and those of Gie by Inll circles and dashed lines, respectively The geometric duals of and Gty nre isomorphic, GSe — Gm- Since the vertices of G29 axe allowed to occupy arbitrary points in the plane in which they are drawn (they are not restricted to definite subsectiims), Gty is called the dual of Gu- Note that in the geometric dual the location of edges and loops obey definite face relations, whereas such relations are not necessarily realised in the dual. 

From eq.(16) one can easily show that any planar simple grs h is the geometnc dual of its geometric dual ... 

Thus, from its geometric dual 7, the original graph G can be regenerated. This statement, however, does not hdd for the duals isomorphic with the geometric duals, as seen from the fact that G29 ... 

In the previous section the geometric dual of a graph was introduced as a kind of a bijective nonisomorphic mapping. A different kind of mapping produces the line graph, C Q) of a graph G = [V,f] and may be derived as follows ... 

We shall name each of the 2 volumes by a Boolean N-vector found from Eq. (18), which represents a vertex on the Boolean N-cube (the geometric dual of the concentration space). Consider the element / = 1. If mj = 0 in Eq. (16) then there will be a directed edge from (011 1) to (111 1) and if mi = 1, the orientation of this edge will be reversed. Similarly the edge joining (001 1) and (101 1) is directed from the first to the latter vertex if mi = 1, and if mi = 0 the orientation of this edge is reversed. The remainder of the edges are handled in similar fashion. 

For example, for a cube P-C + F = 8-12 + 6 = 2, as required. Similarly for an octahedron P-C + F = 6-12 + 8=2. The octahedron, moreover, is a geometrical dual of the cube, because the role of P and F are interchanged in the two structures. Thus the 8 ppints of the cube correspond to the 8 faces of the octahedron, and the 6 faces of the cube correspond to the 6 points of the octahedron. The duals share exactly the same value C (=12), and exactly, the same point group symmetry (Oh). For any case where the Descartes-Euler formula applies, duals are defined by the interchange of P and F, C held constant. Thus, Figure 1 indicates the dodecahedron and icosahedron are duals, and the tetrahedron is self-dual. 

Figure 3. The carbon fullerene C o nd its geometrical dual B32H32 which had been suggested long ago [4].

The Euler-Poincare formula invokes the use of Betti numbers [10] which may be calculated as the count of the number of critical points, of various types, associated with the geometrical structure of nanotori. The theory of Morse flmctions [11] relates critical points to topological structure. We shall show, an alternating sum of Betti numbers defines the Euler characteristic of a torus to be zero. This connects the topology of a nanotorus, nanotube, and plan sheet, which have the same Euler characteristic. We show that for every possible carbon nanotorus there is a geometrical dual boron nanotorus. 

Figure 10, Nanotori geometrical duals, Cg o cmd F4S0 (right).

We have shown how a simple idea, the Descartes-Euler formula, defines the boron fullerenes as the geometrical duals of the carbon fullerenes. As the carbon fullerenes are actually existing molecules, the Descartes-Euler formula is immediately suggestive of the possible existence of their boron duals. This is confirmed by quantum chemical molecular orbital calculations which indicate considerable stability for boron flillerene cage and multi-cage geometries. Therefore we encourage attempts at the synthesis of these proposed compounds. 

In the illustration of Fig. 29.4 we regard the matrix X as either built up from n horizontal rows of dimension p, or as built up from p vertical columns x,.of dimension n. This exemplifies the duality of the interpretation of a matrix [9]. From a geometrical point of view, and according to the concept of duality, we can interpret a matrix with n rows and p columns either as a pattern of n points in a p-dimensional space, or as a pattern of p points in an n-dimensional space. The former defines a row-pattern P" in column-space 5, while the latter defines a column-pattern P in row-space S". The two patterns and spaces are called dual (or conjugate). The term dual space also possesses a specific meaning in another... 

The matrix X defines a pattern P" of n points, e.g. x, in which are projected perpendicularly upon the axis v. The result, however, is a point s in the dual space S". This can be understood as follows. The matrix X is of dimension nxp and the vector V has dimensions p. The dimension of the product s is thus equal to n. This means that s can be represented as a point in S". The net result of the operation is that the axis v in 5 is imaged by the matrix X as a point s in the dual space 5". For every axis v in 5 we will obtain an image s formed by X in the dual space. In this context, we use the word image when we refer to an operation by which a point or axis is transported into another space. The word projection is reserved for operations which map points or axes in the same space [11]. The imaging of v in S into s in S" is represented geometrically in Fig. 29.9a. Note that the patterns of points P" and P are represented schematically by elliptic envelopes. 

In a general way, we can state that the projection of a pattern of points on an axis produces a point which is imaged in the dual space. The matrix-to-vector product can thus be seen as a device for passing from one space to another. This property of swapping between spaces provides a geometrical interpretation of many procedures in data analysis such as multiple linear regression and principal components analysis, among many others [12] (see Chapters 10 and 17). 

The geometrical interpretation of MLR is given in Fig. 29.10. The n rows (objects) of X form a pattern P" of points (represented by x,) which is projected upon an (unknown) axis b. This causes the axis b in S to be imaged by X in the dual space S" at the point y. The vector of observed measurements y has dimension n and, hence, is also represented as a point in 5". Is it possible then to define an axis b in S " such that the predicted y coincides with the observed y Usually this will not be feasible. One may propose finding the best possible b such that y comes as close to y as possible. A criterion for closeness is to ask for the distance between y and y, which is equal to the normlly - yll, to be as small as possible. 

From the above we conclude that the product of a matrix with a vector can be interpreted geometrically as an operation by which a pattern of points is projected upon an axis. This projection produces an image of the axis at a point in the dual space. The concept can be extended to the product of a matrix with another matrix. In this case we can conceive of the latter as a set of axes, each of which produces image points in the dual space. In the special case when this matrix has only two columns, the product can be regarded as a projection of a pattern of points upon the plane formed by the two axes. As a result one obtains two image points (one for each axis that defines the plane of projection) in the dual space. 

The same geometrical considerations can be applied to the dual representation of the column-pattern in row-space S" (Fig. 31.2b). Here u, is the major axis of symmetry of the equiprobability envelope. The projection of theyth column Xy of X upon u, is at a distance from the origin given by ... 

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

The success of the dual approach and the form of the objective function and constraints suggest geometric (posynomial) programming as an alternative optimization technique. In the absence of the so-called reverse constraints, the posynomial program takes the following form ... 

C5 cyclization requires stricter geometric conditions than aromatization. This is in favor of the dual-site mechanism of C5 cyclic reactions (25). All metals catalyzing it have an fee lattice, and their atomic diameter lies between 0.269 and 0.277 nm. These two criteria must be fulfilled simultaneously. With such a distance between the two sites, the screening of the C—C bond adjacent to the preferably adsorbed tertiary C atom becomes evident. Figure... 

This approach is therefore based in rigorous and general geometric tensor theory. The PL vector dual to turns out to be the light-like invariant ... 

The pseudovector element dfi represents the same surface element as dfjL. and, geometrically, is a pseudovector normal to the surface element and equal in magnitude to the area of the element. In 4-space, such a pseudovector cannot be constructed from an antisymmetric tensor such as dfpv. However, the dual pseudotensor can be defined by [10] ... 

The dual axial vector in 4-space is constructed geometrically from the integral over a hypersurface, or manifold, a rank 3-tensor in 4-space antisymmetric in all three indices [101]. In three-dimensional space, the volume of the parallelepiped spanned by three vectors is equal to the determinant of the third rank formed from the components of the vectors. In four dimensions, the projections can be defined analogously of the volume of the parallelepiped (i.e., areas of the hypersurface) spanned by three vector elements < dl, dx and dx". They are given by the determinant... 

---