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Torus

A torus is a surface of revolution in three-dimensional space, generated by rotating a circle of radius a about an axis in its plane that lies at a distance c from the circle's center, typically forming a doughnut-like shape when c > a.^[1] Topologically, the two-dimensional torus is defined as the Cartesian product of two circles, T^2 = S^1 \times S^1, where S^1 denotes the unit circle, making it a compact orientable surface of genus one with a single hole.^[2] This structure distinguishes it from simpler surfaces like the sphere (genus zero) and endows it with fundamental group \pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}, reflecting its periodic nature in two independent directions.^[2] In geometry, the torus admits various embeddings and metrics; the standard ring torus has parametric equations x = (c + a \cos v) \cos u, y = (c + a \cos v) \sin u, z = a \sin v for parameters u, v \in [0, 2\pi), yielding a surface area of $4\pi^2 a c and an enclosed volume of $2\pi^2 a^2 c for the solid torus.^[1] Depending on the ratio c/a, it can manifest as a non-self-intersecting ring torus (c > a), a horn torus tangent to itself (c = a), or a self-intersecting spindle torus (c < a).^[3] The torus generalizes to higher dimensions as T^n = (S^1)^n, playing a central role in algebraic topology, dynamical systems, and geometry, where it models phenomena like periodic orbits and flat manifolds.^[2] Beyond mathematics, the term "torus" denotes specific structures in other fields, such as the torus palatinus in human anatomy—a bony ridge on the palate—but these derive from the geometric archetype.^[4] Its study has influenced modern applications in computer graphics, physics simulations, and network topologies.

Etymology and History

Etymology

The term "torus" derives from the Latin torus, denoting a swelling, bulge, or rounded protuberance, often likened to a cushion or knot.^[5] This root reflects the shape's characteristic rounded, bulging form, initially applied in architecture from the 1560s to describe a large convex molding at the base of a column.^[5] In mathematical contexts, the word was extended to describe the surface generated by revolving a circle about an axis in its plane but external to it, marking a shift from descriptive to geometric terminology.^[6] An early documented English usage appears in 1860, in William Johnson's The Practical Draughtsman's Book of Industrial Design, defining the torus as "a solid, generated by a circle, revolving about an axis in its own plane."^[6] Prior to this standardization, the shape was commonly known in English as an "anchor ring," evoking its resemblance to a nautical ring.^[1] The related term "toroid" emerged in the late 19th century, formed by combining "torus" with the suffix "-oid" (from Greek -oeidēs, meaning "resembling"), to denote solids or surfaces of toroidal form, often in physics and engineering contexts.^[7] Terminology in other languages mirrors this Latin derivation while adapting phonetically; German retains "Torus," while French employs "tore," reflecting the same geometric connotation.^[8]

Historical Development

The mathematical study of the torus traces its origins to ancient Greece in the 3rd century BCE, where the mathematician Dionysodorus provided the first known formula for the volume of a spindle-shaped torus in his work On the Tore, as referenced by Hero of Alexandria.^[9] This early calculation highlighted the torus as a surface of revolution, building on contemporary explorations of rotational solids by Archimedes, whose methods for volumes of spheres and cylinders influenced subsequent geometric investigations.^[10] In the 18th century, the torus received more formal mathematical treatment through the contributions of Leonhard Euler and Johann Heinrich Lambert, who developed parametric descriptions and analyzed its curvature properties as part of broader studies on surfaces of revolution. Euler's work in the 1770s on surfaces of revolution laid groundwork for later parametric descriptions of the torus, while Lambert advanced studies in hyperbolic geometry.^[10] These efforts marked a shift toward systematic analysis, integrating the torus into the emerging field of differential geometry. The 19th century saw significant advancements in the topological understanding of the torus, driven by Bernhard Riemann's 1851 habilitation lecture, which introduced Riemann surfaces and positioned the torus as a prototypical example of a compact Riemann surface corresponding to elliptic curves. In 1837, Gabriel Lamé analyzed the properties of the torus, contributing to its geometric understanding.^[1] Building on this, Henri Poincaré's late-19th-century work on algebraic topology, including his development of homology groups and the classification of compact surfaces, established the torus as the unique orientable surface of genus 1, distinguishing it from spheres and higher-genus surfaces through invariants like the Euler characteristic of zero.^[11] Around the same time, in the 1850s, the torus became integral to differential geometry through Riemann's manifold concept, enabling studies of intrinsic geometry on curved spaces.^[12] In the 20th century, Heinz Hopf's 1931 discovery of the Hopf fibration revolutionized fiber bundle theory, decomposing the 3-sphere into circle fibers over the 2-sphere and revealing deep connections to toroidal structures in higher-dimensional topology.^[13] Key milestones included the 1960s exploration of lattice tori in physics, such as Morikazu Toda's integrable Toda lattice model for one-dimensional crystals, which employed periodic boundary conditions to simulate infinite systems.^[14] Post-1980s developments in string theory further elevated the torus, with toroidal compactifications providing a framework for reducing extra dimensions while preserving supersymmetry, as seen in heterotic string models on tori.^[15]

Geometric Properties

Standard Torus in Three Dimensions

The standard torus in three dimensions is defined as the surface generated by revolving a circle of radius r (the minor radius), centered at (R, 0, 0) in the xz-plane, around the z-axis, where R > r > 0 and R is the major radius representing the distance from the tube's center to the axis of revolution.^[1] This construction embeds the torus as a surface of revolution in Euclidean 3-space, forming a closed surface without self-intersections when R > r.^[1] Visually, the torus resembles a doughnut or ring, featuring an inner equator of radius R - r (the smallest parallel circle), an outer equator of radius R + r (the largest parallel circle), and meridional circles of radius r that lie in planes containing the z-axis.^[1] These meridional circles trace the tube's cross-section as it revolves, creating a toroidal tube encircling a central hole.^[1] The solid torus encloses a volume given by

V = 2\pi^2 R r^2,

which arises from the product of the cross-sectional area \pi r^2 and the path length $2\pi R traveled by its centroid during revolution, per Pappus's centroid theorem.^[1] The surface area of the torus is

A = 4\pi^2 R r,

derived similarly from the arc length of the generating circle $2\pi r multiplied by the centroid's path $2\pi R.^[1] The Gaussian curvature K of the torus, expressed in toroidal coordinates with poloidal angle \theta, is

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

where K > 0 on the outer half (convex region, \theta \in (-\pi/2, \pi/2)), K < 0 on the inner half (saddle-like region, \theta \in (\pi/2, 3\pi/2)), and K = 0 along the upper and lower equators (asymptotic lines).^[1] This variation highlights the torus's mixed curvature properties, distinguishing it from surfaces of constant curvature like the sphere.^[1]

Parametric Representation

The standard parametric representation of a torus in three-dimensional Euclidean space uses two angular parameters u and v, both ranging over [0, 2\pi), to describe points on the surface. Let R denote the distance from the center of the torus to the center of the tube (major radius), and r the tube radius (minor radius), with R > r > 0. The coordinates are given by

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

^[1] This parameterization arises from revolving a circle of radius r centered at (R, 0, 0) in the xz-plane, whose equation is (x - R)^2 + z^2 = r^2, around the z-axis. Parameterize the circle as x = R + r \cos v, z = r \sin v; rotating this point by angle u around the z-axis yields the full surface equations above.^[16] An equivalent implicit equation for the torus, azimuthally symmetric about the z-axis, is

\left( \sqrt{x^2 + y^2} - R \right)^2 + z^2 = r^2.

^[1] This form eliminates the parameters and directly relates Cartesian coordinates to the surface. These representations facilitate computations in fields such as computer graphics and differential geometry. In rendering, the parametric form enables efficient generation of surface points for ray tracing or polygon meshing, as seen in procedural modeling where tube and major radii define the geometry directly.^[17] For integration, such as computing arc length along curves on the surface, the parameterization allows evaluation of the first fundamental form, yielding the metric ds^2 = (R + r \cos v)^2 du^2 + r^2 dv^2; the arc length of a curve \gamma(t) = (u(t), v(t)) is then \int \sqrt{(R + r \cos v)^2 (u')^2 + r^2 (v')^2} \, dt.^[18] Similarly, geodesics—shortest paths on the surface—can be parametrized using Clairaut's relation derived from this metric, often resulting in helical paths unwrapped on the torus's fundamental domain.^[18] Variations of the torus depend on the ratio of R to r. When R > r, the standard ring torus forms a non-self-intersecting doughnut shape.^[1] The horn torus occurs at R = r, where the surface touches itself at the origin without crossing.^[19] For R < r, a spindle torus emerges, self-intersecting along a circle in the xy-plane.^[20]

Toroidal Coordinates

Toroidal coordinates provide a three-dimensional orthogonal curvilinear coordinate system particularly suited to problems exhibiting toroidal symmetry, extending the two-dimensional bipolar coordinate system through rotation about the z-axis. The coordinates are denoted as (\sigma, \tau, \phi), where \sigma \geq 0 is the poloidal coordinate, $0 \leq \tau < 2\pi is the toroidal angle, and $0 \leq \phi < 2\pi is the azimuthal angle. The parameter a > 0 represents the scale factor, corresponding to the radius of the focal circle in the xy-plane. These coordinates map points in space to Cartesian coordinates via the relations:

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

^[21] The scale factors for the coordinate differentials are h_\sigma = \frac{a}{\cosh \sigma - \cos \tau}, h_\tau = \frac{a}{\cosh \sigma - \cos \tau}, and h_\phi = \frac{a \sinh \sigma}{\cosh \sigma - \cos \tau}.^[21] Surfaces of constant \sigma form tori with major radius a \coth \sigma and minor radius a / \sinh \sigma, while constant \tau surfaces are apple-shaped spheroids degenerating to spheres as \tau \to \pm \pi/2. In the meridional plane (\phi = constant), the system reduces to bipolar coordinates, with the limit as a \to 0 recovering the bipolar form. Laplace's equation \nabla^2 \Psi = 0 is partially separable in toroidal coordinates, particularly for axisymmetric problems independent of \phi (i.e., m=0), allowing separation into ordinary differential equations involving toroidal harmonics.^[22] Complete separation into three independent equations does not occur due to the coupling in the non-axisymmetric case.^[21] This separability facilitates analytical solutions for boundary value problems in toroidal geometries. Applications include solving for the electrostatic potential around charged toroidal conductors or rings, where toroidal functions expand the potential in series to satisfy boundary conditions on the surface.^[23] In acoustics, toroidal coordinates model wave propagation and resonance frequencies for toroidal bubbles or cavities, enabling computation of scattering and radiation patterns.^[24] These coordinates are also employed in fluid dynamics for vortex ring motion, leveraging the natural adaptation to ring-like structures.^[21]

Topological Properties

Fundamental Group and Homology

The fundamental group of the 2-dimensional torus T^2, denoted \pi_1(T^2), is isomorphic to \mathbb{Z} \times \mathbb{Z}, the free abelian group on two generators.^[25] These generators correspond to loops traversing the meridional and longitudinal directions on the torus, which can be visualized as the two circles in the product structure T^2 = S^1 \times S^1.^[25] Although embeddings of the torus in three-dimensional space may introduce non-commutativity in the loops due to linking, the intrinsic fundamental group of the torus is abelian, reflecting its classification as an orientable surface of genus one.^[25] The homology groups of the torus capture its topological structure through singular or simplicial chains. The integer homology groups are H_0(T^2; \mathbb{Z}) = \mathbb{Z}, H_1(T^2; \mathbb{Z}) = \mathbb{Z} \oplus \mathbb{Z}, H_2(T^2; \mathbb{Z}) = \mathbb{Z}, and H_k(T^2; \mathbb{Z}) = 0 for k > 2.^[25] The corresponding Betti numbers, which are the ranks of these free abelian groups, are b_0 = 1, b_1 = 2, and b_2 = 1.^[25] These can be computed using a CW-complex decomposition of the torus, consisting of one 0-cell, two 1-cells labeled a and b (corresponding to the generators of \pi_1), and one 2-cell attached along the commutator path aba^{-1}b^{-1}.^[25] In the cellular chain complex, the boundary map from the 2-chain to the 1-chains is zero due to the relation, yielding the direct sum structure for H_1, while H_2 arises from the kernel in degree 2.^[25] The Euler characteristic of the torus is \chi(T^2) = 0, computed as the alternating sum of Betti numbers b_0 - b_1 + b_2 = 1 - 2 + 1 = 0, or directly from the CW-complex as the number of cells: one 0-cell minus two 1-cells plus one 2-cell.^[25] This value distinguishes the torus among closed orientable surfaces, where \chi = 2 - 2g with genus g = 1.^[25] In the context of differential geometry, the de Rham cohomology of the torus aligns with its singular homology via de Rham's theorem. Specifically, the first de Rham cohomology group H^1_{dR}(T^2) is generated by the cohomology classes of closed 1-forms, such as the constant forms d\theta and d\phi in toroidal coordinates, which are non-exact and span a 2-dimensional real vector space isomorphic to \mathbb{R}^2.^[26]

Covering Spaces

The universal covering space of the 2-dimensional torus T^2 is the Euclidean plane \mathbb{R}^2, which is simply connected.^[27] The covering projection p: \mathbb{R}^2 \to T^2 is defined by p(x, y) = (e^{2\pi i x}, e^{2\pi i y}), or equivalently after rescaling coordinates to the unit interval, by identifying points differing by integer translations.^[27] The deck transformation group consists of the integer lattice \mathbb{Z}^2 acting on \mathbb{R}^2 via translations (x, y) \mapsto (x + m, y + n) for m, n \in \mathbb{Z}, which faithfully represents the fundamental group \pi_1(T^2) \cong \mathbb{Z}^2. Connected covering spaces of T^2 are classified by subgroups of \pi_1(T^2) = \mathbb{Z}^2; an n-sheeted cover corresponds to a subgroup of index n. Since all subgroups of \mathbb{Z}^2 are free abelian of rank at most 2, every connected finite-sheeted cover of T^2 is homeomorphic to a torus.^[28] For example, there are three non-isomorphic 2-sheeted covers of T^2, arising from the three distinct subgroups of index 2 in \mathbb{Z}^2: $2\mathbb{Z} \times \mathbb{Z}, \mathbb{Z} \times 2\mathbb{Z}, and the subgroup generated by (2,0) and (1,1).^[28] The torus T^2 itself serves as the orientation double cover of the Klein bottle, providing a 2-sheeted orientable cover of this non-orientable surface.^[29] Explicitly, the Klein bottle can be obtained as the quotient of T^2 by the fixed-point-free involution that reverses orientation along one circle factor while preserving the other.^[29] Since T^2 is orientable, its own orientation double cover is trivial, meaning T^2 is its own orientable double cover. In complex analysis, covering spaces of the torus as a Riemann surface enable the study of monodromy for multi-valued functions like elliptic integrals.^[30] The universal cover \mathbb{C} \to T^2 = \mathbb{C}/\Lambda (where \Lambda is a lattice) allows analytic continuation along paths in T^2, with the monodromy action given by the lattice translations representing loops in \pi_1(T^2).^[30] For the flat metric on T^2, the developing map is the local isometry \mathrm{dev}: \mathbb{R}^2 \to \mathbb{R}^2 (the identity), composed with the covering projection to T^2, where the holonomy representation identifies the deck transformations with the lattice of translations preserving the flat structure.^[31]

Genus and Classification

The torus is a fundamental example of an orientable compact surface with genus g=1, characterized by its Euler characteristic \chi = 2 - 2g = 0.^[32] This distinguishes it from the sphere (g=0, \chi=2) and higher-genus surfaces (g>1, \chi < 0). In the broader classification of compact surfaces, the torus represents the unique orientable case for g=1, where the genus counts the number of "handles" or tori in the connected sum decomposition.^[33] According to Poincaré's uniformization theorem, every simply connected Riemann surface is conformally equivalent to the complex plane, the unit disk, or the Riemann sphere, and compact Riemann surfaces arise as quotients by discrete group actions. For the torus, this manifests as the quotient \mathbb{C} / \Lambda, where \Lambda is a lattice in the complex plane generated by two linearly independent vectors.^[34] This elliptic case (parabolic uniformization) contrasts with the spherical case for the sphere and the hyperbolic case for surfaces of genus g > 1.^[35] The classification theorem for compact surfaces states that all orientable compact surfaces of genus 1 are homeomorphic to the torus, up to diffeomorphism in the smooth category.^[36] This uniqueness follows from invariants like the Euler characteristic and orientability; for higher genera (g > 1), surfaces are hyperbolic and admit more varied geometries, but all genus-1 orientable surfaces share the toroidal topology.^[32] Complex tori are parameterized by the modular parameter \tau \in \mathbb{H}, the upper half-plane, where the lattice is \Lambda = \mathbb{Z} + \mathbb{Z}\tau with \operatorname{Im}(\tau) > 0. The special linear group \mathrm{SL}(2, \mathbb{Z}) acts on \mathbb{H} via Möbius transformations \tau \mapsto \frac{a\tau + b}{c\tau + d} for \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2, \mathbb{Z}), identifying tori that are isomorphic as complex manifolds. The fundamental domain for this action is the region \{ \tau \in \mathbb{H} : |\operatorname{Re}(\tau)| \leq 1/2, |\tau| \geq 1 \}, modulo boundary identifications.^[37] While the torus is orientable, its non-orientable analog with the same Euler characteristic \chi = 0 is the Klein bottle, which arises as the connected sum of two real projective planes and differs in fundamental group and homology invariants. In contrast, the real projective plane, a non-orientable surface of crosscap number 1, has \chi = 1 and is not homeomorphic to the torus.^[38]

Higher-Dimensional Generalizations

n-Dimensional Torus

The n-dimensional torus, denoted T^n, is defined as the Cartesian product of n copies of the circle S^1, that is, T^n = S^1 \times \cdots \times S^1 (n times).^[39] It can also be realized as the quotient space \mathbb{R}^n / \mathbb{Z}^n, where points in Euclidean space are identified if they differ by an integer vector.^[39] Parametrized by angular coordinates \theta_1, \dots, \theta_n \in [0, 2\pi), it forms a compact n-dimensional manifold without boundary.^[2] Specific cases illustrate this construction: the 1-torus T^1 is simply the circle S^1; the 2-torus T^2 is the familiar surface torus; and the 3-torus T^3 is a hypersurface embedded in 6-dimensional Euclidean space.^[39] For unit circles (radius 1), the n-dimensional volume of T^n is (2\pi)^n, derived from the product of the circumferences of the component circles.^[2] More generally, if the circles have radii r_1, \dots, r_n, the volume is the product \prod_{i=1}^n (2\pi r_i).^[2] Although T^n can be embedded in \mathbb{R}^{2n} using the map (\theta_1, \dots, \theta_n) \mapsto (\cos \theta_1, \sin \theta_1, \dots, \cos \theta_n, \sin \theta_n), it is typically studied in the abstract as a topological space rather than a specific embedding.^[39] As a Lie group, T^n is compact, connected, and abelian, with the group operation given componentwise by addition modulo $2\pi.^[40] It admits a unique normalized Haar measure, which is the pushforward of the Lebesgue measure on [0, 2\pi)^n under the quotient map, providing an invariant volume form for integration over the space.

Flat Tori and Metrics

A flat torus in two dimensions, denoted T^2, can be constructed as the quotient space \mathbb{R}^2 / \Lambda, where \Lambda is a discrete subgroup of \mathbb{R}^2 isomorphic to \mathbb{Z}^2, known as a lattice. This construction inherits the standard Euclidean metric from \mathbb{R}^2, given by the Riemannian metric ds^2 = dx^2 + dy^2, making T^2 a flat Riemannian manifold with zero Gaussian curvature everywhere. Lattices are typically generated by two linearly independent vectors, often represented in the complex plane as \Lambda = \mathbb{Z} + \tau \mathbb{Z} with \tau \in \mathbb{C} and \Im(\tau) > 0, ensuring the quotient is compact and orientable.^[41]^[42] The conformal classification of flat tori up to similarity transformations (which include scalings, rotations, and reflections) is captured by the moduli space \mathbb{H} / \mathrm{SL}(2, \mathbb{Z}), where \mathbb{H} is the upper half-plane and \mathrm{SL}(2, \mathbb{Z}) acts via Möbius transformations. This space parametrizes equivalence classes of lattices, with each point corresponding to a complex modulus \tau = (a + bi)/c (for integers a, b, c with b > 0 and \gcd(a, b, c) = 1) that determines the shape of the basis vectors up to similarity. The fundamental domain for this action is the region where |\tau| \geq 1 and -\frac{1}{2} \leq \Re(\tau) \leq \frac{1}{2}, highlighting the non-trivial geometry of conformal structures on the torus. The Teichmüller space for the torus, which marks the space of hyperbolic metrics up to diffeomorphisms isotopic to the identity, is one-dimensional and isomorphic to \mathbb{H}, often parametrized by \tau with |\tau| > 1 in the standard fundamental domain to avoid redundancy under the modular group action.^[43]^[44]^[45] In higher dimensions, a flat n-torus T^n is similarly defined as the quotient \mathbb{R}^n / \Gamma, where \Gamma is an n-dimensional lattice, a discrete cocompact subgroup isomorphic to \mathbb{Z}^n. The flat metric on T^n is induced from the Euclidean metric on \mathbb{R}^n, and the full isometry group of such manifolds corresponds to n-dimensional crystallographic groups, which are discrete subgroups of the Euclidean motion group \mathrm{E}(n) = \mathbb{R}^n \rtimes \mathrm{O}(n) acting properly discontinuously and freely. These groups classify all compact flat Riemannian manifolds with Euclidean holonomy. Regarding rigidity, the Bieberbach theorems establish that any such compact flat manifold is isometric to a quotient \mathbb{R}^n / \Gamma by a lattice \Gamma, and the manifold's geometry is uniquely determined up to affine equivalence by its fundamental group under certain embeddings into Euclidean space, ensuring no non-trivial deformations preserve the flat structure.^[46]^[47]^[48]

Configuration Spaces

In physical systems, the n-dimensional torus T^n frequently arises as a configuration space, parameterizing the possible states of particles or rigid bodies subject to periodic constraints. This topological structure captures the periodicity inherent in circular or rotational degrees of freedom, where positions wrap around without boundary. For instance, the configuration space of n distinguishable particles confined to a circle S^1 is the product space (S^1)^n, which is homeomorphic to the n-torus T^n.^[49] If the particles are indistinguishable, the space becomes the quotient T^n / S_n under the action of the symmetric group, though the full T^n suffices for labeled or distinguishable cases.^[49] For a rigid body in three dimensions, the configuration space of orientations is the special orthogonal group SO(3), which can be parametrized by Euler angles (\phi, \theta, \psi), though this parametrization introduces singularities at \theta = 0, \pi where the map degenerates.^[50] This covering resolves the non-simply connected nature of SO(3), allowing consistent description of rotational dynamics while accounting for the topological identification of equivalent orientations.^[50] In quantum mechanics, the torus T^n models the configuration space for particles in periodic potentials, such as electrons in a crystal lattice. Wavefunctions on T^n take the form of plane waves modulated by periodic functions, as per Bloch's theorem: for a Hamiltonian H = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) with periodic V(\mathbf{r} + \mathbf{T}) = V(\mathbf{r}), the solutions are \psi_{\mathbf{k}}(\mathbf{r}) = e^{i \mathbf{k} \cdot \mathbf{r}} u_{\mathbf{k}}(\mathbf{r}), where u_{\mathbf{k}} shares the lattice periodicity and \mathbf{k} lies in the Brillouin zone, topologically a torus.^[51] These Bloch waves underpin band theory in solids, with the torus structure enabling quantized momentum states under periodic boundary conditions. For a single particle on a torus, the Schrödinger equation yields energy eigenfunctions like e^{i (m \theta + n \phi)} in toroidal coordinates, incorporating geometric effects such as curvature.^[52] In classical mechanics, systems with n periodic degrees of freedom, such as particles on circles, have a phase space that is the cotangent bundle T^* T^n \cong T^{2n}, where positions lie on T^n and conjugate momenta also form an n-torus due to angular periodicity. The Hamiltonian dynamics on this T^{2n} phase space follows Hamilton's equations, generating flows that are often integrable for free or weakly interacting particles, with trajectories winding densely on invariant tori by the Kolmogorov-Arnold-Möser theorem.^[53] Applications of toroidal configuration spaces extend to fundamental phenomena. The Aharonov-Bohm effect manifests on a torus when electrons encircle a confined magnetic flux, inducing a phase shift \Delta \phi = \frac{e}{\hbar} \oint \mathbf{A} \cdot d\mathbf{l} despite zero field in the particle's path; this was experimentally confirmed using a superconducting toroidal magnet, where flux quantization in units of h/2e produced observable interference shifts of 0 or \pi.^[54] In string theory, compactification on T^n reduces the 10-dimensional spacetime to four dimensions by wrapping extra dimensions on the torus, preserving supersymmetry and yielding a moduli space of flat metrics parameterized by the torus's shape and size, with winding modes contributing to the low-energy effective theory.^[55]

Discrete and Polyhedral Variants

Toroidal Polyhedra

A toroidal polyhedron is a polyhedron whose surface is topologically equivalent to a torus, satisfying the Euler characteristic \chi = V - E + F = 0, where V, E, and F denote the number of vertices, edges, and faces, respectively.^[56] This distinguishes it from spherical polyhedra, which have \chi = 2. Such polyhedra approximate the continuous torus surface through discrete vertex-edge-face structures and can be convex or nonconvex, provided they embed without self-intersections in three-dimensional space. One of the simplest and most notable toroidal polyhedra is the Császár polyhedron, discovered by Ákos Császár in 1949.^[57] It consists of 7 vertices, 21 edges, and 14 triangular faces, realizing a simplicial embedding of the complete graph K_7 on a toroidal surface without crossings.^[58] Its dual, the Szilassi polyhedron, was constructed by Lajos Szilassi in 1977 and features 7 irregular hexagonal faces, 21 edges, and 14 vertices, with the property that every pair of faces shares an edge.^[59] These polyhedra are unique as the only known pairs (besides the tetrahedron and its dual) where all vertices (or faces) are mutually adjacent, highlighting extremal graph embedding on the torus.^[60] Regular toroidal polyhedra extend concepts from spherical constructions, such as Goldberg polyhedra, to genus-1 surfaces using primarily hexagonal and pentagonal faces to achieve near-uniform curvature.^[61] These meshes balance the Gaussian curvature of the torus—zero on average but varying locally—through arrangements where pentagons introduce positive curvature and hexagons maintain flatness, enabling infinite families of such polyhedra with trivalent vertices. For instance, constructions analogous to fullerenes yield toroidal carbon-like structures consisting only of hexagons (zero pentagons) in the simplest cases, or with pentagons compensated by heptagons, ensuring topological closure.^[62] Archimedean solids generalize to tori as uniform polyhedra, forming infinite families characterized by Schläfli symbols extended for toroidal symmetry, such as \{4,4\}_{(m,n)} for quadrilateral-faced variants or \{3,6\} and \{6,3\} for triangular and hexagonal tessellations embeddable as polyhedra.^[63] These include prismatic and antiprismatic-like forms wrapped toroidally, with vertex figures that are regular polygons and faces that are congruent regular polygons meeting in identical configurations at each vertex. Seminal classifications identify several such infinite families, beyond the 13 finite Archimedean solids on the sphere. Toroidal polyhedra find applications in computer graphics for meshing complex surfaces, where their topology supports seamless texture mapping and subdivision without singularities, as in modeling donuts or rings in animations.^[64] In finite element analysis, they discretize toroidal structures like pressure vessels or magnetic coils, enabling simulations of stress, buckling, and fluid dynamics on non-spherical geometries with high fidelity.^[65]

de Bruijn Tori

A de Bruijn torus is a toroidal array of symbols from a finite alphabet of size k, arranged in an r \times s grid, such that every possible m \times n subarray over the alphabet appears exactly once when considering the periodic boundary conditions of the torus, with the total number of cells satisfying r \cdot s = k^{m \cdot n}.^[66] This structure generalizes the one-dimensional de Bruijn sequence to two dimensions, ensuring comprehensive coverage of all substrings in a compact, cyclic form.^[67] For instance, a binary (k=2) de Bruijn torus with $2 \times 2 windows requires a $4 \times 4 grid to contain all 16 possible binary $2 \times 2 subarrays exactly once.^[68] Construction of de Bruijn tori often relies on Eulerian paths in appropriately defined de Bruijn graphs, where nodes represent overlapping subarrays and edges dictate transitions that wrap around the toroidal boundaries to ensure periodicity.^[69] One common approach involves recursive decomposition, building higher-dimensional or larger tori from smaller ones by combining term products of existing tori or layering de Bruijn families to fill the array while maintaining the uniqueness property.^[67] For the binary $4 \times 4 example with $2 \times 2 windows, explicit constructions such as the "clockwise" or "counterclockwise" arrays achieve this by systematically rotating and mapping submatrices to cover all combinations without repetition.^[68] Algorithms for generating de Bruijn tori adapt hierarchical assembly techniques from one-dimensional sequences, such as stacking rotated de Bruijn sequences into rows or columns with periodic alignments to enforce toroidal shifts.^[69] These methods, including the use of alternating de Bruijn sequences combined via Eulerian cycles in modified graphs, ensure efficient computation while satisfying the exact-once condition for all windows, often under constraints like alphabet size and window dimensions.^[69] The discrete grid topology of the torus facilitates these adaptations by naturally supporting wrap-around operations akin to a flat torus.^[66] Applications of de Bruijn tori include DNA sequencing assembly, where they aid in overlapping and merging multidimensional sequence fragments analogous to one-dimensional de Bruijn graphs.^[68] They also serve in error-correcting codes, enabling the design of array codes that detect and correct bursts or patterns in two-dimensional data storage and transmission.^[70] Additionally, de Bruijn tori find use in VLSI testing for generating pseudorandom test patterns that cover all possible fault configurations in chip arrays efficiently.^[71] Generalizations extend de Bruijn tori to higher dimensions, forming d-dimensional arrays where every n_1 \times \cdots \times n_d subarray appears exactly once in a periodic r_1 \times \cdots \times r_d grid with \prod r_i = k^{\prod n_i}, supporting multi-dimensional shift registers and applications in volumetric data processing.^[67] Inductive constructions layer lower-dimensional tori to build these higher-dimensional variants, as demonstrated for 3D binary cases like $16 \times 4 \times 4 arrays covering all $2 \times 2 \times 2 subcubes.^[68]

Symmetries and Applications

Automorphism Groups

The diffeomorphism group of the 2-dimensional torus T^2, denoted \mathrm{Diff}(T^2), is an infinite-dimensional Fréchet manifold that classifies all smooth self-maps of the torus up to isotopy. Its group of connected components, the mapping class group \mathrm{MCG}(T^2), is isomorphic to the modular group \mathrm{PSL}(2,\mathbb{Z}), which arises as the quotient \mathrm{SL}(2,\mathbb{Z})/\{\pm I\}. This finite-index subgroup of \mathrm{SL}(2,\mathbb{Z}) captures the homotopy classes of orientation-preserving diffeomorphisms, with elements classified as elliptic (finite order), parabolic (unipotent), or hyperbolic (expanding). The group \mathrm{PSL}(2,\mathbb{Z}) acts faithfully on the Teichmüller space T_1 of the torus, identified with the hyperbolic plane \mathbb{H}^2, via Möbius transformations \tau \mapsto (a\tau + b)/(c\tau + d) for \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{PSL}(2,\mathbb{Z}), with the quotient \mathbb{H}^2 / \mathrm{PSL}(2,\mathbb{Z}) forming the moduli space of flat tori up to isomorphism.^[45]^[72] For the standard embedded torus in \mathbb{R}^3 as a surface of revolution (ring torus), the isometry group consists of continuous rotations around the axis of revolution and reflections through meridional planes, forming the infinite group O(2). However, when considering the flat torus T^2 = \mathbb{R}^2 / \Lambda with the Euclidean metric induced from a lattice \Lambda, the isometry group is a crystallographic group depending on the lattice geometry. For the square lattice \Lambda = \mathbb{Z}^2, this group is the semidirect product D_4 \ltimes T^2, where D_4 is the dihedral group of order 8 generated by 90-degree rotations and reflections preserving the lattice, and T^2 accounts for translational isometries. In contrast, for a rhombic lattice with acute angle not equal to \pi/2, the rotational subgroup is smaller, isomorphic to \mathbb{Z}_2 \times \mathbb{Z}_2 or \mathbb{Z}_4, reducing the order of the finite symmetry component.^[73]^[74] In higher dimensions, the automorphism group \mathrm{Aut}(T^n) of the n-torus T^n = \mathbb{R}^n / \mathbb{Z}^n, considered as orientation-preserving homeomorphisms up to homotopy, includes the maximal torus (S^1)^n of rotational symmetries acting coordinatewise, the discrete translational symmetries \mathbb{Z}^n, and reflections extending to the full affine group T^n \rtimes \mathrm{GL}(n,\mathbb{Z}). For the flat metric, the isometry group restricts to those automorphisms preserving the Euclidean structure, forming a semidirect product of the n-torus of translations with the crystallographic point group O(n) \cap \mathrm{GL}(n,\mathbb{Z}), which incorporates orthogonal integer matrices including reflections. This structure generalizes the 2-dimensional case, with the finite part depending on the lattice; for the cubic lattice, it includes the full octahedral group of order 48.^[75]^[76] Fixed-point-free actions on the torus relate to its role as a quotient space, where discrete groups acting freely and properly discontinuously on \mathbb{R}^n produce covering spaces over T^n. Such actions are precisely the crystallographic groups with no rotational elements of finite order greater than 1, ensuring the deck transformation group acts without fixed points and yields the torus as the orbit space. These actions underpin the classification of flat tori as quotients of Euclidean space by translation lattices, connecting symmetries to fundamental group representations.^[77]

Graph Colorings on Tori

A toroidal graph is a graph that can be embedded on the surface of a torus without edge crossings, meaning it has topological genus 1.^[78] Such embeddings allow for the study of graph properties influenced by the toroidal topology, including colorings where adjacent vertices receive distinct colors.^[79] The chromatic number of a toroidal graph, denoted \chi(G), satisfies \chi(G) \leq 7, as established by Heawood's theorem, which provides an upper bound of \left\lfloor \frac{7 + \sqrt{1 + 48g}}{2} \right\rfloor for graphs embeddable on an orientable surface of genus g \geq 1.^[80]^[78] This bound is sharp for the torus, since the complete graph K_7 embeds on the torus and has \chi(K_7) = 7.^[79] In the context of map coloring, where regions (faces of a planar embedding) must receive different colors if adjacent, the torus requires up to 7 colors, as demonstrated by configurations of seven mutually adjacent countries.^[81] The Heawood graph, a cubic (3-regular) graph with 14 vertices, serves as the point-line incidence graph of the Fano plane and realizes a 7-coloring of a toroidal map, confirming the necessity of 7 colors.^[82] Lattice structures on the flat torus provide concrete examples of toroidal graphs and their colorings. The \mathbb{Z}^2 grid embedded periodically on a flat torus yields the toroidal grid graph C_m \square C_n, where C_k denotes the cycle graph on k vertices; this graph has maximum degree \Delta = 4 and is 4-colorable by Brooks' theorem, which states that for a connected graph that is neither complete nor an odd cycle, \chi(G) \leq \Delta.^[83] When both m and n are odd, the graph is non-bipartite and requires more than 2 colors, achieving \chi = 4 in certain cases due to the absence of a 3-coloring.^[84] For the hexagonal lattice on the torus, which corresponds to a 6-regular graph from the triangular tiling with periodic boundaries, proper 3-colorings exist and exhibit rigid structures, such as height functions aligning colors across the surface.^[85]^[86] Brooks' theorem finds direct application in bounding chromatic numbers of toroidal graphs beyond lattices, particularly for regular graphs of low degree embedded on the torus. For instance, 6-regular toroidal triangulations often have \chi = 3 or 4, with the theorem ensuring \chi \leq 6 unless the graph is K_7 or an odd cycle.^[83]^[78] There also exist non-3-colorable toroidal graphs, including triangle-free examples with edge-width less than 6 that require 4 colors, as well as higher-chromatic ones like certain 6-regular grids with \chi = 5.^[87]^[88] Snarks, which are bridgeless cubic graphs non-3-edge-colorable, can embed on the torus and illustrate challenges in related coloring problems, though their vertex chromatic numbers are typically 3 or 4.^[89]

Cutting and Embedding Tori

Cutting the torus along a single closed curve parallel to a meridian (a curve encircling the tube) or a longitude (a curve encircling the central hole) results in a cylindrical surface.^[90] A second cut along the complementary direction—meridian if the first was longitude, or vice versa—further opens the cylinder into a rectangle, representing the fundamental domain of the torus.^[90] These cuts preserve the topology while allowing the surface to be flattened, though the standard curved torus requires approximation due to its non-zero Gaussian curvature. For a flat torus, exact unfolding to the plane is achieved via the fundamental domain, typically a parallelogram in the Euclidean plane quotiented by a lattice, enabling isometric mapping without distortion.^[91] In contrast, the curved torus is not developable overall, as its Gaussian curvature varies (positive on the outer equator, negative on the inner, zero elsewhere), necessitating approximations with developable patches like ruled cylinders or cones. Algorithms for such approximations minimize distortion by segmenting the torus into strips along generating curves, with minimal cuts often limited to two for coarse unfoldings, though more are needed for high-fidelity manufacturing.^[92] The standard 2-torus embeds smoothly and without self-intersection in \mathbb{R}^3 as a surface of revolution generated by rotating a circle offset from an axis.^[93] Higher-genus surfaces generally require immersion in \mathbb{R}^3 with self-intersections, as exemplified by Boy's surface for the real projective plane. For the n-torus T^n, embeddings are possible in \mathbb{R}^{n+1}, extending the low-dimensional cases via parametric constructions or implicit functions that avoid intersections.^[94] Toroidal knots, which lie on the surface of an embedded torus, have an unknotting number given by \frac{(p-1)(q-1)}{2} for the (p,q)-torus knot with coprime p and q, representing the minimal crossing changes needed to unknot it; this bound is tight and proven using gauge theory.^[95] Seifert surfaces for these knots, which are orientable surfaces bounded by the knot, can be constructed from the spanning disk of the unknot augmented by twisted bands corresponding to the knot's crossings, facilitating genus computations and slice properties. In manufacturing, cutting and embedding techniques enable fabrication of toroidal parts by unfolding approximations from flat sheets into developable sectors, which are then bent and welded, as seen in deployable structures using modified Kresling patterns.^[96] Computational geometry algorithms for toroidal meshes often involve cutting along non-separating curves to map the topology to a planar domain for processing, such as remeshing or parameterization, while preserving embedding properties in higher dimensions.^[97]

References

[1]
Torus -- from Wolfram MathWorld
A torus is a surface with a single hole, often shaped like a donut, and can be constructed from a rectangle by gluing opposite edges.
[2]
[PDF] topology, geometry, and dynamical system of torus - UChicago Math
Aug 28, 2024 · A torus of dimension n, considered as a topological space, is defined to be the product Tn = S1 × ... × S1 (with n factors). For n = 1, the ...
[3]
Torus
A torus is a surface having Genus 1, and therefore possessing a single ``Hole.'' The usual torus in 3-D space is shaped like a donut.
[4]
https://www.merriam-webster.com/dictionary/torus
[5]
Torus - Etymology, Origin & Meaning
From Latin torus, meaning "swelling or bulge," the word originated in the 1560s in architecture as a large rounded molding at a column's base.Missing: mathematics Lambert
[6]
Earliest Known Uses of Some of the Words of Mathematics (T)
... torus is given [DSB]. An early use of torus as a mathematical term in English is in 1860 in The Practical Draughtsman's Book of Industrial Design by William ...
[7]
TOROID - Definition & Meaning - Reverso English Dictionary
toroid: solid shape enclosed by a toroidal surface solid shape enclosed by a toroidal surface. Origin of toroid. Greek, toros (bulge) + -oid (resembling) ...
[8]
Torus - Translation into French - examples English | Reverso Context
The Torus tube contains many mathematical formulas and equations. Le Tore contient de nombreuses formules et équations mathématiques.
[9]
Who was the first individual that used the word "torus" to refer to ...
Nov 9, 2017 · The mathematical usage comes from the usage in architecture. The word is of latin origin "torus" and not of greek origin, and has a number of ...Missing: Lambert | Show results with:Lambert
[10]
[PDF] Outline of a History of Differential Geometry: I
His demonstration starts with an arbitrary plane section through a point of the surface, then proceeds to an expres- sion of the radius of curvature for this ...
[11]
[PDF] History of Riemann surfaces
Oct 11, 2005 · This is a short survey about the history of Riemann surfaces and the devel- opment of such surfaces from Bernard Riemann's doctoral thesis and ...
[12]
[PDF] THE POINCARÉ CONJECTURE 1. Introduction The topology of two ...
In the two-dimensional case, each smooth compact surface can be given a beauti- ful geometrical structure, as a round sphere in the genus zero case, as a flat ...
[13]
[PDF] The Concept of Manifold, 1850-1950
There was, of course, still another line of research, more closely linked to differential geometry, where manifolds played an essential role, and purely ...
[14]
[PDF] The Hopf fibration—seven times in physics
The Hopf fibration as a purely mathematical idea has been around since 1931 when it allowed Hopf1 to determine the third homotopy group of the 2-sphere and ...
[15]
1960s Physics Achievement Chronology
Morikazu Toda introduces his “Toda Lattice” as a simple model for a one ... Roger Penrose proposes Twister Theory as a general platform for physics.
[16]
[hep-th/0405115] String Theory and the Fuzzy Torus - arXiv
May 12, 2004 · We outline a brief description of non commutative geometry and present some applications in string theory. We use the fuzzy torus as our guiding ...
[17]
The Mathematics of Torus Triptych - Brown Math
Oct 8, 2000 · We can produce parametric equations for such a torus of revolution as follows: if we consider a circle of radius b in the xz-plane, centered ...<|separator|>
[18]
[PDF] Joining Interactive Graphics and Procedural Modeling for Precise ...
May 14, 2021 · Behind the scenes, the parametric equations used to create a torus with handle radius a and large radius c are x = (c + a cos v) cos u, y = (c ...
[19]
[PDF] Geodesics on the Torus and other Surfaces of Revolution ... - arXiv
Dec 26, 2012 · Thus we have a 1-parameter family of arc length parametrized geodesics which interpolate between the outer equator (horizontal initial direction) ...
[20]
Horn Torus -- from Wolfram MathWorld
One of the three standard tori given by the parametric equations x = a(1+cosv)cosu (1) y = a(1+cosv)sinu (2) z = asinv, (3) corresponding to the torus with ...
[21]
Spindle Torus -- from Wolfram MathWorld
A spindle torus is one of the three standard tori given by the parametric equations x = (c+acosv)cosu (1) y = (c+acosv)sinu (2) z = asinv (3) with c.
[22]
[PDF] COORDINATE SYSTEMS , ox ox - Math@LSU
equation" is not completely separable in toroidal coordinates. ... 2.13.1 Show that the surface area of a toroid defined by Flg. 2.15 is (27m) X (27Th).<|control11|><|separator|>
[23]
[PDF] arXiv:1802.03484v3 [math-ph] 20 Dec 2019
Dec 20, 2019 · The sphere r = a corresponds to σ = 소π. B. Toroidal harmonics. Laplace's equation is partially separable in toroidal coordinates, meaning that ...
[24]
An application of toroidal functions in electrostatics - AIP Publishing
Aug 1, 2007 · The toroidal expansion has applications to coil design used in MRI magnets,9 transformer coils, and circular cylindrical coils with a ...
[25]
[PDF] Acoustic Resonance Frequencies of Underwater Toroidal Bubbles
Toroidal bubbles are found in nature and in applications. Such bubbles are ... acoustic applications such as acoustic trapping of particles. [26,27].
[26]
[PDF] Algebraic Topology - Cornell Mathematics
The Homotopy Extension Property 14. Chapter 1. The Fundamental Group ... torus as a square with opposite edges identified and divides the square into two ...
[27]
[PDF] 4.5 de Rham's theorem - Purdue Math
Consider the torus T = Rn/Zn. We will show later that: Every de Rham cohomology class on T contains a unique form with constant coefficients. This will ...
[28]
[PDF] Section 53. Covering Spaces
Jan 11, 2018 · The space T = S1 × S1 is the torus. The product map p × p : R × R → S1 × S1 is a covering map of T by R2 by Theorem 53.5 where p is ...
[29]
[PDF] ( eX, ˜x 0) → (X, x 0) is an n-sheeted covering space - OU Math
May 2, 2006 · (b) Find two 2-sheeted covering spaces of the torus S1 × S1 that are not isomorphic to each other. (as covering spaces). Can you find three ...
[30]
[PDF] Exercise Sheet 4 (Solution) - metaphor
May 15, 2023 · Q4 Find an orientable two-sheeted covering space of the Klein bottle. Which well-known space do you get? Proof. We obtain a 2-torus T to be from ...
[31]
[PDF] A concise course in complex analysis and Riemann surfaces ...
This course covers complex analysis basics, the Riemann mapping theorem, Riemann surface theory, and elementary aspects like the Cauchy integral theorem.
[32]
[PDF] THE AFFINE STRUCTURES ON THE REAL TWO-TORUS (I)
Aug 10, 1973 · Technically, the great difficulty lies in establishing the fact that the develop- ing map from the universal covering space of an affine torus ...
[33]
[PDF] A Guide to the Classification Theorem for Compact Surfaces
Jan 8, 2025 · The topic of this book is the classification theorem for compact surfaces. We present the technical tools needed for proving rigorously the ...
[34]
[PDF] Classification of Surfaces - ETH Zürich
Jun 7, 2023 · Every compact surface is homeomorphic to either a sphere, or to a connected sum of tori, or to a connected sum of projective planes.
[35]
[PDF] Uniformization of Riemann Surfaces
Every simply connected Riemann surface is isomorphic to the complex plane, the open unit disc, or the Riemann sphere. And one can even find proofs in the ...
[36]
[PDF] On Poincarés Uniformization Theorem - DiVA portal
Poincarés classical theorem states that a hyperbolic polygon satisfying certain conditions is a fundamental domain for a Fuchsian group acting on the hyperbolic ...
[37]
[PDF] The Classification Theorem: Informal Presentation
In addition to being orientable or nonorientable, surfaces may have boundaries. For example, the first surface obtained by slicing a torus shown in Fig. 1.6 ...
[38]
246B, Notes 3: Elliptic functions and modular forms - Terence Tao
Feb 2, 2021 · ... fundamental domain up to boundary for that torus), obeying the boundary periodicity conditions ... {\tau} in the upper half-plane {{\mathbf H}} , ...
[39]
[PDF] classification of surfaces - UChicago Math
Aug 26, 2011 · Abstract. The sphere, torus, Klein bottle, and the projective plane are the classical examples of orientable and non-orientable surfaces.Missing: real | Show results with:real
[40]
torus in nLab
Feb 14, 2024 · In this fashion each torus canonically carries the structure of an abelian group, in fact of an abelian Lie group. ... The character group of a ...
[41]
[PDF] 2 Lie groups and algebraic groups. - UCSD Math
A compact n-torus is a Lie group isomorphic with (S1)n (the n-fold product. 9. Page 10. group). We will write Tn for the group (S1)n. We can realize Tn as the.
[42]
[PDF] Talk IV: Flat 2-torus in E3
Jan 6, 2014 · – A flat torus is a quotient E2/Λ where Λ = ZU ⊕ ZV ⊂ E2 is a lattice. This quotient is called a square flat torus if it is isometric to a.
[43]
[PDF] Lattices and Manifolds of Classes of Flat Riemannian Tori
The associated torus is the quotient set TG = R2/G, which inherits the canonical flat connection and orientation of R2, as well as the usual Rieman- nian ...Missing: ds² = dx² + dy² Λ
[44]
[PDF] Moduli Spaces of Flat tori with Prescribed Holonomy
A leaf Fθ(M) when n = 2 shall be thought of as a generalisation of the mod- ular surface H/PSL(2,Z) which is the moduli space of regular flat tori. Veech's.
[45]
A note on moduli spaces of conformal classes for flat tori of higher ...
Feb 8, 2021 · Furthermore, we study the class of conformally equivalent lattices and describe the moduli space of the corresponding tori and use a similar ...
[46]
[PDF] Introduction to Teichmüller theory - IMJ-PRG
The technical term for such a space is a hyperbolic orbifold. M1 is called the moduli space of tori. T 1 = H2 is called the Teichmüller space of tori. Our ...
[47]
https://webhomes.maths.ed.ac.uk/~v1ranick/papers/stark.pdf
[48]
[PDF] Topological Rigidity Theorems
Bieberbach's solution to Hilbert's Eighteenth Problem on crystallo- graphic groups begins with metric data, namely a flat Riemannian metric, and asserts that a ...
[49]
[PDF] Math 683 - Collapsing Riemannian Manifolds
Introduction. L. Bieberbach in two fundamental papers [2], [3] established the fundamental theorems for the crystallographic groups or Raumgruppen.
[50]
[PDF] CONFIGURATION SPACES Contents 1. Introduction to Linkages ...
The configuration space of n labeled points on a circle is a n-torus. It is possible to make the configuration space any Euclidean manifold by identifying ...Missing: particles | Show results with:particles
[51]
[PDF] Rigid body dynamics on the Poisson torus
define the same orientation of the body. In that sense the configuration space T3 is a double cover of the space SO(3) of Euler angles. We may formally ...
[52]
[PDF] Bloch's theorem - Rutgers Physics
THE QUANTUM MECHANICS OF PARTICLES IN A PERIODIC POTENTIAL: BLOCH'S THEOREM. As the Born-von Karman plane waves are an orthogonal set of functions, the ...
[53]
[PDF] quantum mechanics of particle on a torus knot - arXiv
Sep 4, 2020 · The present paper deals with quantum motion of a spinless particle along a torus knot. The connection between the abstract mathematical Knot ...
[54]
Torus as phase space: Weyl quantization, dequantization, and ...
Aug 23, 2016 · In this paper we consider the quantization of systems having the torus 𝕋2 = ℝ2/ℤ2 as classical phase space. The classical mechanics of ...
[55]
Experimental confirmation of Aharonov-Bohm effect using a toroidal ...
The electron holography technique was employed to make a crucial test of the existence of the Aharonov-Bohm (AB) effect. The relative phase shift was measured ...Missing: torus | Show results with:torus
[56]
[PDF] Toroidal compactification of closed bosonic string theory 1 Motivation
Toroidal compactification obtains 4D physics by considering a theory in a curved background, using a compact manifold, like a torus, as the internal space.
[57]
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 ...
[58]
Higherdimensional Analogues of Csaszar's Torus
Apr 18, 2013 · 87, 462–472 (1965). Article MathSciNet MATH Google Scholar. A. Csaszar, A polyhedron without diagonals, Acta Sci. Math. (Szeged). 13, 140–142 ...
[59]
Császár Polyhedron -- from Wolfram MathWorld
The Császár polyhedron is a polyhedron that is topologically equivalent to a torus which was discovered in the late 1940s by Ákos Császár (Gardner 1975).Missing: 1965 | Show results with:1965
[60]
Szilassi polyhedron - Polytope Wiki
The Szilassi polyhedron is a toroidal polyhedron. With seven irregular hexagonal faces, it has the least number of faces out of any toroid.
[61]
The Császár Polyhedron - Futility Closet
Dec 10, 2015 · In 1949, Hungarian topologist Ákos Császár found the specimen above, which has 7 vertices, 14 faces, and 21 edges. But so far these two are the ...Missing: 1965 | Show results with:1965
[62]
[PDF] Construction of Polyhedra with Tetravalent Nodes as an Analogue to ...
We study tetravalent analogues to fullerene systems which include Goldberg polyhedra (genus 0) and toroidal polyhedra (genus 1), where each node is connected to ...
[63]
Extending Goldberg's method to parametrize and control the ... - NIH
Aug 10, 2022 · Goldberg polyhedra are three-dimensional structures made up of planar hexagons and pentagons with exactly three faces that meet at each ...<|separator|>
[64]
Schläfli Symbol -- from Wolfram MathWorld
A symbol of the form {p,q,r,...} used to describe regular polygons, polyhedra, and their higher-dimensional counterparts.
[65]
Table of Infinite Regular Polyhedra
Regular polyhedra are often represented with a notation called Schläfli symbols which consist of two numbers between curly braces. The first number is the ...
[66]
[PDF] Polyhedral modeling - Hal-Inria
Feb 14, 2018 · Polyhedral meshes are used for visualization, computer graphics or geometric modeling purposes and result from many applications.
[67]
Hydroforming and buckling of toroids with polyhedral sections
The nonlinear finite-element method was used to investigate the hydroforming and buckling properties. The experimental results and numerical estimations were ...
[68]
Review on de Bruijn shapes in one, two and three dimensions
De Bruijn graph indicating a Eulerian cycle B(4,2). 3. De Bruijn Tori: two dimensions. If the concept of de Bruijn sequences is extended to higher dimensions ...
[69]
[PDF] New Constructions for De Bruijn Tori - Lehigh University
Abstract. A De Bruijn torus is a periodic d−dimensional k−ary array such that each n1 × ··· × nd k−ary array appears exactly once with the same period. We.Missing: algorithm seminal
[70]
[PDF] De Bruijn Sequences of Higher Dimension - UCSB Math
This paper briefly reviews an algorithm for generating De. Bruijn sequences before discussing De Bruijn torii, which extend the properties of De Bruijn ...Missing: seminal | Show results with:seminal
[71]
[PDF] Using Alternating de Bruijn sequences to construct de Bruijn tori
Jun 2, 2023 · We present a novel method to generate de Bruijn tori with rectangular windows by combining two variants de Bruijn sequences called 'Alternating ...
[72]
[PDF] On de Bruijn Array Codes Part II: Linear Codes - arXiv
Aug 19, 2025 · Such generalizations were considered for various structures such as error-correcting codes [2], [30], burst-correcting codes [3], [4], [11], [12] ...
[73]
On de Bruijn Array Codes—Part I: Nonlinear Codes
Feb 1, 2025 · A de Bruijn array code is a set of $r \times s$ binary doubly-periodic arrays such that each binary.
[74]
[PDF] arXiv:1801.01812v2 [math.GT] 11 Dec 2021
Dec 11, 2021 · We show that every C1-diffeomorphism of Teichmüller space to itself ... However, the mapping class group of the torus is PSL(2,Z). The proof ...
[75]
[PDF] arXiv:2110.13686v3 [math.DS] 26 Feb 2024
Feb 26, 2024 · as the isometry group of the torus. With Dk denoting the dihedral group on k elements, it is well known that. Iso T2 ∼= D4 ⋉ T. 2. The ...
[76]
[PDF] Riemannian metrics - IME-USP
subgroup of the square torus at an arbitrary point is isomorphic to the dihedral group D4 (or order. 8) whereas the isotropy subgroup of the hexagonal torus at ...
[77]
[PDF] Symmetries and reversing symmetries of toral automorphisms
The toral automorphisms of the n-torus T n. := R n/Zn can be represented by the unimodular n × n-matrices with integer coefficients which form the group GL(n, Z) ...Missing: rotations | Show results with:rotations
[78]
[PDF] arXiv:2208.10750v3 [math.DG] 14 Sep 2023
Sep 14, 2023 · The isometry group of a cocompact euclidean orbifold R3/Γ is a compact Lie group whose identity component is a Torus of dimension. Page 21 ...
[79]
[PDF] Covering spaces and Delaunay triangulations of the 2D flat torus
Jan 26, 2015 · Group Actions and Covering Spaces. ... (Technically, this follows from the fact that the action is discontinuous and fixed point free; See [12] ...
[80]
[PDF] arXiv:2102.06993v2 [math.CO] 7 Jun 2021
Jun 7, 2021 · Chromatic number, choice number, toroidal graph, triangulation, regular graph. ... Heawood proved that H(g), the so called Heawood number, is an ...<|control11|><|separator|>
[81]
4-Colorable 6-regular toroidal graphs - ScienceDirect.com
For example K7 can be embedded as an Eulerian triangulation of the torus, but has chromatic number 7. On the other hand, it is recently proved by Hutchinson et ...
[82]
Torus Coloring -- from Wolfram MathWorld
The number of colors sufficient for map coloring on a surface of genus g is given by the Heawood conjecture, chi(g)=|_1/2(7+sqrt(48g+1))_|.
[83]
Susan Goldstine's Seven-Color Tori
Here is a collection of maps on surfaces that are topological tori, each map having seven countries all of which touch all of the others.
[84]
Heawood Graph -- from Wolfram MathWorld
The Heawood graph corresponds to the seven-color torus map on 14 nodes illustrated above. The Heawood graph is the point/line Levi graph on the Fano plane ( ...
[85]
Brooks's theorem (Chapter 2) - Topics in Chromatic Graph Theory
May 5, 2015 · In this chapter only simple graphs are considered. Brooks's theorem relates the chromatic number to the maximum degree of a graph. In modern ...
[86]
The list chromatic number of some special toroidal grid graphs
Jul 5, 2019 · The list chromatic number of some special toroidal grid graphs ; (C3◻C5) and ; (C5◻C5). I conjecure both of them are 3.Breaking frustrated loops in list coloring problem - MathOverflowchromatic number of a simple graph whose length of the longest odd ...More results from mathoverflow.netMissing: dimensions | Show results with:dimensions
[87]
[PDF] Topological interpretation of color exchange invariants - SciPost
Feb 18, 2021 · Figure 1: Example of a 3-coloring of a hexagonal lattice, with periodic boundary conditions (torus geometry). Any 2-colored loop can be oriented ...
[88]
Rigidity of 3-colorings of the discrete torus
### Summary of Colorings of the Discrete Torus
[89]
Characterization of 4-critical triangle-free toroidal graphs
We show every triangle-free graph drawn in the torus with edge-width at least six is 3-colorable, a key property used in an efficient 3-colorability algorithm.
[90]
[PDF] FOUR-COLORING SIX-REGULAR GRAPHS ON THE TORUS
A classic 3-color theorem attributed to Heawood [10, 11] (also ob- served by Kempe [16]) states that every even triangulation of the plane is 3-chromatic; by ...
[91]
Three-edge-coloring (Tait coloring) cubic graphs on the torus - arXiv
May 11, 2025 · This means every (non-trivial) toroidal snark can be obtained from several copies of the Petersen graph using the dot product operation. The ...
[92]
[PDF] Introduction to Surfaces
Jun 21, 2019 · For example, either one of the following circles can be cut out of a torus, but not both. ... does not disconnect T cuts it into a cylinder, which ...
[93]
[PDF] A Universal Triangulation for Flat Tori - DROPS
Every flat torus has a modulus in the fundamental domain of this quotient as shown in Figure 1. 0. 1. H2 e2ip/3 eip/3 i. Figure 1 A point ...
[94]
[PDF] Approximation Algorithms for Developable Surfaces
A developable surface is a surface which can be unfolded (developed) into a plane without stretching or tearing. Mathematically speaking, there is a mapping ...
[95]
[PDF] Isometric C embedding of flat tori into R
Smooth embeddings of the torus into Euclidean space cannot pre- serve its flat metric. But one can obtain a C1 embedding which is isometric, by repeatedly ...
[96]
Embedding of n-Torus in $\mathbb{R}^{n+1}$ via implicit function $T ...
Feb 18, 2013 · ×S1 can be embedded into Rn+1 by giving a function f:Rn+1→R such that the Torus is described by all points satisfying f(x1,…,xn)=0.Embedding torus in Euclidean space - Math Stack ExchangeIsometric embedding of flat torus $\mathbb{R}^n/\Lambda$ into ...More results from math.stackexchange.com
[97]
Lower bounds for the unknotting numbers of certain torus knots
all knots and links are in S3. The unknotting number of a knot is the minimum number of crossings which must be changed to make the knot trivial.
[98]
Deployable toroidal structures based on modified Kresling pattern
The main objectives of this study are; to introduce a step by step design algorithm for developing foldable/deployable toroidal structures from flat sheets ...
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[PDF] An algorithm for embedding graphs in the torus
The torus embedding algorithm presented in this paper has a great advantage to bypass that step by using a recent result of Fiedler, Huneke, Richter, and ...