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Toroidal

Toroidal refers to any shape, structure, or configuration that resembles or is mathematically associated with a torus, a doughnut-like surface generated by rotating a circle of radius r about an axis in its plane located at a distance R > r from the circle's center, ensuring the surface forms a ring without self-intersection.^[1] This geometric form, characterized by its genus of one—meaning it has a single hole—distinguishes it topologically from simpler surfaces like the sphere and serves as a foundational object in three-dimensional space.^[2] In mathematics, toroidal structures underpin various coordinate systems and topological studies; for instance, toroidal coordinates (\sigma, \tau, \phi) describe points in space using confocal tori and spheres, with meridional planes, facilitating solutions to Laplace's equation in axisymmetric problems.^[3] The torus also appears in dynamical systems, such as geodesic flows on flat tori, where it models periodic behaviors and irrational rotations on the surface T^2 = S^1 \times S^1.^[4] These properties make the torus essential for understanding embeddings, knots, and higher-dimensional analogs in algebraic geometry and topology. In physics and engineering, toroidal geometries enable efficient confinement and energy transfer; in plasma physics, tokamaks employ toroidal magnetic fields to stably confine high-temperature plasmas for nuclear fusion experiments, leveraging the closed-loop topology to minimize particle and energy losses.^[5] Similarly, toroidal moments in electrodynamics represent an alternative to traditional multipoles, arising from current distributions that produce field patterns with enhanced near-field control for applications in antennas and metamaterials.^[6] In electrical engineering, toroidal transformers and inductors utilize ring-shaped cores to achieve low magnetic leakage, high efficiency, and compact designs, outperforming traditional E-I cores in power electronics by reducing electromagnetic interference.^[7] Toroidal forms also appear in biological contexts, such as the packaging of DNA into toroids in viruses and sperm chromatin.^[8]

Geometry

Definition and Basic Shapes

A toroid is a doughnut-shaped surface generated by revolving a closed plane curve about an axis lying in its plane without intersecting the curve.^[9] This surface of revolution features a central hole, distinguishing it from simpler shapes like spheres or cylinders.^[10] While the terms toroid and torus are often used interchangeably, a precise distinction exists: a torus specifically refers to a toroid formed by revolving a circle, whereas a toroid more generally arises from any closed curve, such as a square or rectangle.^[10] For the standard torus, two key parameters define its geometry: the major radius R, which is the distance from the center of the tube to the center of the torus, and the minor radius r, which is the radius of the tube itself.^[10] The term "toroid" was coined in the late 19th century, derived from the Latin torus meaning "swelling" or "bulge," with the suffix -oid indicating resemblance; its first recorded use dates to 1895–1900.^[9] Visually, toroidal shapes evoke everyday objects like doughnuts or life preservers, which illustrate the genus-1 topology of a torus—a compact orientable surface with exactly one hole.^[10]^[11]

Toroidal Surfaces and Volumes

A toroidal surface is generated by revolving a closed curve in a plane around an axis in that plane that does not intersect the curve. For a circular torus, formed by rotating a circle of radius r around an axis at distance R from its center, the surface area S is given by S = 4\pi^2 R r.^[12]^[13] This formula arises from integrating the arc length elements along the path of revolution, assuming R > r to prevent self-intersection.^[14] The volume V enclosed by this surface is V = 2\pi^2 R r^2.^[12]^[15] This result follows from the method of disks or washers applied to the solid of revolution, where cross-sections perpendicular to the axis of rotation yield circular areas scaled by the distance from the axis.^[16] These formulas generalize through Pappus's centroid theorem, which states that the surface area of a solid of revolution is S = 2\pi R C, where C is the length of the generating curve and R is the distance from the axis of rotation to the centroid of that curve; similarly, the volume is V = 2\pi R A, where A is the area enclosed by the generating curve.^[14]^[12] For the circular torus, the generating circle has C = 2\pi r and A = \pi r^2, recovering the specific formulas above. This theorem extends to non-circular cross-sections; for instance, a toroid with a square cross-section of side length a has volume V = 2\pi R (\ a^2\ ) and surface area S = 2\pi R (4a), where the centroid lies at distance R from the axis.^[14]^[16] The standard formulas apply under the condition R > r for circular toroids, ensuring a ring-like shape without self-intersection; when R < r, the surface forms a spindle torus with intersecting regions along the axis.^[17]^[18]^[19]

Mathematical Properties

Parametric Equations and Metrics

The standard parametrization of a torus of major radius R and minor radius r (with R > r > 0) embedded in \mathbb{R}^3 is given by

\begin{align*} x &= (R + r \cos \theta) \cos \phi, \\ y &= (R + r \cos \theta) \sin \phi, \\ z &= r \sin \theta, \end{align*}

where \theta, \phi \in [0, 2\pi).^[20] This representation arises from revolving a circle of radius r in the xz-plane, centered at (R, 0, 0), around the z-axis.^[20] To derive the first fundamental form, or metric tensor, compute the partial derivatives of the parametrization with respect to \theta and \phi. The tangent vectors are \mathbf{r}_\theta = (-r \sin \theta \cos \phi, -r \sin \theta \sin \phi, r \cos \theta) and \mathbf{r}_\phi = (-(R + r \cos \theta) \sin \phi, (R + r \cos \theta) \cos \phi, 0). Their dot products yield the coefficients E = \mathbf{r}_\theta \cdot \mathbf{r}_\theta = r^2, F = \mathbf{r}_\theta \cdot \mathbf{r}_\phi = 0, and G = \mathbf{r}_\phi \cdot \mathbf{r}_\phi = (R + r \cos \theta)^2, resulting in the line element

ds^2 = r^2 \, d\theta^2 + (R + r \cos \theta)^2 \, d\phi^2.

^[20] This orthogonal metric describes the intrinsic geometry of the surface, enabling calculations of arc lengths and angles without reference to the embedding space.^[21] The surface area of the torus is $4\pi^2 R r, obtained by integrating \sqrt{EG - F^2} = r (R + r \cos \theta) over [0, 2\pi) \times [0, 2\pi). The volume of the solid torus is $2\pi^2 R r^2.^[1] The Gaussian curvature K of the torus, computed via the formula K = (LN - M^2)/(EG - F^2) using the second fundamental form coefficients L, M, N, is

K = \frac{\cos \theta}{r (R + r \cos \theta)}.

^[22] K > 0 along the outer equator (\theta = 0), indicating elliptic points, while K < 0 along the inner equator (\theta = \pi), indicating hyperbolic points; K = 0 at the upper and lower equators (\theta = \pm \pi/2).^[22] This variation underscores the non-constant curvature of the torus, distinguishing it from spheres or planes.^[20] The metric facilitates geodesic computations on the torus by determining the Christoffel symbols of the second kind, which appear in the geodesic equations \theta'' + \Gamma^\theta_{\theta\theta} (\theta')^2 + \cdots = 0 and similarly for \phi.^[21] These symbols, derived from partial derivatives of the metric coefficients (e.g., \Gamma^\theta_{\phi\phi} = -(R + r \cos \theta) \sin \theta / r), allow solving for shortest paths, such as closed geodesics winding around the torus.^[21] For higher-genus toroids (surfaces of genus g > 1), generalizations involve hyperbolic metrics of constant negative curvature -1, as guaranteed by the uniformization theorem, contrasting the flat or variable-curvature metrics of the genus-1 torus.^[23] Such metrics can be constructed via Fuchsian groups acting on the hyperbolic plane, providing a uniform framework for studying their intrinsic geometry.^[23]

Topology and Polyhedra

The standard torus is a compact orientable surface of genus g=1, characterized by a single hole that distinguishes it topologically from the sphere (genus 0).^[1] This genus assignment reflects the surface's connectivity, where the torus cannot be continuously deformed into a sphere without cutting or gluing.^[24] The Euler characteristic \chi of the torus is 0, satisfying the formula \chi = V - E + F = 0 for any cellular embedding on the surface, where V, E, and F denote the numbers of vertices, edges, and faces, respectively.^[25] The fundamental group of the torus is the free abelian group \mathbb{Z} \times \mathbb{Z}, capturing the surface's one-dimensional holes through homotopy classes of loops.^[26] This group is generated by two commuting elements corresponding to the meridional loop (encircling the tube) and the longitudinal loop (encircling the hole through the torus).^[26] Toroidal polyhedra realize the torus's topology in three-dimensional space as discrete approximations with genus at least 1. The Császár polyhedron, discovered in 1949, is a toroidal polyhedron of genus 1 with 7 vertices, 21 edges, and 14 triangular faces, embedding the complete graph K_7 without crossings.^[27] Its dual, the Szilassi polyhedron from 1977, also has genus 1 and features 7 hexagonal faces where each pair shares an edge, with 14 vertices and 21 edges.^[27] These examples achieve the minimal vertex and face counts for embedded toroidal polyhedra, illustrating the constraints of genus-1 embeddings.^[28] Graphs embeddable on the torus without edge crossings are termed toroidal graphs, with simple toroidal graphs (no loops or multiple edges) supporting at most $3V$ edges, as derived from Euler's formula and the assumption of triangular faces in maximal embeddings.^[29] This bound is attained by triangulations like the Császár polyhedron's skeleton. Higher-genus toroids generalize the standard torus to g-holed surfaces, such as the double torus for g=2, with Euler characteristic \chi = 2(1 - g).^[30] These surfaces exhibit increasingly complex topology, where each additional handle reduces \chi by 2, enabling embeddings of denser graphs or polyhedra compared to the genus-1 case.^[30]

Physics Applications

Toroidal Coordinates

Toroidal coordinates provide an orthogonal curvilinear system in three-dimensional space, obtained by rotating the two-dimensional bipolar coordinate system about the z-axis, resulting in coordinate surfaces consisting of tori, spheres, and azimuthal planes. The coordinates are denoted (σ, τ, φ), where σ ∈ [0, 2π] is the poloidal angle, τ ≥ 0 is the toroidal parameter determining the scale of the toroid, and φ ∈ [0, 2π] is the azimuthal angle; the system is based on torus geometry to facilitate solutions for problems exhibiting toroidal symmetry. The focal distance a > 0 separates the two foci on the z-axis at z = ±a, and the coordinates cover the entire space excluding the degenerate circle of radius a in the xy-plane.^[31]^[32] The transformation to Cartesian coordinates is given by

\begin{align} x &= \frac{a \sinh \tau \cos \phi}{\cosh \tau - \cos \sigma}, \\ y &= \frac{a \sinh \tau \sin \phi}{\cosh \tau - \cos \sigma}, \\ z &= \frac{a \sin \sigma}{\cosh \tau - \cos \sigma}, \end{align}

where the denominator ensures the geometry aligns with bipolar origins.^[31]^[33] The scale factors, essential for the metric and differential operators, are derived from the line element ds^2 = h_\sigma^2 d\sigma^2 + h_\tau^2 d\tau^2 + h_\phi^2 d\phi^2, computed via the partial derivatives of the position vector \mathbf{r}(\sigma, \tau, \phi). Specifically, h_i = \left| \frac{\partial \mathbf{r}}{\partial u^i} \right| for u^i = \{\sigma, \tau, \phi\}, yielding

h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos \sigma}, \quad h_\phi = \frac{a \sinh \tau}{\cosh \tau - \cos \sigma}.

These are obtained by direct differentiation: for instance, \frac{\partial \mathbf{r}}{\partial \phi} = \left( -\frac{a \sinh \tau \sin \phi}{\cosh \tau - \cos \sigma}, \frac{a \sinh \tau \cos \phi}{\cosh \tau - \cos \sigma}, 0 \right), whose magnitude is h_\phi; similar calculations for the others confirm the forms, with the equality h_\sigma = h_\tau arising from the rotational symmetry in the bipolar base. The Jacobian \sqrt{g} = h_\sigma h_\tau h_\phi = \frac{a^2 \sinh \tau}{(\cosh \tau - \cos \sigma)^2} follows from the product of scale factors.^[33]^[32] Orthogonality is established by verifying the metric tensor g_{ij} = \frac{\partial \mathbf{r}}{\partial u^i} \cdot \frac{\partial \mathbf{r}}{\partial u^j} has vanishing off-diagonal elements (g_{ij} = 0 for i \neq j), a property inherited from the orthogonal bipolar coordinates and preserved under rotation about the symmetry axis. This ensures the coordinate vectors \mathbf{e}_i = \frac{1}{h_i} \frac{\partial \mathbf{r}}{\partial u^i} are mutually perpendicular.^[32]^[33] The Laplace equation \nabla^2 \Psi = 0 separates in toroidal coordinates due to the form of the Laplacian in orthogonal systems,

\nabla^2 \Psi = \frac{1}{h_\sigma h_\tau h_\phi} \left[ \frac{\partial}{\partial \sigma} \left( \frac{h_\tau h_\phi}{h_\sigma} \frac{\partial \Psi}{\partial \sigma} \right) + \frac{\partial}{\partial \tau} \left( \frac{h_\sigma h_\phi}{h_\tau} \frac{\partial \Psi}{\partial \tau} \right) + \frac{\partial}{\partial \phi} \left( \frac{h_\sigma h_\tau}{h_\phi} \frac{\partial \Psi}{\partial \phi} \right) \right] = 0,

allowing solutions of the form \Psi(\sigma, \tau, \phi) = S(\sigma) T(\tau) \Phi(\phi), leading to ordinary differential equations in each variable after substitution and separation of constants. The azimuthal part yields \Phi(\phi) \propto e^{i m \phi} for integer m, while the \sigma and \tau equations involve associated Legendre functions: solutions are toroidal harmonics P_{n-m}^m (\cos \sigma) Q_{n-m}^m (i \sinh \tau) e^{i m \phi}, where P and Q are Legendre functions of the first and second kinds. These are applied in electrostatics to compute potentials for ring-shaped conductors or charged rings, where constant-\tau surfaces align with the boundary.^[32]^[33]

Electromagnetic and Plasma Confinement

In electromagnetism, toroidal multipole moments represent a distinct class of multipolar expansions separate from traditional electric and magnetic dipoles, emerging from current distributions with azimuthal symmetry, such as loops of current flowing in toroidal patterns. These moments arise in the vector spherical harmonic decomposition of the electromagnetic field, capturing contributions from poloidal or toroidal current configurations that do not reduce to standard dipole terms. A key example is the toroidal dipole moment, defined for a localized current density \mathbf{J}(\mathbf{r}, t) as \mathbf{T} = \frac{1}{10c} \int \left[ (\mathbf{r} \cdot \mathbf{J}) \mathbf{r} - 2 r^2 \mathbf{J} \right] d^3 r, where c is the speed of light.^[34] Toroidal configurations find practical application in devices like toroidal solenoids, where a tightly wound coil in a doughnut shape produces a magnetic field that is uniform and strong inside the core but approaches zero outside, due to the symmetric cancellation of field lines beyond the torus. This containment arises from Ampère's law, \nabla \times \mathbf{B} = \mu_0 \mathbf{J}, applied to the azimuthal current, ensuring the field lines form closed loops confined within the solenoid's volume. Such designs are advantageous for generating localized fields without external interference, as the geometry minimizes fringing effects compared to linear solenoids.^[35] In plasma physics, toroidal geometries enable effective confinement for fusion research, particularly in tokamaks, which employ a strong toroidal magnetic field B_\phi generated by external coils, combined with a poloidal field B_p from the plasma current, to create helical field lines that trap charged particles. Stability against macroscopic instabilities, such as kink modes, is quantified by the safety factor q = \frac{r B_\phi}{R B_p}, where r is the minor radius and R the major radius; values of q > 1 at the plasma edge prevent field line reconnection and particle loss. The toroidal shape inherently minimizes field leaks by closing magnetic surfaces, reducing end-losses inherent in linear devices and leveraging Ampère's law to sustain equilibrium via \nabla \times \mathbf{B} = \mu_0 \mathbf{J} within the confined volume.^[36] The tokamak concept originated in the 1950s, proposed by Soviet physicists Igor Tamm and Andrei Sakharov as a toroidal system to achieve controlled thermonuclear fusion through magnetic confinement. Their design addressed limitations of earlier straight-cylinder approaches by curving the plasma into a torus, enabling steady-state operation without axial losses. A prominent modern implementation is the ITER project, an international collaboration under construction in France as of 2025, targeting first plasma in 2034 and deuterium-tritium operations beginning in 2039 to demonstrate net energy gain from fusion.^[37]^[38]

Engineering Applications

Electrical Devices

Toroidal inductors consist of a doughnut-shaped magnetic core, typically made from materials such as ferrite or iron powder, around which conductive wire is wound to form multiple turns.^[39] These cores confine the magnetic field within the ring structure, resulting in low leakage flux that minimizes electromagnetic interference in surrounding circuits.^[40] The inductance L of a toroidal inductor is given by the formula

L = \frac{\mu N^2 A}{2\pi R},

where \mu = \mu_0 \mu_r is the permeability of the core material, \mu_0 is the permeability of free space, \mu_r is the relative permeability of the core, N is the number of turns, A is the cross-sectional area of the core, and R is the mean radius of the toroid; this approximation holds for thin toroids where the inner and outer radii are close.^[40] Toroidal transformers, which extend this design to include primary and secondary windings on the same core, offer reduced electromagnetic interference due to the closed magnetic path that contains nearly all flux within the core.^[41] Their efficiency often exceeds 95%, attributed to lower core losses and minimal stray magnetic fields, making them suitable for applications in audio equipment and power supplies where clean signal transmission is essential.^[42] For instance, in high-fidelity audio systems, toroidal transformers reduce hum and noise by suppressing external magnetic coupling.^[43] Winding techniques for toroidal cores emphasize uniform distribution of turns around the ring to ensure even flux density and prevent hotspots that could lead to uneven heating or saturation.^[44] Specialized machinery or manual methods, such as distributing wire evenly under controlled tension, are employed to achieve this, particularly with high-permeability materials like ferrite for high-frequency operation or iron powder for broader frequency ranges and higher current handling.^[39] Compared to solenoidal inductors, which have cylindrical cores with open ends, toroidal designs provide advantages in compactness and the absence of end effects that cause flux fringing in solenoids, allowing for higher inductance in a smaller volume.^[45] However, toroidal inductors are more challenging to wind due to the core's geometry, often requiring automated equipment, and they may be prone to saturation if the core material is not selected appropriately for the operating flux levels.^[46] The toroidal transformer concept emerged in the late 19th century, with early implementations by the Ganz Works in Hungary around the early 1880s as part of advancements in alternating current technology.^[47] In modern applications, such as switch-mode power supplies (SMPS), toroidal components are favored for their high efficiency and low EMI, enabling compact, reliable designs in consumer electronics and renewable energy systems.^[48]

Mechanical and Fluid Dynamics

Toroidal propellers, also known as ducted fans with looped blade designs, represent an innovative approach in marine propulsion systems. These propellers feature interconnected blades forming a continuous toroidal shape, which significantly reduces tip vortex formation and cavitation compared to conventional open propellers.^[49] Independent testing on vessels equipped with Sharrow toroidal propellers has demonstrated fuel efficiency improvements ranging from 20% to 30% at cruising speeds, attributed to minimized energy losses from vortex shedding and enhanced thrust generation (as of 2022).^[50] For instance, on a 32-foot catamaran with twin outboards, the design achieved up to 36% better fuel economy at planing speeds while eliminating visible cavitation trails.^[50] In fluid dynamics, toroidal structures manifest as vortex rings, which are self-propagating coherent structures formed during pulsed jet ejections and governed by the Navier-Stokes equations. These rings arise from the roll-up of shear layers in unsteady flows, enabling efficient momentum transfer with minimal dissipation.^[51] In bio-inspired applications, such as mimicking squid locomotion, toroidal vortex rings facilitate pulsed jet propulsion, where short-duration ejections (formation number ≈4) maximize propulsive efficiency up to 80% by augmenting thrust through entrained fluid mass.^[52] Engineering adaptations, including synthetic jet actuators, leverage these dynamics for compact propulsion in underwater vehicles, reducing drag and improving maneuverability over steady jets.^[53] Toroidal roller bearings, such as SKF's CARB series, are engineered for high-load rotational applications where shaft misalignment or deflection occurs (specifications as of 2023). These bearings consist of a single row of barrel-shaped rollers between concave inner and outer ring raceways, allowing axial displacement up to several millimeters while supporting radial loads exceeding 2,000 kN in large sizes.^[54] The design accommodates up to 0.5° misalignment without increased stress, making them suitable for heavy machinery like crushers and wind turbine gearboxes, where they enable smoother rotation under combined high radial and moderate axial loads.^[54] A key advantage of toroidal geometries in mechanical systems is the closed-loop path, which minimizes frictional and leakage losses by maintaining continuous contact and fluid containment. In toroidal continuously variable transmissions (CVTs) for vehicles, this configuration supports high torque capacities up to 500 Nm while achieving efficiencies of 86-97% across operating ratios (as of 1981 design studies), outperforming belt-driven CVTs in electric and hybrid drivetrains.^[55] NASA design studies highlight how the recirculating power path in dual-toroidal setups reduces spin losses and enables seamless ratio changes without clutches.^[55] Recent developments post-2020 have advanced additive manufacturing for toroidal mechanical components, enabling complex geometries unattainable by traditional methods. For example, 3D-printed toroidal propellers using fused deposition modeling or metal binder jetting have been tested for marine drones, offering customized blade interconnectivity that further reduces noise and cavitation while streamlining production for low-volume applications (as of 2025).^[56]

Biological and Chemical Contexts

Molecular and Cellular Structures

In chemistry, toroidal molecules exhibit doughnut-like topologies, often achieved through interlocked structures such as catenanes, where two or more macrocyclic rings are mechanically linked without covalent bonds.^[57] Jean-Pierre Sauvage pioneered template-directed synthesis of such catenanes in the 1980s using copper(I) ions to coordinate phenanthroline units, enabling efficient formation of these topologically complex molecules, a breakthrough recognized by the 2016 Nobel Prize in Chemistry for molecular machines.^[57] Cyclophanes, large cyclic oligomers of benzene rings, can also adopt toroidal conformations when stacked or linked, providing scaffolds for supramolecular assemblies with toroidal symmetry.^[58] At the cellular level, toroidal structures appear in viral DNA packaging, notably in bacteriophage φ29, where double-stranded DNA is compacted into a toroidal configuration within the icosahedral capsid during maturation.^[59] Cryo-electron microscopy reconstructions reveal this DNA toroid as a tightly wound spool with a radius of approximately 22 nm and a central hole, stabilized by electrostatic interactions and facilitating high-pressure ejection during infection.^[60] In eukaryotic cells, mitochondrial cristae—invaginations of the inner membrane—often form tubular or lamellar folds that enhance surface area for oxidative phosphorylation while maintaining compartmentalization of respiratory complexes.^[61] Supercoiled DNA compaction into toroids is a key mechanism in biological storage, observed in sperm protamine-packaged chromatin and viral genomes, where negatively supercoiled plasmids or chromosomes fold into stable doughnut shapes to fit confined spaces.^[62] In bacteria, the E. coli chromosome adopts a toroidal topology in vivo, with an overall toroid radius on the order of 450 nm, as imaged in live cells, allowing efficient segregation and transcription regulation.^[63] Boron clusters, known as borospherenes, represent a milestone in cluster chemistry; the B₄₀^- anion was predicted in 2014 as the first all-boron fullerene analog with a buckyball-like cage structure with D_2d symmetry, featuring 40 boron atoms. Experimental observation via anion photoelectron spectroscopy confirmed its stability in the gas phase that year.^[64] Subsequent studies exploring derivatives for potential applications in nanomaterials. The stability of toroidal conformations in these molecules and structures relies on non-covalent interactions, including van der Waals forces that promote close packing of aromatic or electron-rich surfaces, and hydrogen bonding that bridges polar groups to prevent unfolding.^[65] In DNA toroids, these forces, alongside electrostatic attractions between phosphates, minimize bending energy and resist dissociation, ensuring compact yet reversible assembly under physiological conditions.^[65]

Hypothetical and Natural Formations

Hypothetical toroidal planets represent a speculative class of celestial bodies shaped like a doughnut, where gravitational self-stability demands exceptionally rapid rotation to counteract the inward collapse of the ring structure. Theoretical models indicate that such configurations could form if a protoplanetary disk collapses into a toroidal shape, but they would violate standard Roche limit considerations for tidal stability unless the rotation period is on the order of hours, providing centrifugal support against gravitational instability. A 2006 study on magnetized tori around Kerr black holes provides analogous insights into the dynamics of rotating toroidal structures in strong gravity, suggesting that similar principles might apply to planetary scales, though no natural examples have been observed. These concepts have been explored in science fiction, such as Larry Niven's Ringworld series, but remain unfeasible under current astrophysical formation mechanisms without artificial intervention. In natural formations, toroidal smoke rings, or vortex rings, emerge during volcanic eruptions when gas and ash are expelled through a confined vent, creating stable, doughnut-shaped vortices that propagate through the atmosphere. Observations at volcanoes like Mount Etna and Stromboli demonstrate these rings forming due to the interaction of eruptive jets with ambient air, with diameters reaching up to 100 meters and persisting for seconds to minutes before dissipating. A 2021 geophysical study analyzed the axial dynamics and acoustic signatures of these volcanic vortex rings, highlighting their potential for real-time eruption monitoring via infrasound detection, as the rings emit consistent low-frequency sounds during flight. Under confinement, bacterial colonies can also exhibit toroidal patterns, as seen in biofilm experiments where rod-shaped cells align circularly in toroidal geometries, driven by surface curvature and nutrient gradients. A 2015 investigation into biofilm formation in curved confinements, including toroidal setups, showed that motile bacteria like Bacillus subtilis organize into ring-like structures to optimize collective motility and resource access. In astrophysical contexts, toroidal accretion disks surround black holes, where matter orbits in a ring-shaped configuration influenced by the Kerr metric's frame-dragging effects, leading to warped, thick tori near the event horizon. These disks are modeled as geometrically thick due to viscous heating and magnetic stresses, with simulations revealing inward spiraling flows that power relativistic jets. A seminal 2006 analysis of magnetized tori in the Kerr spacetime demonstrated how differential rotation amplifies magnetic fields, stabilizing the torus against hydrodynamic instabilities while facilitating accretion rates up to the Eddington limit. Similarly, neutron stars often harbor strong toroidal magnetic fields, confined within the stellar interior by the crust's elasticity, with field strengths exceeding 10^15 gauss in magnetars. Recent 2022 magnetohydrodynamic simulations of rotating neutron stars with mixed poloidal-toroidal fields confirmed that these configurations enhance stability against non-axisymmetric instabilities, contributing to observed pulsar glitches and magnetic field evolution over gigayears. Geological examples of toroidal formations are rare but include subsurface volcanic reservoirs, such as the low-density toroidal deposit identified beneath Tofua Volcano in the Tonga arc, formed by ring fractures preceding caldera collapse. This structure, delineated by seismic tomography, spans several kilometers and acts as a magma storage zone, influencing eruptive patterns similar to those preceding the 2022 Hunga Tonga event. A 2025 geophysical study mapped this toroidal reservoir using gravity and seismic data, revealing its role in sustaining episodic volcanism through buoyant magma ascent. Forming and maintaining these toroidal structures pose significant challenges, primarily due to inherent instabilities without sufficient angular momentum or external support. Computational models post-2020, including fluid dynamics simulations on toroidal geometries, underscore that non-rotating tori collapse under self-gravity, requiring energy inputs equivalent to 10^40 ergs for planetary-scale stabilization—far beyond typical astrophysical processes. These simulations highlight that spin-induced oblateness is essential, but even then, tidal interactions or differential rotation can trigger fragmentation within millions of years.

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[PDF] geometric registration of high-genus surfaces - UCLA Mathematics
Figure 6.1(A) and (B) show two genus-1 torus, denoted by S1 and S2 respectively, with different intensity functions defined on each of them. The two surfaces ...
[23]
Torus -- from Wolfram MathWorld
The single-holed "ring" torus is known in older literature as an "anchor ring." It can be constructed from a rectangle by gluing both pairs of opposite edges ...
[24]
Genus -- from Wolfram MathWorld
Roughly speaking, it is the number of holes in a surface. The genus of a surface, also called the geometric genus, is related to the Euler characteristic chi .
[25]
Euler Characteristic -- from Wolfram MathWorld
The only compact closed surfaces with Euler characteristic 0 are the Klein bottle and torus (Dodson and Parker 1997, p. 125). The following table gives the ...
[26]
fundamental group - PlanetMath.org
Mar 22, 2013 · π1(Rn)≅{0} π 1 ⁢ ( ℝ n ) ≅ { 0 } for each n∈N n ∈ ℕ . · π1(S1)≅Z π 1 ⁢ ( S 1 ) ≅ ℤ . · π1(T)≅Z⊕Z π 1 ⁢ ( T ) ≅ ℤ ⊕ ℤ , where T T is the torus.
[27]
[PDF] Properties of 3-Triangulations for p-Toroid - Athens Journal
The Császár's polyhedron is an example of a 1-toroid from class T1. An example of a regular 1-toroid from class T2 is given in Figure 2, marked here with. P9 ...Missing: academic
[28]
Szilassi Polyhedron - Interactive Mathematics Miscellany and Puzzles
The Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with 14 vertices, 21 edges, and 7 hexagonal faces. It was discovered by the Hungarian ...Missing: genus source
[29]
[PDF] FOUR-COLORING SIX-REGULAR GRAPHS ON THE TORUS
The embedding is a triangulation if and only if e = 3v, and then the average degree equals 6. Thus a graph on the torus either contains a vertex of degree less ...
[30]
[PDF] Mostly Surfaces Richard Evan Schwartz - Brown Math Department
χ(S)=2 − 2g. Here χ is the Euler characteristic, as discussed in §3.4. Thus, a torus has genus 1, and the octagon surface has genus 2. In general, a genus g ...
[31]
Toroidal Coordinates -- from Wolfram MathWorld
. The toroidal coordinates are defined by. x, = (asinhvcosphi)/(coshv-cosu) ... (7). The scale factors are. h_u, = a/(coshv-cosu). (8). h_v, = a/(coshv-cosu). (9).Missing: orthogonal | Show results with:orthogonal
[32]
Methods of theoretical physics : Morse, Philip M ... - Internet Archive
Nov 11, 2022 · Methods of theoretical physics. by: Morse, Philip M. (Philip McCord), 1903-1985. Publication date: 1953. Topics: Mathematical physics. Publisher ...
[33]
[PDF] COORDINATE SYSTEMS , ox ox - LSU Math
(a) Calculate the spherical polar coordinate scale factors: h" h{J, and hQ>'. (b) Check your calculated scale factors by the relation ds/,= hi dq/. 2.2 ...<|control11|><|separator|>
[34]
https://doi.org/10.1515/nanoph-2017-0017
[35]
Magnetic Field inside a Toroid — Collection of Solved Problems
Aug 2, 2019 · The magnetic field is homogeneous inside the toroid and zero outside the toroid. We asses the direction of the magnetic B-field by the right- ...
[36]
Rotational transform - FusionWiki - CIEMAT
Jan 26, 2023 · In tokamak research, the quantity q = 2π/ι is preferred (called the "safety factor"). In a circular tokamak, the equations of a field line on ...Missing: authoritative | Show results with:authoritative
[37]
from A D Sakharov to the present (the 60-year history of tokamaks)
The 60-year history of plasma research work in toroidal devices with a longitudinal magnetic field suggested by Andrei D Sakharov and Igor E Tamm in 1950
[38]
ITER Council endorses updated project schedule
Nov 21, 2016 · The schedule, which covers the period up to First Plasma in 2025 and on to Deuterium-Tritium Operation in 2035, was approved by all Members.
[39]
Exploring the Winding Methods for Toroidal Cores: Best Practices ...
Nov 15, 2023 · Toroidal cores are ring-shaped magnetic cores made from highly permeable materials like iron powder, ferrite, or laminated silicon steel.Missing: hotspots | Show results with:hotspots
[40]
Inductance of a Toroid - HyperPhysics
The inductance of the toroid is approximately L = Henry = mH. This is a single purpose calculation which gives you the inductance value when you make any ...
[41]
Save More, Power Better: Benefits of Toroidal Transformers
Efficiency: The toroidal transformer's higher electrical efficiency reduces power consumption, resulting in lower operating costs over time. · Lower Maintenance ...
[42]
Toroidal vs E-I Core Transformers - DigiKey TechForum
Nov 21, 2022 · Toroidal transformers feature a doughnut-shaped core, offering higher efficiency, compact size, and reduced electromagnetic interference (EMI).
[43]
High-Efficiency Toroidal Inductors for Power & EMI | Custom Coils Blog
Jun 16, 2025 · Toroidal inductors are highly favored in power electronics and energy-efficient applications due to their compactness and high performance.
[44]
What is, and why using Iron Powder Cores? - Caracol Tech
For toroids, winding should be fully and evenly distributed throughout the core to minimize the leakage inductance. Iron powder toroidal cores will always have ...Missing: techniques uniform
[45]
What is the difference between a toroidal inductor and a solenoid ...
Aug 21, 2025 · In terms of size and space efficiency, toroidal inductors have an advantage over solenoid inductors. The toroidal shape allows for a more ...
[46]
https://it.farnell.com/inductors-demystified-exploring-their-function-and-applications-trc-ar
[47]
History of Transformers - Edison Tech Center
Early designs came from Austro-Hungarian and English inventors. William Stanley built the first reliable commercial transformer in 1886, with earlier forms ...Missing: 1890s | Show results with:1890s
[48]
Custom Switch Mode Power Supply Transformers - Amgis
Amgis can develop a wide range of custom transformers and energy storing inductors that can be used exclusively in switch mode power supplies (SMPS).
[49]
A Study on the Effect of Toroidal Propeller Parameters on Efficiency ...
According to Sikirica's research findings, the toroidal propeller has a better advantage in lowering the influence of the cavitation phenomena; the simulation ...
[50]
Sharrow Propeller™ - BoatTEST
The Sharrow Propeller is a remarkable new, patented loop propeller design that CEO and inventor, Greg Sharrow, claims can out-perform conventional propellers.
[51]
https://doi.org/10.1063/1.1565095
[52]
https://doi.org/10.1017/S0022112098008645
[53]
[PDF] Vortex Rings in Bio-inspired and Biological Jet Propulsion
Recent investigations of squid and jellyfish locomotion have demonstrated vortex ring formation during pulsed jet propulsion ... Propulsive efficiency of squid ...Missing: Navier- | Show results with:Navier-
[54]
[PDF] CARB toroidal roller bearings - SKF
Rotate the inner or outer ring to align the rollers during mounting. Where this is not feasible, use a bearing handling tool or other device to keep the bearing ...
[55]
[PDF] Design Studies of Continuously Variable Transmissions for Electric ...
The design studies were performed under contracts to NASA for DOE by. Garrett/AiResearch (toroidal traction CVT), Battelle Columbus Labs (steel V-belt CVT).
[56]
Numerical investigation on the influence of toroidal propeller on the ...
The study reveals that the toroidal propeller with 45° blade angle is found to improve the convection heat transfer coefficient to a maximum of 45 % over the ...
[57]
Press release: The 2016 Nobel Prize in Chemistry - NobelPrize.org
Oct 5, 2016 · The Nobel Prize in Chemistry 2016 is awarded to Jean-Pierre Sauvage, Sir J. Fraser Stoddart and Bernard L. Feringa for their design and production of molecular ...
[58]
Chemical Topology: Complex Molecular Knots, Links, Entanglements
... chemistry, molecular topology, and metal–organic frameworks. Jean-Pierre Sauvage. Download MS PowerPoint Slide. Jean-Pierre Sauvage was born in Paris (1944).
[59]
Structural Ensemble and Dynamics of Toroidal-like DNA Shapes in ...
May 7, 2013 · In the bacteriophage ϕ29, DNA is packed into a preassembled capsid from which it ejects under high pressure. A recent cryo-EM reconstruction ...
[60]
Structure of toroidal DNA collapsed inside the phage capsid - PNAS
Jun 9, 2009 · The DNA bundle is twisted, and the toroid can be seen as a series of constrained hexagonal domains separated by twist walls. In addition, at ...
[61]
Mitochondrial Cristae Architecture and Functions - NIH
From a functional point of view, cristae structure was proposed to increase the inner membrane surface and thus to enhance the capacity of oxidative ...
[62]
Deciphering the structure of DNA toroids - ResearchGate
Aug 6, 2025 · Toroids are small donut shaped organizational units within sperm chromatin and viruses containing DNA and protein.
[63]
Direct imaging of the circular chromosome in a live bacterium - Nature
May 16, 2019 · Here we directly image the circular chromosome in live E. coli with a broadened cell shape. We find that it exhibits a torus topology.
[64]
B-DNA structure and stability: the role of hydrogen bonding, π–π ...
Our analyses provide detailed insight into the role and relative importance of the various types of interactions, such as, hydrogen bonding, π–π stacking ...