We present Clifford fully homomorphic encryption (FHE), the first ring-learning with errors (RLWE)-based FHE scheme with native support for all Clifford algebra operations, enabling privacy-preserving computation on geometric data. Unlike conventional FHE schemes that flatten geometric structure into scalar operations, our approach maintains the algebraic properties of multi-vectors through homomorphic geometric products. Building on this foundation, we introduce geometric neural networks that operate directly on encrypted multi-vectors, achieving the first-ever demonstration of privacy-preserving geometric deep learning. Our construction encrypts three-dimensional point clouds as Cl(3,0) multi-vectors and performs encrypted classification with less than 1% accuracy loss compared to plaintext, completing inference in under 60 s. These results demonstrate that geometric algebra (GA) provides unique advantages for both cryptographic constructions (enabling structure-preserving encryption) and machine learning (ML) (enabling privacy-preserving geometric learning), opening new pathways at the intersection of cryptography, ML and applied mathematics. This article is part of the theme issue 'Modern applications of geometric algebra'.
We introduce splossoms, the spherical analogue of polynomial blossoms, extending Ramshaw's theory of blossoming into spherical geometry. We note that the classical blossom axioms require spherical reinterpretation: strict symmetry does not hold on S2, and we identify the appropriate spherical analogues that the splossom satisfies. Splossoms provide a structured approach to designing spherical curves, generalizing Shoemake's SLERP interpolation. In addition, we define spolynomials, iterated spherical interpolants that mirror polynomial structures such as Bézier and B-spline forms, and we study their continuity. Potential applications include robotics, CNC milling, virtual/augmented reality and computer animation, where orientation curves on spheres are central. The splossom is naturally expressed in the framework of geometric algebra (GA), where rotors and spinors generalize quaternionic rotation and make spherical curve construction both simpler and more powerful. We argue that while splines on Lie groups and algebras are of general mathematical interest, GA provides a direct and computationally efficient setting for implementation. This article is part of the theme issue 'Modern applications of geometric algebra'.
Several implementations of geometric algebra are available, and many have been approved in the past and can be found in various software codes. For low-dimensional algebras, representations as a sum of basis blades are typically the most suitable. However, isomorphic or irreducible matrix representations are also options. Techniques such as compression on a per-coordinate, per-grade or per-group basis, as well as the use of sparse matrices, can help reduce memory consumption. In the case of high-dimensional algebras, factorized representations can significantly decrease the memory needed for blades and versors, reducing it to O(n2), as opposed to the O(2n) required to represent a complete basis of blades. This is particularly relevant in the context of quantum computing, where high dimensionality and memory usage are increasingly important considerations. This paper reviews various implementation strategies, classifies the corresponding software tools and libraries and highlights recent advancements aimed at reducing memory consumption within the tool Gaalop. This article is part of the theme issue 'Modern applications of geometric algebra'.
Null geometric algebra (NGA) refers to a basis-free version of Clifford algebra where the base vector space is spanned by vectors whose inner product with itself equals zero. It provides powerful tools for manipulating expressions in conformal geometric algebra (CGA) when making symbolic reasoning in classical geometry. This paper develops several new techniques in NGA and uses them to explore the geometric interpretations of basic algebraic objects in NGA-null monomials, their scalar parts and pseudo-scalar parts, centred null binomials-and illustrates how these objects are used in making geometric reasoning for problems in classical geometry. This article is part of the theme issue 'Modern applications of geometric algebra'.
This paper describes the geometric algebra for quadrics (GAQ) with a focus on elements representing Euclidean transformations, particularly translations and rotations. We provide general methods of deriving the generators of such transformations and verify their correctness on examples. We also show how to compute the versors numerically, yet their exact form is rather technical and extensive because of the high dimension of GAQ. Thus, we introduce a convenient C++ implementation. This article is part of the theme issue 'Modern applications of geometric algebra'.
The integration of data-driven and knowledge-driven approaches in generative geospatial modelling (GGM) is often hindered by their mathematical incompatibilities. Here, we propose a geometric algebra (GA)-based framework that employs a unified multi-vector representation to fuse heterogeneous data and diverse knowledge. The framework facilitates structured reasoning and hypothesis generation through a task-adaptable, five-stage cycle: representation, reasoning, generation, synthesis and computation. We illustrate this design through three case studies covering constrained trajectory reconstruction, typhoon intensity prediction and large language model-based GA code generation, which instantiate different components and implementation levels of the proposed framework. By offering a cohesive mathematical perspective, our work provides a conceptual and methodological framework for interpretable and constraint-aware GGM. This article is part of the theme issue 'Modern applications of geometric algebra'.
We introduce various parametric techniques to construct and manipulate triply orthogonal curvilinear lattices. Using principal contact tangent elements and the rationalization of spheres through conformal transformations, we detail their constraints and provide an inverse mapping at the boundary. We develop our approach from the perspective of a three-dimensional (3D) practitioner, working with geometric primitives to express various forms with this system. This article is part of the theme issue 'Modern applications of geometric algebra'.
In geometric algebra (GA), real and complex integral transforms (Fourier transforms, linear canonical transforms (LCTs), wavelets, fractional Fourier transforms, special affine Fourier transforms, etc.) can be generalized to holistic higher-dimensional quaternionic, octonionic and Clifford GA versions. This review introduces the foundations of these generalizations, provides representative examples of these new types of transforms and refers to some of their applications in mathematics, physics and engineering. This article is part of the theme issue 'Modern applications of geometric algebra'.
The Brauer-Weyl theorem (Brauer, Weyl. 1935 Am. J. Maths. 57, 425-449. (doi:10.2307/2371218)) states that the algebra of tensor products of spinors is isomorphic to the Clifford (geometric) algebra, in any number of space-time dimensions. The algebraic relation between fermions and bosons seen in the standard model agrees with the Brauer-Weyl theorem, but not with supersymmetry. It is commonly argued that string theory requires supersymmetry, on the grounds that non-supersymmetric bosonic string theory (i) does not admit fermions, and (ii) has an unstable tachyonic ground state. This paper rebuts those standard objections, pointing out that bosonic string theory admits fermions lying on a D-brane boundary of open strings, and that the open-string tachyon has all the properties of a Higgs field. The original 1970's formulation of bosonic string theory, re-envisaged as a theory of all the forces, not just the strong force, fits the standard model beautifully. This article is part of the theme issue 'Modern applications of geometric algebra'.
This work explores the application of quantum entanglement to the analysis of cooperative three-player games. The interactions between players influence their behaviour and the overall outcomes of the transferable utility (TU) cooperative game. The mathematical framework employed is the quantum register algebra (QRA), which algebraizes the Dirac formalism. By using QRA, we can simulate the game and explore how different levels of entanglement impact players. This article is part of the theme issue 'Modern applications of geometric algebra'.
When a lattice in Rn is superimposed on a rotated copy with coincident origins, coincidence points may arise for specific rotations, forming a coincidence site lattice (CSL). In crystallography, particularly in two and three dimensions, CSL theory is fundamental for describing low-energy grain boundaries. Geometric (Clifford) algebra provides a natural framework for this problem, as orthogonal transformations can be expressed as products of reflections. This article presents a unified account of the geometric algebra approach to coincidence isometries, including a Cartan-type decomposition of coincidence rotations for hypercubic lattices Zn. We then summarize the analytical characterization of coincidence rotations, indices and bases for planar hexagonal lattices. The main new result is a rigorous extension of these results to honeycomb structures, such as graphene, which are periodic but not Bravais lattices. We prove that the coincidence structure of two rotated honeycombs is completely determined by the coincidence lattice of their associated hexagonal Bravais lattices. As an application, structures near the first magic angle in twisted bilayer graphene are analysed, yielding quantitative agreement with experimentally observed Moiré lattice parameters. This article is part of the theme issue 'Modern applications of geometric algebra'.
Abstract This paper presents the very basics of W. K. Clifford's geometric algebras (GAs), which encompass Euclidean GA, as well as of the most important ones for engineering, graphics, virtual reality, signal processing, encryption, neural networks (deep learning), crystallography, physics and geographic information systems etc.: quaternions, conformal GA (CGA), projective (plane) GA (PGA), space-time GA and spinor algebras for physics, quantum register algebra (QRA) for qubit computations and algebras for computing with conics and quadrics. This article is part of the theme issue ‘Modern applications of geometric algebra’.
Computing the transformation between noisy collections of geometric primitives is a classic problem in three-dimensional computer vision and geometric computing. In this paper, we consider minimum sets of noisy objects and provide closed-form formulae to compute the Euclidean transformation that best transforms from one of these sets to another set. We enumerate these for all primitive object groups that appear in common three-dimensional computer vision problems and describe example applications. This article is part of the theme issue 'Modern applications of geometric algebra'.
This paper studies the intersection of multiple spherical shells and presents a dimension-independent algorithm based on conformal geometric algebra (CGA), providing both a decision test and a characterization of the feasible region. At a high level, the method proceeds in two stages. First, it tightens each shell's radius interval by comparing it with the result of the intersection of spheres. Second, it reconciles shells pairwise to ensure mutual compatibility, updating bounds until either emptiness is certified or a consistent family of intervals is obtained. The output is a compact, interpretable description of the solution set that integrates directly with standard CGA pipelines. This formulation is especially useful when multiple shells must be enforced simultaneously. Examples in R3 and R6 illustrate how the approach captures interval-based constraints without resorting to dimension-specific constructions or ad hoc geometric casework. This article is part of the theme issue 'Modern applications of geometric algebra'.
Hypercomplex number systems, such as complex numbers and quaternions, have found many uses in the area of signal analysis. Clifford algebras, along with the embedding of (usually) a real vector space, can be used to represent some of the commonly used hypercomplex algebras through subalgebras. Concepts used in hypercomplex algebras can be easily further extended to Clifford algebra. An overview of methods along with their advantages and disadvantages based on Clifford algebra, mostly used in signal analysis, is given in this review, with a focus on applications in neural networks. The use of the described methods is discussed in the context of medical imaging. This article is part of the theme issue 'Modern applications of geometric algebra'.
This review provides a comprehensive overview of the applications of geometric algebra (GA) in electrical engineering, with a particular focus on power systems. The paper covers the fundamental concepts of GA and its advantages over traditional mathematical approaches in power system analysis. Key topics explored include: an introduction to GA and its relevance to electrical engineering; AC circuit theory using GA, covering GA formulations for power flow analysis in non-sinusoidal conditions including harmonic and interharmonic analysis for single-phase and multi-phase systems; power-free current compensation techniques that move beyond traditional reactive power concepts; parameter identification in electrical circuits using GA; GA-based perspectives on electrical transformations (Clarke or Fortescue) and the analysis of electrical curves and their geometric invariants. The review synthesizes recent advancements, including contributions from the authors and colleagues, discusses future directions and highlights the potential of GA to address current and emerging challenges in power system analysis, control, smart grids and renewable energy systems. We aim to provide both a theoretical foundation and practical insights for researchers and practitioners in the field. This article is part of the theme issue 'Modern applications of geometric algebra'.
Abstract We consider several aspects of data analysis that are underemphasized in most presentations of statistical theory and practice. We illustrate some of these with a simple example of Bayesian workflow and conclude by emphasizing shared aspects of Bayesian and non-Bayesian data analysis workflows. The audience for this paper includes statisticians and applied researchers who might not be aware of these commonalities across apparently opposing statistical philosophies. This article is part of the theme issue ‘Statistical workflow’.
Abstract This special issue examines how natural and artificial intelligences (AIs) model the world, and what this modelling reveals about cognition and relationships between life and mind. Rather than adopting a single definition, the collection considers how world models function and emerge in biological and artificial systems, exploring a diverse range of world modelling including causal, self-referential, individual goal-directed, collective and narrative forms. A recurring theme is the extent to which current AI systems trained on vast quantities of data learn the context-sensitive, temporally embedded, value-laden dimensions of world modelling that characterize diverse biological intelligences, or whether their impressive capabilities arise primarily from statistical surface regularities. The contributions also raise broader issues concerning embodiment, complexity, learning architectures and the social and scientific contexts in which world models operate. With this collection, we hope to clarify the conceptual landscape, identify key points of similarity and divergence between natural and artificial minds, and outline questions that may guide future research on the forms of world modelling that support grounded understanding, robust agency and potentially human-like general intelligence. This article is part of the theme issue ‘World models in natural and artificial intelligence’.
Abstract The phenomena occurring at interfaces are complex, and elucidating their influence on catalytic performance remains a major challenge. Among the many physicochemical processes active at surfaces and interfaces, the formation and evolution of chemical bonds during catalytic reactions constitute a fundamental aspect of catalytic function. Here, we briefly revisit established models of chemical bonding, which include Dewar–Chatt–Duncanson (DCD), Blyholder, Newns–Anderson and the d-band models, together with real-space density-based frameworks such as atoms in molecules (AIM) and the electron localization function (ELF). We highlight their latest applications in experimental and theoretical studies, demonstrating their continued relevance. We then outline illustrative examples of emerging directions in this area. While the contributions in this special issue span a broad spectrum of topics within surface and interface science, this introductory article provides a unifying conceptual foundation that complements the diverse studies showcased. This article is part of the theme issue ‘Surfaces, interfaces and heterogeneous catalysis’.
