Geometric Algebra

This paper proposes a reinterpretation of imaginary numbers, suggesting that they should not be viewed solely as numerical units (with the property i 2 =-1), but rather as entities with a dual role-one as a numeric value and the other as a directional identity. By separating the imaginary component into magnitude and direction, we aim to provide greater clarity and analytical capability in fields such as complex analysis, physics, and engineering.

We present different methods for symbolic computer algebra computations in higher dimensional (≥ 9) Clifford algebras using the CLIFFORD and Bigebra packages for Maple R . This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.

We present different methods for symbolic computer algebra computations in higher dimensional (≥ 9) Clifford algebras using the CLIFFORD and Bigebra packages for Maple R . This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.

We present different methods for symbolic computer algebra computations in higher dimensional (≥ 9) Clifford algebras using the CLIFFORD and Bigebra packages for Maple R . This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.

CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C (B) -the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUMbased on Chevalley's recursive formula, and cmulRS -based on a non-recursive Rota-Stein sausage. Graßmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.

We present different methods for symbolic computer algebra computations in higher dimensional (≥ 9) Clifford algebras using the CLIFFORD and Bigebra packages for Maple R . This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.

CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C (B) -the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUMbased on Chevalley's recursive formula, and cmulRS -based on a non-recursive Rota-Stein sausage. Graßmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.

Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.

Geometric Algebra can be considered as a very useful language for dealing with mathematics, physics and computer science. Among the various algebras commonly used by practitioners of GA, Conformal Geometric Algebra (CGA) is of special interest for its powerful transformations and its ability to represent any hypersphere or hyperplane. Moreover, CGA is an algebra capable of representing pencils of hyperspheres. This paper presents an approach for constructing Voronoi diagrams by using pencils of CGA to build cell interfaces from centroid positions. Power, multiplicatively weighted Voronoi and multiplicatively weighted power diagrams are also supported by the presented method. This allows the use of circles as centroids to add control to the weights of the cells and to the curvatures of their borders.

Geometric Algebra can be considered as a language that unifies mathematics, physics and computer sciences etc. Among other, CGA is of special interest for its powerful transformations and its ability to represent any hypersphere or hyperplane. Moreover, CGA is an algebra capable of representing pencils of spheres. This paper presents a reinterpretation of every objects of 3D CGA as pencils of spheres and introduces set operators on its elements (i.e. union, intersection, complement, etc). As an application, these operators are used to find the smallest tangent sphere of two skew lines. 2010 MSC: primary 34K06, 34K25; secondary 39.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

We introduce the quadric conformal geometric algebra (QCGA) inside the algebra of R 9,6 . In particular, this paper presents how three-dimensional quadratic surfaces can be defined by the outer product of conformal geometric algebra points in higher dimensions, or alternatively by a linear combination of basis vectors with coefficients straight from the implicit quadratic equation. These multivector expressions code all types of quadratic surfaces in arbitrary scale, location, and orientation. Furthermore, we investigate two types of definitions of axis aligned quadric surfaces, from contact points and dually from linear combinations of R 9,6 basis vectors.

This paper presents an efficient implementation of geometric algebra, based on a recursive representation of the algebra elements using binary trees. The proposed approach consists in restructuring a state of the art recursive algorithm to handle parallel optimizations. The resulting algorithm is described for the outer product and the geometric product. The proposed implementation is usable for any dimensions, including high dimension (e.g. algebra of dimension 15). The method is compared with the main state of the art geometric algebra implementations, with a time complexity study as well as a practical benchmark. The tests show that our implementation is at least as fast as the main geometric algebra implementations.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The new applications of Clifford's geometric algebra surveyed in this paper include kinematics and robotics, computer graphics and animation, neural networks and pattern recognition, signal and image processing, applications of versors and orthogonal transformations, spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces.

Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it does not reflect the practical usage of geometric algebra. Indeed, in applications related to geometry, multivectors are likely to be full homogeneous, having their non-zero elements over a single grade. In this paper, we provide a complete computational study on geometric algebra products of two full homogeneous multivectors, that is, the outer, inner, and geometric products of two full homogeneous multivectors. We show tight bounds on the number of the arithmetic operations required for these products. We also show that algorithms exist that achieve this number of arithmetic operations.

Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly e cient and useful to deal with computer graphics solutions. For example, the Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow e cient realizations of these objects. is paper presents a Geometric Algebra implementation dedicated for both low and high dimensions. e proposed method is a hybrid solution for precomputed code with fast execution and runtime computations with low memory requirement. More speci cally, the proposed method combines a precomputed table approach with a recursive method using binary trees. Some rules are de ned to select the most appropriate choice, according to the dimension of the algebra and the type of multivectors involved in the product. e resulting implementation is well suited for high dimensional spaces (e.g. algebra of dimension 15) as well as for lower dimensional space. is paper details the integration of this hybrid method as a plug-in into Gaalop, which is a very advanced optimizing code generator. is paper also presents some benchmarks to show the performances of our method, especially in high dimensional spaces.

Quadratic surfaces gain more and more attention in the geometric algebra community and some frameworks to represent, transform, and intersect these quadratic surfaces have been proposed. To the best of our knowledge, however, no framework has yet proposed that supports all the operations required to completely handle these surfaces. Some existing frameworks do not allow the construction of quadratic surfaces from control points while some do not allow to transform these quadratic surfaces. Although a framework does not exit that covers all the required operations, if we consider all already proposed frameworks together, then all the operations over quadratic surfaces are covered there. This paper presents an approach that transversely uses different frameworks. We employ a framework to represent any quadratic surfaces either using control points or the coefficients of its implicit form and then map the representation into another framework so that we can transform them and compute their intersection. Our approach also allows us to easily extract some geometric properties.

The new applications of Clifford's geometric algebra surveyed in this paper include kinematics and robotics, computer graphics and animation, neural networks and pattern recognition, signal and image processing, applications of versors and orthogonal transformations, spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces.

Manipulating objects using geometric algebra may involve several associative products in a single expression. For example, an object can be constructed by the outer product of multiple points. This number of products can be small for some conformal algebra and high for higher dimensional algebras such as quadric conformal geometric algebras. In these situations, the order of products (i.e. the choice of the parenthesis in the expression) should not change the final result but may change the overall computational cost, according to the grade of the intermediate multivectors. Indeed, the usual left to right way to evaluate the expression may not be most computationally efficient. Studies on the number of arithmetic operations of geometric algebra expressions have been limited to products of only two homogeneous multivectors. This paper shows that there exists an optimal order in the evaluation of an expression involving geometric and outer products, and presents a dynamic programming ...

We introduce the quadric conformal geometric algebra (QCGA) inside the algebra of R 9,6 . In particular, this paper presents how three-dimensional quadratic surfaces can be defined by the outer product of conformal geometric algebra points in higher dimensions, or alternatively by a linear combination of basis vectors with coefficients straight from the implicit quadratic equation. These multivector expressions code all types of quadratic surfaces in arbitrary scale, location, and orientation. Furthermore, we investigate two types of definitions of axis aligned quadric surfaces, from contact points and dually from linear combinations of R 9,6 basis vectors.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

This paper presents an efficient implementation of geometric algebra, based on a recursive representation of the algebra elements using binary trees. The proposed approach consists in restructuring a state of the art recursive algorithm to handle parallel optimizations. The resulting algorithm is described for the outer product and the geometric product. The proposed implementation is usable for any dimensions, including high dimension (e.g. algebra of dimension 15). The method is compared with the main state of the art geometric algebra implementations, with a time complexity study as well as a practical benchmark. The tests show that our implementation is at least as fast as the main geometric algebra implementations.

This paper presents both a recursive scheme to perform Geometric Algebra operations over a prefix tree, and Garamon, a C++ library generator implementing these recursive operations. While for low dimension vector spaces, precomputing all the Geometric Algebra products is an efficient strategy, it fails for higher dimensions where the operation should be computed at run time. This paper describes how a prefix tree can be a support for a recursive formulation of Geometric Algebra operations. This recursive approach presents a much better complexity than the usual run time methods. This paper also details how a prefix tree can represent efficiently the dual of a multivector. These results constitute the foundations for Garamon, a C++ library generator synthesizing efficient C++ / Python libraries implementing Geometric Algebra in both low and higher dimensions, with any arbitrary metric. Garamon takes advantage of the prefix tree formulation to implement Geometric Algebra operations on high dimensions hardly accessible with state-of-the-art software implementations. Garamon is designed to produce easy to install, easy to use, effective and numerically stable libraries. The design of the libraries is based on a data structure using precomputed functions for low dimensions and a smooth transition to the new recursive products for higher dimensions.

Using GA’s capacities for formulating refl ections, we solve an interesting
multiple-tangency problem. The solution is obtained in two ways, the
easiest of which transforms the relevant refl ections into a single rotation.
The solution is validated via a GeoGebra worksheet.

Four critical elementary mathematical mistakes in Joy Christian's counterexample to Bell's theorem are presented. Consequently, Joy Christian's hidden variable model cannot reproduce any quantum mechanics results and cannot be used as a counterexample to Bell's theorem. The mathematical investigation is followed by a short discussion about the possibility to construct other hidden variable theories. A tutorial section on relevant Clifford algebra topics was added at the end to help interested readers decide for themselves the validity of Joy Christian's claims. Also an appendix section discusses recent developments.

In this paper we present a new representation for 3D free-form contours in the conformal geometric algebra G 4,1 . This new representation allows to extract local geometrical feature information which is used to solve the correspondence problem for pose estimation applications. Under perspective projection, local features are extracted from a projected contour segment and compared with image features obtained from the monogenic signal. We tested our approach using synthetical and real data for several pose estimation algorithms.

An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Maxwell's equations can be expressed in a single multivector equation using 3D Clifford algebra (isomorphic to Pauli algebra spinorial formulation of electromagnetism). Subsequently one can more easily solve for the time evolution of both the electric and magnetic field simultaneously in terms of the fields evaluated only on a 3D hypersurface. The form of the special "characteristic" surfaces for which the time derivative of the fields can be singular are quickly deduced with little effort.

Attention is given to the interface of mathematics and physics, specifically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to the meta-monogenic (massive) Dirac equation is constructed which form a Hilbert space. A new integral solution is proposed which involves application of a kernel to the right side of the function, instead of to the left as usual. This allows for the introduction of a multivector generalization of the Feynman Path Integral formulation, which shows that particular "geometric groupings" of solutions evolve in the manner to which we ascribe the term "quantum particle". Further, it is shown that the role of usual i is subplanted by the unit time basis vector, applied on the right side of the functions.

The inverse problem, to reconstruct the general multivector wave function from the observable quadratic densities, is solved for 3D geometric algebra. It is found that operators which are applied to the right side of the wave function must be considered, and the standard Fierz identities do not necessarily hold except in restricted situations, corresponding to the spin-isospin superselection rule. The Greider idempotent and Hestenes quaterionic spinors are included as extreme cases of a single superselection parameter.

Standard formulation is unable to distinguish between the (+ + +-) and (---+) spacetime metric signatures. However, the Clifford algebras associated with each are inequivalent, R(4) in the first case (real 4 by 4 matrices), H(2) in the latter (quaternionic 2 by 2). Multivector reformulations of Dirac theory by various authors look quite inequivalent pending the algebra assumed. It is not clear if this is mere artifact, or if there is a right/wrong choice as to which one describes reality. However, recently it has been shown that one can map from one signature to the other using a tilt transformation . The broader question is that if the universe is signature blind, then perhaps a complete theory should be manifestly tilt covariant. A generalized multivector wave equation is proposed which is fully signature invariant in form, because it includes all the components of the algebra in the wavefunction (instead of restricting it to half) as well as all the possibilities for interaction terms.

In this paper we embed m-dimensional Euclidean space in the geometric algebra Cl m to extend the operators of incidence in R m to operators of incidence in the geometric algebra to generalize the notion of separator to a decision boundary hyperconic in the Clifford algebra of hyperconic sections denoted as Cl (V 2 ). This allows us to extend the concept of a linear perceptron or the spherical perceptron in conformal geometry and introduce the more general conic perceptron, namely the elliptical perceptron. Using Clifford duality a vector orthogonal to the decision boundary hyperplane is determined. Experimental results are shown in 2-dimensional Euclidean space where we separate data that are naturally separated by some typical plane conic separators by this procedure. This procedure is more general in the sense that it is independent of the dimension of the input data and hence we can speak of the hyperconic elliptic perceptron.

Throughout high school, the real-world applications of mathematical concepts, from basic arithmetic to complex equations, are often overlooked. Relationships such as those used in circuit analysis are frequently underestimated despite their significance. In classrooms, students memorize formulas without fully understanding their deep connections.
For instance, number theory is a foundational tool in modern technology. One of its critical applications today is securing communications through algorithms and encryption by implementing different techniques, such as using prime numbers as a code for messages. Mathematical tools describe the epicenter of physics, and without substantially knowing them, we will face obstacles as engineers.
This research aims to bridge the gap between theoretical mathematics and its real-world applications in electromagnetism. These tools range from basic manipulations in Algebra to advanced evaluations in Calculus, such as the proof of Maxwell’s equations. I aim to provide a fresh perspective and deep insight into this field by thoroughly explaining the concepts using history, visuals, and in-depth analysis and evaluation. My objectives include:
• Clarifying Electromagnetism: explaining advanced topics while connecting them to basic and advanced mathematics.
• Showcasing Practical Applications: Demonstrating how mathematical tools solve real-world problems like circuit analysis, wave tunning, and the motion of particles in an electric field.
• Explaining Interdisciplinary Connections: The relationships between mathematics, number theory, and electromagnetism in the broader context of modern technology.
This paper is particularly relevant for high school students pursuing STEM careers. I aim to inspire the next generation of scientists and engineers by providing precise and concise explanations of these concepts. The chronology and history of this paper are compiled from scholarly sources, with independent analysis conducted using methods introduced throughout the research. Finally, calculations and visualizations are performed using GeoGebra and Desmos to ensure accuracy.

The 7th International Workshop on Advanced Computational Intelligence and Intelligent Informatics (IWACIII2021) Beijing, China, Oct.31-Nov.3, 2021 1 Abstract: The conventional automatic drug dispenser can significantly contribute to numerous services such as medicine industries, pharmacy shops and medical assistance by reducing the workload of medical workers and allowing patients to take their timely medication. In this paper, we use a segmentation technique to identify the location of drug pillboxes from the medicine shelves with the automatic counting of spiral pillboxes. The proposed method is a solution to a dynamic segmentation framework for recognizing objects from large datasets. Firstly, the picture is filtered to remove the noise to recognize the boundary line. Following that, projection lines are drawn in the center to divide the segments of pillboxes by utilizing the midpoint formula, which supports smooth counting of line detection among objects. Besides, we applied the...

Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.

This thesis is a study of geometric algebra and its applications to relativistic physics. Geometric algebra (or real Clifford algebra) serves as an efficient language for describing rotations in vector spaces of arbitrary metric signature, including Lorentzian spacetime. By adopting the rotor formalism of geometric algebra, we derive an explicit BCHD formula for composing Lorentz transformations in terms of their generators — much more easily than with traditional matrix representations. This is used to straightforwardly derive the composition law for Lorentz boosts and the concomitant Wigner angle. Later, we include a gentle introduction to differential geometry, noting how the Lie derivative and covariant derivative assume compact forms when expressed with geometric algebra. Curvature is studied as an obstruction to the integrability of the parallel transport equations, and we present a surface-ordered Stokes’ theorem relating the ‘enclosed curvature’ in a surface to the holonomy ...

From the beginning of David Hestenes rediscovery of geometric algebra in the 1960s, outermorphisms have been a cornerstone in the mathematical development of GA. Many important mathematical formulations in GA can be expressed as outermorphisms such as versor products, linear projection operators, and mapping between related coordinate frames. Over the last two decades, GA-based mathematical models and software implementations have been developed in many fields of science and engineering. As such, efficient implementations of outermorphisms are of significant importance within this context. This work attempts to shed some light on the problem of optimizing software implementations of outermorphisms for practical prototyping applications using geometric algebra. The approach we propose here for implementing outermorphisms requires orders of magnitude less memory compared to other common approaches, while being comparable in time performance, especially for high-dimensional geometric algebras.

Current models of visual perception typically assume that human vision estimates true properties of physical objects, properties that exist even if unperceived. However, recent studies of perceptual evolution, using evolutionary games and genetic algorithms, reveal that natural selection often drives true perceptions to extinction when they compete with perceptions tuned to fitness rather than truth: Perception guides adaptive behavior; it does not estimate a preexisting physical truth. Moreover, shifting from evolutionary biology to quantum physics, there is reason to disbelieve in preexisting physical truths: Certain interpretations of quantum theory deny that dynamical properties of physical objects have definite values when unobserved. In some of these interpretations the observer is fundamental, and wave functions are compendia of subjective probabilities, not preexisting elements of physical reality. These two considerations, from evolutionary biology and quantum physics, suggest that current models of object perception require fundamental reformulation. Here we begin such a reformulation, starting with a formal model of consciousness that we call a "conscious agent." We develop the dynamics of interacting conscious agents, and study how the perception of objects and space-time can emerge from such dynamics. We show that one particular object, the quantum free particle, has a wave function that is identical in form to the harmonic functions that characterize the asymptotic dynamics of conscious agents; particles are vibrations not of strings but of interacting conscious agents. This allows us to reinterpret physical properties such as position, momentum, and energy as properties of interacting conscious agents, rather than as preexisting physical truths. We sketch how this approach might extend to the perception of relativistic quantum objects, and to classical objects of macroscopic scale.

Despite substantial efforts by many researchers, we still have no scientific theory of how brain activity can create, or be, conscious experience. This is troubling, since we have a large body of correlations between brain activity and consciousness, correlations normally assumed to entail that brain activity creates conscious experience. Here I explore a solution to the mind-body problem that starts with the converse assumption: these correlations arise because consciousness creates brain activity, and indeed creates all objects and properties of the physical world. To this end, I develop two theses. The multimodal user interface theory of perception states that perceptual experiences do not match or approximate properties of the objective world, but instead provide a simplified, species-specific, user interface to that world. Conscious realism states that the objective world consists of conscious agents and their experiences; these can be mathematically modeled and empirically explored in the normal scientific manner.

The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers would only provide dramatic speedups for a few specific problems, for example, factoring integers and breaking cryptographic codes in the conventional quantum computing approach. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In the conventional approach, it is implemented through the tensor product of qubits. In the suggested geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on a three-dimensional sphere, which is very different from the usual Hilbert space scheme.

The current work considers the sprefield wave functions received as special g-qubit solutions of Maxwell equations in the terms of geometric algebra. I will call such g-qubits spreons or sprefields. The purpose of this article is to analyze behavior of such wave functions in scattering and measurements. It is shown that sprefields are defined through the whole three-dimensional space at all values of the time parameter. They instantly change all their values when get scattered, that is subjected to Clifford translation. In "measurements", when a sprefield acts on a static geometric algebra element through the Hopf fibration, sprefield collapses and new geometric algebra non static, rotating element is thereby created.

Maxwell equation in geometric algebra formalism with equally weighted basic solutions is subjected to continuously acting Clifford translation. The received states, operators acting on observables, are analyzed with different values of the Clifford translation time factor and through the observable measurement results.

When quantum mechanical qubits as elements of two dimensional complex Hilbert space are generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space, geometrically formal complex plane becomes explicitly defined as an arbitrary, variable plane in 3D. The result is that the quantum state definition and evolution receive more detailed description, including clear calculations of geometric phase, with important consequences for topological quantum computing.

The research considers wavelike objects that are elements of even subalgebra of geometric algebra in three dimensions. The used formalism particularly eliminates long existing confusion about the reasons behind the appearance of the imaginary unit in quantum mechanics and introduces clear definition of wave functions. When a wave function acts through the Hopf fibration on a localized geometric algebra element, that is executing a measurement, the result can be named as "collapse" of the wave function.

Recently suggested scheme [1] of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors.

The approach initialized in [1], [2] is used for description and analysis of qubits, geometric phase parameters – things critical in the area of topological quantum computing [3], [4]. The used tool, Geometric (Clifford) Algebra [5], [6], is the most convenient formalism for that case. Generalizations of formal " complex plane " to an arbitrary variable plane in 3D, and of usual Hopf fibration to the map generated by an arbitrary unit value element 0 g   3 G are resulting in more profound description of qubits compared to quantum mechanical Hilbert space formalism.

In this paper we start exploring the procongruence completions of three varieties of curve complexes attached to hyperbolic surfaces, as well as their automorphisms groups. The discrete counterparts of these objects, especially the curve complex and the so-called pants complex were defined long ago and have been the subject of numerous studies. Introducing some form of completions is natural and indeed necessary to lay the ground for a topological version of Grothendieck-Teichmüller theory. Here we state and prove several results of fundational nature, among which reconstruction theorems in the discrete and complete settings, which give a graph theoretic characterizations of versions of the curve complex as well as a rigidity theorem for the complete pants complex, in sharp contrast with the case of the (complete) curve complex, whose automorphisms actually define a version of the Grothendieck-Teichmüller group, to be studied elsewhere (see [20]). We work all along with the procongruence completions-and for good reasons-recalling however that the so-called congruence conjecture predicts that this completion should coincide with the full profinite completion.