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Notes on Plücker's relations in geometric algebra

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Abstract

Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up using many different languages of higher mathematics, such as multilinear and tensor algebra, matroid theory, and Lie groups and Lie algebras. Here we explore the Plücker relations in Clifford's geometric algebra. We discover that the Plücker relations can be fully characterized in terms of the geometric product, without the need for a confusing hodgepodge of many different formalisms and mathematical traditions found in the literature.

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... However this is not the case since the terms g ij a i a j of (24) consists of scalars and bivectors. Plücker relations are important in understanding the structure of bivectors [11], particularly bivectors in G 1,n , and in study of conformal mappings [12]. ...
... For x ∈ R n+1 the position vector (11), the convex null n-simplex in R n+1 is defined by S + n := S + n (a 1 , . . . , a n+1 ) = {x ∈ R n+1 | x 1 + · · · + x n+1 = 1, x i ≥ 0}, (50) by the requirement that the coordinates x (s) of x ∈ R n+1 , are homogeneous barycentric coordinates, [16]. ...
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A null vector is an algebraic quantity with square equal to zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by N. The rules of addition and multiplication in N are taken to be the familiar rules of addition and multiplication of real or complex square matrices. A pair of null vectors is positively or negatively correlated if their inner product is positive or negative, respectively. A large class of (Clifford) geometric algebras are isomorphic to real or complex matrix algebras, or a pair of such algebras. I begin the study of the eigenvector-eigenvalue problem of linear operators in the geometric algebra G(1,n) of R^{n+1}, and by restricting to barycentric coordinates, n-simplices whose n+1 vertices are non-zero null vectors. These ideas provide a foundation for a new Cayley-Grassmann Theory of Linear Algebra, with many possible applications in pure-applied areas of science and engineering.
... The work 195 explores Plücker relations in GA that can be fully characterized in terms of the geometric product, without multiple different formalisms and mathematical traditions found in the literature. It suggests a general and simple (practical) algorithm (the idea of which is due to Dung B. Nguyen, who presented it 2020 (online) during the ICCA12 conference in Heifei, China) to test if a given homogeneous multivector is a blade (i.e. could be written as an outer product of vectors). ...
... We thank an anonymous reviewer for these detailed remarks on195 . ...
Article
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We extensively survey applications of Clifford Geometric algebra in recent years (mainly 2019–2022). This includes engineering; electric engineering; optical fibers; geographic information systems; geometry; molecular geometry; protein structure; neural networks; artificial intelligence; encryption; physics; signal, image, and video processing; and software.
... Over the last half century Clifford's geometric algebras have become a powerful geometric language for the study of the atomic structure of matter in elementary particle physics, Einstein's special and general theories of relativity, string theories and super symmetry. In addition, geometric algebras have found their way into many areas of mathematics, computer science, engineering and robotics, and even the construction of computer games, [15,19,20]. ...
... Coordinate geometric matrices for the geometric algebras G 1,1 and G 1,2 were constructed in Section 3. We now show how the construction given in (14) and (15) generalizes to G k,k and G k,k+1 for k > 1. For both G 1,1 and G 1,2 , the standard vector and null vector basis is given by 1 e 1 u 1 ( 1 e 1 ) = 1 a 1 u 1 ( 1 b 1 ) , respectively, where u 1 = 1 2 (1 + e 1 f 1 ) = b 1 a 1 . ...
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Abstract Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in mathematics is the Periodic Table of Geometric Numbers discussed here. In 1878 William Kingdon Clifford discovered the defining rules for what he called geometric algebras. We show how these algebras, and their coordinate isomorphic geometric matrix algebras, fall into a natural periodic table, sidelining the superfluous definitions based upon tensor algebras and quadratic forms. (23) (PDF) Periodic Table of Geometric Numbers. Available from: https://www.researchgate.net/publication/339784011_Periodic_Table_of_Geometric_Numbers [accessed Mar 08 2020].
... Eq. (19)). A thorough study of the representation of the more general Grassmann-Plücker relations in geometric algebra was recently given by Sobczyk [27]. ...
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This paper shows how geometric algebra can be used to derive a novel generalization of Heron’s classical formula for the area of a triangle in the plane to higher dimensions. It begins by illustrating some of the many ways in which the conformal model of three-dimensional Euclidean space yields provocative insights into some of our most basic intuitive notions of solid geometry. It then uses this conceptual framework to elucidate the geometric meaning of Heron’s formula in the plane, and explains in detail how it extends naturally to the volumes of tetrahedra in space. The paper closes by outlining a proof of a previously conjectured extension of the formula to the hyper-volumes of simplices in all dimensions.
... However this is not the case since the terms g ij a i a j of (24) consists of scalars and bivectors. Plücker relations are important in understanding the structure of bivectors [23], particularly bivectors in G 1,n , and in study of conformal mappings [20]. ...
Article
Full-text available
A null vector is an algebraic quantity with the property that its square is zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by N{{\mathcal {N}}}. The rules of addition and multiplication in N{{\mathcal {N}}} are taken to be the same as those for real and complex square matrices. A distinct pair of null vectors is positively or negatively correlated if their inner product is positive or negative, respectively. A basis of n+1 null vectors, with pairwise inner products equal to plus or minus one half, defines the Clifford geometric algebras G1,n{\mathbb {G}}_{1,n}, or Gn,1{\mathbb {G}}_{n,1}, respectively, and provides a foundation for a new Cayley–Grassman linear algebra, a theory of complete graphs, and other applications in pure and applied areas of science.
... Plücker relations are important in understanding the structure of bivectors [11], particularly bivectors in G 1,n , and in study of conformal mappings [12]. ...
Conference Paper
Full-text available
A null vector is an algebraic quantity with square equal to zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by N. The rules of addition and multiplication in N are taken to be the familiar rules of addition and multiplication of real or complex square matrices. A pair of null vectors is positively or negatively correlated if their inner product is positive or negative, respectively. A large class of (Clifford) geometric algebras are isomorphic to real or complex matrix algebras, or a pair of such algebras. I begin the study of the eigenvector-eigenvalue problem of linear operators in the geometric algebra G(1,n) of R^{n+1}, and by restricting to barycentric coordinates, n-simplices whose n+1 vertices are non-zero null vectors. These ideas provide a foundation for a new Cayley-Grassmann Theory of Linear Algebra, with many possible applications in pure-applied areas of science and engineering.
... Given this observation, it is perhaps no surprise that the real algebraic variety of all degenerate (volume = 0) tetrahedra should be intimately related to the Klein quadric K := {x ∈ R 6 | x 1 x 6 − x 2 x 5 + x 3 x 4 = 0} [7]. A thorough study of the representation of the more general Grassmann-Plücker relations in geometric algebra was recently given by Sobczyk [11]. ...
Preprint
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The representation of some elementary geometric concepts in the conformal geometric algebra of three dimensions are reviewed, and their connections to a recently discovered extension of Heron's formula for the area of a triangle to the volume of a tetrahedron are discussed. Mathematics Subject Classification (2010). Primary 51M04; Secondary 11E88, 15A67, 51M25, 51M30, 52A38, 52B10.
... How geometric matrices arise as algebraically isomorphic coordinate matrices, and a practical application to the classical Plücker relations, is explored in [7]. A general introduction to geometric algebras and their coordinate geometric matrices is given in [6]. ...
Conference Paper
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A (Clifford) geometric algebra is usually defined in terms of a quadratic form. A null vector v is an algebraic quantity with the property that v^2 = 0. Null vectors over the real or complex numbers are taken as fundamental and are added and multiplied together according to the familiar rules of real or complex square matrices. In a series of ten definitions, the concepts of a Grassmann algebra, its dual Grassmann algebra, the associated real and complex geometric algebras, and their isomorphic real or complex coordinate matrix algebras are laid down. In a series of ten definitions, the concepts of a Grassmann algebra, its dual Grassmann algebra, the associated real and complex geometric algebras, and their isomorphic real or complex coordinate matrix algebras are laid down. This is followed by a discussion of affine transformations, the horosphere and conformal transformations on pseudoeuclidean spaces.
... which are used with (15) to calculate ...
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A signed graph is a graph in which every pair of vertices is connected by a directed edge. By geometric algebra }, I mean a Clifford Geometric Algebra (GA). A null vector v is an algebraic quantity with the property that v^2=0. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by N. The rules of addition and multiplication in N are taken to be the familiar rules of addition and multiplication of real or complex square matrices. I define a pair of real null vectors to be positively or negatively correlated if their inner product is positive or negative, respectively. In particular, I begin the study of the eigenvector-eigenvalue problem of linear operators in a geometric algebra G_{p,q} of R^n in terms of n-simplices whose n=p+q vertices are named by non-zero null vectors. The Cayley-Grassmann Theorem in Lightcone Projective Graph Geometry LPGG is studied using the powerful tools of projective geometry in suitable geometric algebras. These ideas provide the foundation for a new Cayley-Grassmann Linear Algebra in LPGG, with many expected applications in graph theory, and pure-applied areas of science and engineering.
... Because of the tedious, but not dicult, calculations involved, the ideas presented herein are worked out only for signed complete graphs of order two, three, and four vertices. I also leave open the future additional application of these ideas to graph theory, quantum mechanics and many other areas, see for example [1,2,3,7,8,9,10,11,12]. 2 Correlated pairs of null vectors Denition 1: A pair a, b of real null vectors in N is said to positively, or ...
Preprint
A {\it signed complete graph} is a complete graph in which every pair of vertices is connected by a directed edge. By a {\it geometric algebra}, I mean a Clifford geometric algebra. A null vector v is an algebraic quantity with the property that v2=0v^2=0. I denote the universal algebra generated by taking the sums and products of null vectors over the real or complex numbers by N{\cal N}, \cite{GS2022}. The rules of addition and multiplication in N{\cal N} are taken to be the familiar rules of addition and multiplication of real or complex square matrices. I define a pair of real null vectors to be {\it positively} or {\it negatively} correlated if their inner product is {\it positive} or {\it negative}, respectively. I begin with the study of signed complete graphs Kn±K_n^\pm whose vertices are named by non-zero null vectors. Naming the vertices of a graph with null vectors opens up a way studying any graph, using the powerful tools of {\it projective geometry} in a suitable geometric algebra. The main result of this paper is that each signed convex regular polyhedra defines a {\it light cone} when its vertices are named by real or complex non-zero null vectors \citeite{Havel2022,Tanya2008}. The most famous convex polyhedra in 3 dimensions are the 5 Greek regular solids, but they also exist in higher dimensions \cite{baez}. Welcome to the World of {\it Light Cone Projective Graph Geometry}.
... How geometric matrices arise as algebraically isomorphic coordinate matrices, for the simplest case of 2 × 2 real or complex matrices, and their practical application to the classical Plücker relations is given in [6]. A general introduction to geometric algebras and their coordinate geometric matrices is given in [7]. ...
Preprint
Full-text available
A (Clifford) geometric algebra is usually defined in terms of a quadratic form. A null vector v is an algebraic quantity with the property that v 2 = 0. Null vectors over the real or complex numbers are taken as fundamental and are added and multiplied together according to the familiar rules of real or complex square matrices. In a series of ten definitions, the concepts of a Grassmann algebra, its dual Grassmann algebra, the associated real and complex geometric algebras, and their isomorphic real or complex coordinate matrix algebras are laid down.
Research
Impulse to get ahead © Bernd, Silvester 2024 'Space-Time Isospinors in a Space of Matrix Units' © 2025 by Bernd Schmeikal is licensed under CC BY-SA 4.0 /The Abstract to Space-Time Isospinors in a Space of Matrix Units © 2025 by Bernd Schmeikal is also licensed under CC BY-SA 4.0! Forget metrics, location and topological neighborhood for the time being. Think instead of isotropic darkness, which has an inherent tendency to form symmetry, namely orientation symmetry. As explained in the lecture “On Motion”, the symmetry elements are initially neither identifiable, i.e. indistinguishable, nor allocatable. Only through processes of symmetry breaking do they become visible and localizable.
Chapter
The representation of some elementary geometric concepts in the conformal geometric algebra of three dimensions are reviewed, and their connections to a recently discovered extension of Heron’s formula for the area of a triangle to the volume of a tetrahedron are discussed.
Chapter
A (Clifford) geometric algebra is usually defined in terms of a quadratic form. A null vector v is an algebraic quantity with the property that v2=0v^2=0. The universal algebra generated by taking the sums and products of null vectors over the real or complex numbers is denoted by N\mathcal{N}. The rules of addition and multiplication are taken to be the familiar rules of addition and multiplication of real or complex square matrices. In a series of ten definitions, the concepts of a Grassmann algebra, its dual Grassmann algebra, the associated real and complex geometric algebras, and their isomorphic real or complex coordinate matrix algebras are set down. This is followed by a discussion of affine transformations, the horosphere and conformal transformations on pseudoeuclidean spaces.
Book
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Geometric algebra has been presented in many different guises since its invention by William Kingdon Clifford shortly before his death in 1879. The guiding principle of this book is that it should be fully integrated into the foundations of mathematics, and in this regard nothing is more fundamental than the concept of number itself. Since I acquired this conviction as a graduate student at Arizona State University more than 50 years ago, much work has been done in applying geometric algebra to problems in mathematics, physics, engineering and computer science, in the gradual recognition of importance of geometric algebra and in the truth of this principle.
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1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5. Geometric Algebras of PseudoEuclidean Spaces.- 2 / Differentiation.- 2-1. Differentiation by Vectors.- 2-2. Multivector Derivative, Differential and Adjoints.- 2-3. Factorization and Simplicial Derivatives.- 3 / Linear and Multilinear Functions.- 3-1. Linear Transformations and Outermorphisms.- 3-2. Characteristic Multivectors and the Cayley-Hamilton Theorem.- 3-3. Eigenblades and Invariant Spaces.- 3-4. Symmetric and Skew-symmetric Transformations.- 3-5. Normal and Orthogonal Transformations.- 3-6. Canonical Forms for General Linear Transformations.- 3-7. Metric Tensors and Isometries.- 3-8. Isometries and Spinors of PseudoEuclidean Spaces.- 3-9. Linear Multivector Functions.- 3-10. Tensors.- 4 / Calculus on Vector Manifolds.- 4-1. Vector Manifolds.- 4-2. Projection, Shape and Curl.- 4-3. Intrinsic Derivatives and Lie Brackets.- 4-4. Curl and Pseudoscalar.- 4-5. Transformations of Vector Manifolds.- 4-6. Computation of Induced Transformations.- 4-7. Complex Numbers and Conformal Transformations.- 5 / Differential Geometry of Vector Manifolds.- 5-1. Curl and Curvature.- 5-2. Hypersurfaces in Euclidean Space.- 5-3. Related Geometries.- 5-4. Parallelism and Projectively Related Geometries.- 5-5. Conformally Related Geometries.- 5-6. Induced Geometries.- 6 / The Method of Mobiles.- 6-1. Frames and Coordinates.- 6-2. Mobiles and Curvature 230.- 6-3. Curves and Comoving Frames.- 6-4. The Calculus of Differential Forms.- 7 / Directed Integration Theory.- 7-1. Directed Integrals.- 7-2. Derivatives from Integrals.- 7-3. The Fundamental Theorem of Calculus.- 7-4. Antiderivatives, Analytic Functions and Complex Variables.- 7-5. Changing Integration Variables.- 7-6. Inverse and Implicit Functions.- 7-7. Winding Numbers.- 7-8. The Gauss-Bonnet Theorem.- 8 / Lie Groups and Lie Algebras.- 8-1. General Theory.- 8-2. Computation.- 8-3. Classification.- References.
Article
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Let V denote a finite-dimensional vector space. An s-vector P ∈ Λ s V is called decomposable or simple if it can be written in the form P = u ∧ v ∧ · · · ∧ w for u, v,..., w ∈ V. We shall use in the following both Penrose’s abstract index notation and exterior calculus with the conventions of [3]. Theorem 1. Let P ∈ Λ s V be an s-vector. Then P is decomposable if and only if one of the following conditions holds: 1. i(Φ)P ∧ P = 0 for all Φ ∈ Λ s−1 V ∗. In index notation P [abc···dP e]fg···h = 0. 2. i(iPΨ)P = 0 for all Ψ ∈ Λ s+1 V ∗. 3. iα1∧···∧αs−k P is decomposable for all αi ∈ V ∗ , for any fixed k ≥ 2. 4. i(Ψ)P ∧ P = 0 for all Ψ ∈ Λ s−2 V ∗ In index notation P [abc···dP ef]g···h = 0. 5. i(iPΨ)P = 0 for all Ψ ∈ Λ s+2 V ∗. Proof. (1) These are the well known classical Plücker relations. For completeness’ sake we include a proof. Let P ∈ Λ n V and consider the induced linear mapping ♯P: Λ s−1 V ∗ → V. Its image, W, is contained in each linear subspace U of V with
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The classical relations of Plücker between the invariants and singularities of a plane curve can be expressed as two linear relations and two involving quadratic terms. The linear relations were generalised to curves in n-space already in the nineteenth century, but true generalisations of the others were obtained only in 3-space. In this article, using the classical method of correspondences, we obtain formulae in n-space corresponding to the original ones in the plane.
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Matroids with coefficients in a fuzzy ring have been introduced in [A. W. M. Dress, Adv. in Math.59 (1986), 97–123]. Based on this paper and [A. W. M. Dress and W. Wenzel, Adv. in Math.77 (1989), 1–36] we will show here that a matroid with coefficients of finite rank can also be defined in terms of its associated Grassmann-Plücker map, which—in a way—generalizes classical determinants. More precisely, we will show that such matroids correspond naturally and in a one-to-one fashion to equivalence classes of appropriately defined Grassmann-Plücker maps.
Applications of Geometric Algebra in
L. Dorst, C. Doran, J. Lasenby (Eds.), Applications of Geometric Algebra in Computer Science and Engineering, Birkhäuser, 2002.
Plücker's Relations, Geometric Algebra, and the Electromagnetic Field, unpublished version
  • D B Nguyen
D.B. Nguyen, Plücker's Relations, Geometric Algebra, and the Electromagnetic Field, unpublished version: Sept. 3, 2018.
Hyperbolic numbers revisited
  • G Sobczyk
G. Sobczyk, Hyperbolic numbers revisited, preprint, http://www.garretstar.com /hyprevisited12 -17 -2017.pdf, 2017.
Geometric matrices and the symmetric group
  • G Sobczyk
G. Sobczyk, Geometric matrices and the symmetric group, https://arxiv.org /pdf /1808.02363.pdf, 2018.