Multiple Perspectives and Generalizations of the Desargues Configuration

Prażmowska, Małgorzata

doi:10.1515/dema-2006-0418

Cited by 15 publications

(32 citation statements)

References 6 publications

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“…Next, we invoke the concept of combinatorial Grassmannian (see, for example, [22,23]). Briefly, a combinatorial Grassmannian G k (|X|), where k is a positive integer and X is a finite set, is a point-line incidence structure whose points are k-element subsets of X and whose lines are (k + 1)-element subsets of X, incidence being inclusion.…”

Section: -Nions and A (21 5 35 3 )-Configurationmentioning

confidence: 99%

“…Briefly, a combinatorial Grassmannian G k (|X|), where k is a positive integer and X is a finite set, is a point-line incidence structure whose points are k-element subsets of X and whose lines are (k + 1)-element subsets of X, incidence being inclusion. It is known [22] A pictorial illustration of C 6 ∼ = G 2 (7). Here, the labels of six additional points are only depicted, the rest of the labeling being identical to that shown in the left-hand side figure.…”

Section: -Nions and A (21 5 35 3 )-Configurationmentioning

confidence: 99%

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From Cayley-Dickson Algebras to Combinatorial Grassmannians

2015

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Abstract:Given a 2 N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6, we first observe that the multiplication table of its imaginary units e a , 1 ≤ a ≤ 2 N − 1, is encoded in the properties of the projective space PG(N − 1, 2) if these imaginary units are regarded as points and distinguished triads of them {e a , e b , e c }, 1 ≤ a < b < c ≤ 2 N − 1 and e a e b = ±e c , as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b = c. Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG (N − 1, 2), the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration C N ; in particular, C 3 (octonions) is isomorphic to the Pasch (6 2 , 4 3 )-configuration, C 4 (sedenions) is the famous Desargues (10 3 )-configuration, C 5 (32-nions) coincides with the Cayley-Salmon (15 4 , 20 3 )-configuration found in the well-known Pascal mystic hexagram and C 6 (64-nions) is identical with a particular (21 5 , 35 3 )-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic C N leads to a conjecture that C N is isomorphic to a combinatorial Grassmannian of type G 2 (N + 1).

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Section: -Nions and A (21 5 35 3 )-Configurationmentioning

confidence: 99%

Section: -Nions and A (21 5 35 3 )-Configurationmentioning

confidence: 99%

From Cayley-Dickson Algebras to Combinatorial Grassmannians

2015

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“…If each line of C has three points, a line of V (C) is also of size three and of the form {H , H , H ∆H }, where the symbol ∆ stands for the symmetric difference of the two geometric hyperplanes and an overbar denotes the complement of the object indicated. Our central concept is that of a combinatorial Grassmannian (e. g., [9,10]) G k (|X|), where k is a positive integer and X is a finite set, which is a point-line incidence structure whose points are k-element subsets of X and whose lines are (k + 1)-element subsets of X, incidence being inclusion. It is known [9] …”

Section: Relevant Finite-geometrical Backgroundmentioning

confidence: 99%

“…Our central concept is that of a combinatorial Grassmannian (e. g., [9,10]) G k (|X|), where k is a positive integer and X is a finite set, which is a point-line incidence structure whose points are k-element subsets of X and whose lines are (k + 1)-element subsets of X, incidence being inclusion. It is known [9] …”

Section: Relevant Finite-geometrical Backgroundmentioning

confidence: 99%

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

Санига

2017

Entropy

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It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of a combinatorial Grassmannian of type G 2 (7), V (G 2 (7)). The lines of the ambient symplectic polar space are those lines of V (G 2 (7)) whose cores feature an odd number of points of G 2 (7). After introducing the basic properties of three different types of points and seven distinct types of lines of 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 Q − (5, 2), a hyperbolic quadric Q + (5, 2), and a quadratic cone Q(4, 2) that are centered on the GQ(2, 2). In particular, each point of Q + (5, 2) is represented by a Pasch configuration and its complementary line, the (Schläfli) double-six of points in 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 Q(4, 2).

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“…[2,14]) was a starting point for further generalizations and considering so called combinatorial Grassmann spaces (cf. [11]). …”

Section: Introductionmentioning

confidence: 99%