A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line

It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of combinatorial Grassmannian of type $G_2(7)$, $\mathcal{V}(G_2(7))$. The lines of the ambient symplectic polar space are those lines of $\mathcal{V}(G_2(7))$ whose cores feature an odd number of points of $G_2(7)$. After introducing basic properties of three different types of points and six distinct types of lines of $\mathcal{V}(G_2(7))$, we explicitly show the combinatorial Grassmannian composition of the magic Veldkamp line; we first give representatives of points and lines of its core generalized quadrangle GQ$(2,2)$, and then additional points and lines of a specific elliptic quadric $\mathcal{Q}^{-}$(5,2), a hyperbolic quadric $\mathcal{Q}^{+}$(5,2) and a quadratic cone $\widehat{\mathcal{Q}}$(4,2) that are centered on the GQ$(2,2)$. In particular, each point of $\mathcal{Q}^{+}$(5,2) is represented by a Pasch configuration and its complementary line, the (Schl\"afli) double-six of points in $\mathcal{Q}^{-}$(5,2) comprise six Cayley-Salmon configurations and six Desargues configurations with their complementary points, and the remaining Cayley-Salmon configuration stands for the vertex of $\widehat{\mathcal{Q}}$(4,2).Comment: 6 pages, 2 figure

On an Observer-Related Unequivalence Between Spatial Dimensions of a Generic Cremonian Universe

Given a generic Cremonian space-time, its three spatial dimensions are shown to exhibit an intriguing, "two-plus-one" partition with respect to standard observers. Such observers are found to form three distinct, disjoint groups based on which one out of the three dimensions stands away from the other two. These two subject-related properties have, to our knowledge, no analogue in any of the existing physical theories of space-time.Comment: 4 pages, no figures, submitted to CS

Cremonian Space-Time(s) as an Emergent Phenomenon

It is shown that the notion of fundamental elements can be extended to_any_, i.e. not necessarily homaloidal, web of rational surfaces in a three-dimensional projective space. A Cremonian space-time can then be viewed as an_emergent_ phenomenon when the condition of "homaloidity" of the corresponding web is satisfied. The point is illustrated by a couple of particular types of "almost-homaloidal" webs of quadratic surfaces. In the first case, the quadrics have a line and two distinct points in common and the corresponding pseudo-Cremonian manifold is endowed with just two spatial dimensions. In the second case, the quadrics share six distinct points, no three of them collinear, that lie in quadruples in three different planes, and the corresponding pseudo-Cremonian configuration features three time dimensions. In both the cases, the limiting process of the emergence of generic Cremonian space-times is explicitly demonstrated.Comment: 5 pages, no figures, submitted to CS

Combinatorial Intricacies of Labeled Fano Planes

Given a seven-element set $X = \{1,2,3,4,5,6,7\}$, there are 30 ways to define a Fano plane on it. Let us call a line of such Fano plane, that is to say an unordered triple from $X$, ordinary or defective according as the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of order $s$, $0 \leq s \leq 3$, if there are $s$ {\it defective} lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown, in particular, that no labeled Fano plane can have all points of zeroth order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out.Comment: 5 pages, 2 figure

The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

Given a hyperbolic quadric of PG(5,2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the $(28_6, 56_3)$-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type $G_2(8)$. It is also pointed out that a set of seven points of $G_2(8)$ whose labels share a mark corresponds to a Conwell heptad of PG(5,2). Gradual removal of Conwell heptads from the $(28_6, 56_3)$-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).Comment: 4 pages, 4 table

On an Intriguing Signature-Reversal Exhibited by Cremonian Spacetimes

It is shown that a generic quadro-quartic Cremonian spacetime, which is endowed with one spatial and three time dimensions, can continuously evolve into a signature-reversed configuration, i.e. into the classical spacetime featuring one temporal and three space dimensions. An interesting cosmological implication of this finding is mentioned.Comment: 3 pages, 1 table, no figures; submitted to Chaos, Solitons & Fractal