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62-474: In interiors, the frieze of a room is the section of wall above the picture rail and under the crown moldings or cornice. By extension, a frieze is a long stretch of painted , sculpted or even calligraphic decoration in such a position, normally above eye-level. Frieze decorations may depict scenes in a sequence of discrete panels. The material of which the frieze is made of may be plasterwork , carved wood or other decorative medium. More loosely, "frieze"

124-585: A Riemannian manifold , as well as the structure of an abelian Lie group. Perhaps the simplest example of this is when L = Z 2 {\displaystyle \mathbb {Z} ^{2}} : R 2 / Z 2 {\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}} , which can also be described as the Cartesian plane under the identifications ( x , y ) ~ ( x + 1, y ) ~ ( x , y + 1) . This particular flat torus (and any uniformly scaled version of it)

186-449: A closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding. If

248-514: A fiber bundle over S (the Hopf bundle ). The surface described above, given the relative topology from R 3 {\displaystyle \mathbb {R} ^{3}} , is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into R 3 {\displaystyle \mathbb {R} ^{3}} from

310-545: A frieze has been generalized in the mathematical construction of frieze patterns . Picture rail Moulding ( British English ), or molding ( American English ), also coving (in United Kingdom, Australia), is a strip of material with various profiles used to cover transitions between surfaces or for decoration. It is traditionally made from solid milled wood or plaster , but may be of plastic or reformed wood. In classical architecture and sculpture,

372-593: A maximal torus ; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori T have a controlling role to play in theory of connected G . Toroidal groups are examples of protori , which (like tori) are compact connected abelian groups, which are not required to be manifolds . Automorphisms of T are easily constructed from automorphisms of the lattice Z n {\displaystyle \mathbb {Z} ^{n}} , which are classified by invertible integral matrices of size n with an integral inverse; these are just

434-752: A product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem , giving: A = ( 2 π r ) ( 2 π R ) = 4 π 2 R r , V = ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 . {\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr,\\[5mu]V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}} These formulas are

496-409: A torus ( pl. : tori or toruses ) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut . If the axis of revolution does not touch

558-521: A 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical. These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model musical triads . A flat torus

620-416: A core with a cement-based protective coating. Synthetic mouldings are a cost-effective alternative that rival the aesthetic and function of traditional profiles. Common mouldings include : At their simplest, mouldings hide and help weather seal natural joints produced in the framing process of building a structure. As decorative elements, they are a means of applying light- and dark-shaded stripes to

682-492: A flat torus into 3-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} was found. It is a flat torus in the sense that, as a metric space, it is isometric to a flat square torus. It is similar in structure to a fractal as it is constructed by repeatedly corrugating an ordinary torus at smaller scales. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals , yielding

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744-429: A rectangle together, choosing the other two sides instead will cause the same reversal of orientation. The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian ). The 2-torus is a twofold branched cover of the 2-sphere, with four ramification points . Every conformal structure on the 2-torus can be represented as such

806-406: A regular torus. For example, in the following map: If R and P in the above flat torus parametrization form a unit vector ( R , P ) = (cos( η ), sin( η )) then u , v , and 0 < η < π /2 parameterize the unit 3-sphere as Hopf coordinates . In particular, for certain very specific choices of a square flat torus in the 3-sphere S , where η = π /4 above, the torus will partition

868-417: A shadow that is lighter at the top and darker at the bottom. Other varieties of concave moulding are the scotia and congé and other convex mouldings the echinus , the torus and the astragal. Placing an ovolo directly above a cavetto forms a smooth s -shaped curve with vertical ends that is called an ogee or cyma reversa moulding. Its shadow appears as a band light at the top and bottom but dark in

930-454: A so-called "smooth fractal". The key to obtaining the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths". (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics. In

992-544: A sphere — by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space is a sphere with three points each having less than 2π total angle around them. (Such a point is termed a "cusp", and may be thought of as the vertex of a cone, also called a "conepoint".) This third conepoint will have zero total angle around it. Due to symmetry, M* may be constructed by glueing together two congruent geodesic triangles in

1054-411: A structural object without having to change the material or apply pigments . Depending on their function they may be primarily a means of hiding or weather-sealing a joint, purely decorative, or some combination of the three. As decorative elements the contrast of dark and light areas gives definition to the object. If a vertical wall is lit at an angle of about 45 degrees above the wall (for example, by

1116-461: A torus is a closed surface defined as the product of two circles : S  ×  S . This can be viewed as lying in C and is a subset of the 3-sphere S of radius √2. This topological torus is also often called the Clifford torus . In fact, S is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S as

1178-405: A torus is any topological space that is homeomorphic to a torus. The surface of a coffee cup and a doughnut are both topological tori with genus one. An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Klein bottle ). Torus

1240-425: A torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of

1302-426: A torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, R , the distance from the center of the coordinate system, and θ and φ , angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in

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1364-563: A torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below). A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows: where R and P are positive constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric . It can not be analytically embedded ( smooth of class C , 2 ≤ k ≤ ∞ ) into Euclidean 3-space. Mapping it into 3 -space requires one to stretch it, in which case it looks like

1426-469: A two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points . In fact, the conformal type of the torus is determined by the cross-ratio of the four points. The torus has a generalization to higher dimensions, the n-dimensional torus , often called the n -torus or hypertorus for short. (This is the more typical meaning of

1488-459: Is R n {\displaystyle \mathbb {R} ^{n}} modulo the action of the integer lattice Z n {\displaystyle \mathbb {Z} ^{n}} (with the action being taken as vector addition). Equivalently, the n -torus is obtained from the n -dimensional hypercube by gluing the opposite faces together. An n -torus in this sense is an example of an n- dimensional compact manifold . It

1550-1005: Is a Latin word for "a round, swelling, elevation, protuberance". A torus of revolution in 3-space can be parametrized as: x ( θ , φ ) = ( R + r cos ⁡ θ ) cos ⁡ φ y ( θ , φ ) = ( R + r cos ⁡ θ ) sin ⁡ φ z ( θ , φ ) = r sin ⁡ θ {\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \\\end{aligned}}} using angular coordinates θ , φ ∈ [ 0 , 2 π ) , {\displaystyle \theta ,\varphi \in [0,2\pi ),} representing rotation around

1612-461: Is a member of the Lie group SO(4). It is known that there exists no C (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of

1674-408: Is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings , non-inflatable lifebuoys , ring doughnuts , and bagels . In topology , a ring torus is homeomorphic to the Cartesian product of two circles : S 1 × S 1 {\displaystyle S^{1}\times S^{1}} , and the latter is taken to be

1736-411: Is a torus with the metric inherited from its representation as the quotient , R 2 {\displaystyle \mathbb {R} ^{2}} / L , where L is a discrete subgroup of R 2 {\displaystyle \mathbb {R} ^{2}} isomorphic to Z 2 {\displaystyle \mathbb {Z} ^{2}} . This gives the quotient the structure of

1798-457: Is also an example of a compact abelian Lie group . This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication. Toroidal groups play an important part in the theory of compact Lie groups . This is due in part to the fact that in any compact Lie group G one can always find

1860-447: Is known as the "square" flat torus. This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. It is flat in the same sense that the surface of a cylinder is flat. In 3 dimensions, one can bend a flat sheet of paper into a cylinder without stretching the paper, but this cylinder cannot be bent into

1922-662: Is sometimes used for any continuous horizontal strip of decoration on a wall, containing figurative or ornamental motifs. In an example of an architectural frieze on the façade of a building, the octagonal Tower of the Winds in the Roman agora at Athens bears relief sculptures of the eight winds on its frieze. A pulvinated frieze (or pulvino ) is convex in section. Such friezes were features of 17th-century Northern Mannerism , especially in subsidiary friezes, and much employed in interior architecture and in furniture. The concept of

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1984-545: Is the n -fold product of the circle, the n -torus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, T n = ( S 1 ) n {\displaystyle \mathbb {T} ^{n}=(\mathbb {S} ^{1})^{n}} . The configuration space of unordered , not necessarily distinct points is accordingly the orbifold T n / S n {\displaystyle \mathbb {T} ^{n}/\mathbb {S} _{n}} , which

2046-491: Is the quotient of the torus by the symmetric group on n letters (by permuting the coordinates). For n = 2, the quotient is the Möbius strip , the edge corresponding to the orbifold points where the two coordinates coincide. For n = 3 this quotient may be described as a solid torus with cross-section an equilateral triangle , with a twist ; equivalently, as a triangular prism whose top and bottom faces are connected with

2108-400: Is the standard 2-torus, T 2 {\displaystyle \mathbb {T} ^{2}} . And similar to the 2-torus, the n -torus, T n {\displaystyle \mathbb {T} ^{n}} can be described as a quotient of R n {\displaystyle \mathbb {R} ^{n}} under integral shifts in any coordinate. That is, the n -torus

2170-454: The Euler characteristic of the n -torus is 0 for all n . The cohomology ring H ( T n {\displaystyle \mathbb {T} ^{n}} ,  Z ) can be identified with the exterior algebra over the Z - module Z n {\displaystyle \mathbb {Z} ^{n}} whose generators are the duals of the n nontrivial cycles. As the n -torus

2232-478: The hyperbolic plane along their (identical) boundaries, where each triangle has angles of π/2, π/3, and 0. (The three angles of a hyperbolic triangle T determine T up to congruence.) As a result, the Gauss-Bonnet theorem shows that the area of each triangle can be calculated as π - (π/2 + π/3 + 0) = π/6, so it follows that the compactified moduli space M* has area equal to π/3. The other two cusps occur at

2294-454: The square root gives a quartic equation , ( x 2 + y 2 + z 2 + R 2 − r 2 ) 2 = 4 R 2 ( x 2 + y 2 ) . {\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).} The three classes of standard tori correspond to

2356-433: The " moduli space " of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space M may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π (radians) around them: One has total angle = π and the other has total angle = 2π/3. M may be turned into a compact space M* — topologically equivalent to

2418-432: The 'cornice cavetto' and 'papyriform columns' appearing in ancient Egyptian architecture , while Greek and Roman practices developed into the highly the regulated classical orders . Necessary to the spread of Classical architecture was the circulation of pattern books , which provided reproducible copies and diagrammatic plans for architects and builders. Works containing sections and ratios of mouldings appear as early as

2480-406: The 3-sphere into two congruent solid tori subsets with the aforesaid flat torus surface as their common boundary . One example is the torus T defined by Other tori in S having this partitioning property include the square tori of the form Q ⋅ T , where Q is a rotation of 4-dimensional space R 4 {\displaystyle \mathbb {R} ^{4}} , or in other words Q

2542-524: The Classical tradition the combination and arrangement of mouldings are primarily done according to preconceived compositions. Typically, mouldings are rarely improvised by the architect or builder, but rather follows established conventions that define the ratio, geometry, scale, and overall configuration of a moulding course or entablature in proportion to the entire building. Classical mouldings have their roots in ancient civilizations, with examples such

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2604-473: The Roman era with Vitruvius and much later influential publications such as Giacomo Barozzi da Vignola's , Five Orders of Architecture , and James Gibbs's , Rules for Drawing the Several Parts of Architecture . Pattern books can be credited for the regularization and continuity of classical architectural mouldings across countries and continents particularly during the colonial era, contributing to

2666-400: The circle, the surface has a ring shape and is called a torus of revolution , also known as a ring torus . If the axis of revolution is tangent to the circle, the surface is a horn torus . If the axis of revolution passes twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus ). If the axis of revolution passes through the center of

2728-437: The circle, the surface is a degenerate torus, a double-covered sphere . If the revolved curve is not a circle, the surface is called a toroid , as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings , inner tubes and ringette rings . A torus should not be confused with a solid torus , which is formed by rotating a disk , rather than a circle, around an axis. A solid torus

2790-476: The definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space , but another way to do this is the Cartesian product of the embedding of S 1 {\displaystyle S^{1}} in the plane with itself. This produces a geometric object called the Clifford torus , a surface in 4-space . In the field of topology ,

2852-469: The global occurrence of Classical mouldings and elements. Pattern books remained common currency amongst architects and builders up until the early 20th century, but soon after mostly disappeared as Classical architecture lost favor to Modernist and post-war building practices that conscientiously stripped their buildings of mouldings. However, the study of formalized pattern languages, including mouldings, has since been revived through online resources and

2914-406: The integral matrices with determinant ±1. Making them act on R n {\displaystyle \mathbb {R} ^{n}} in the usual way, one has the typical toral automorphism on the quotient. The fundamental group of an n -torus is a free abelian group of rank n . The k -th homology group of an n -torus is a free abelian group of rank n choose k . It follows that

2976-427: The interior. Similarly, a cavetto above an ovolo forms an s with horizontal ends, called a cyma or cyma recta . Its shadow shows two dark bands with a light interior. Together the basic elements and their variants form a decorative vocabulary that can be assembled and rearranged in endless combinations. This vocabulary is at the core of both classical architecture and Gothic architecture . When practiced in

3038-693: The moulding is often carved in marble or other stones . In historic architecture, and some expensive modern buildings, it may be formed in place with plaster . A "plain" moulding has right-angled upper and lower edges. A "sprung" moulding has upper and lower edges that bevel towards its rear, allowing mounting between two non-parallel planes (such as a wall and a ceiling), with an open space behind. Mouldings may be decorated with paterae as long, uninterrupted elements may be boring for eyes. Decorative mouldings have been made of wood , stone and cement . Recently mouldings have been made of extruded polyvinyl chloride (PVC) and expanded polystyrene (EPS) as

3100-430: The north pole of S . The torus can also be described as a quotient of the Cartesian plane under the identifications or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA B . The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself: Intuitively speaking, this means that

3162-518: The popularity of new classical architecture in the early 21st century. The middle ages are characterized as a period of decline and erosion in the formal knowledge of Classical architectural principles. This eventually resulted in an amateur and 'malformed' use of moulding patterns that eventually developed into the complex and inventive Gothic style . While impressive and seemingly articulate across Europe, Gothic architecture remained mostly regional and no comprehensive pattern books were developed at

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3224-431: The same as for a cylinder of length 2π R and radius r , obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. Expressing the surface area and the volume by the distance p of an outermost point on

3286-491: The spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r , and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles". In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices. Topologically ,

3348-449: The study of Riemann surfaces , one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The Uniformization theorem guarantees that every Riemann surface is conformally equivalent to one that has constant Gaussian curvature . In the case of a torus, the constant curvature must be zero. Then one defines

3410-447: The sun) then adding a small overhanging horizontal moulding, called a fillet moulding, will introduce a dark horizontal shadow below it. Adding a vertical fillet to a horizontal surface will create a light vertical shadow. Graded shadows are possible by using mouldings in different shapes: the concave cavetto moulding produces a horizontal shadow that is darker at the top and lighter at the bottom; an ovolo ( convex ) moulding makes

3472-1057: The surface of the torus to the center, and the distance q of an innermost point to the center (so that R = ⁠ p + q / 2 ⁠ and r = ⁠ p − q / 2 ⁠ ), yields A = 4 π 2 ( p + q 2 ) ( p − q 2 ) = π 2 ( p + q ) ( p − q ) , V = 2 π 2 ( p + q 2 ) ( p − q 2 ) 2 = 1 4 π 2 ( p + q ) ( p − q ) 2 . {\displaystyle {\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q),\\[5mu]V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}.\end{aligned}}} As

3534-439: The term " n -torus", the other referring to n holes or of genus n . ) Just as the ordinary torus is topologically the product space of two circles, the n -dimensional torus is topologically equivalent to the product of n circles. That is: The standard 1-torus is just the circle: T 1 = S 1 {\displaystyle \mathbb {T} ^{1}=\mathbb {S} ^{1}} . The torus discussed above

3596-426: The three possible aspect ratios between R and r : When R ≥ r , the interior ( x 2 + y 2 − R ) 2 + z 2 < r 2 {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}<r^{2}} of this torus is diffeomorphic (and, hence, homeomorphic) to

3658-429: The time, but instead likely circulated through pilgrimage and the migration of trained Gothic masons. These medieval forms were later imitated by prominent Gothic Revivalists such as Augustus Pugin and Eugène Viollet-le-Duc who formalized Gothic mouldings, developing them into its own systematic pattern books which could be replicated by architects with no native Gothic architecture. Torus In geometry ,

3720-484: The torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the Nash-Kuiper theorem , which was proven in the 1950s, an isometric C embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding. In April 2012, an explicit C (continuously differentiable) isometric embedding of

3782-534: The torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2. An implicit equation in Cartesian coordinates for a torus radially symmetric about the z {\displaystyle z} - axis is ( x 2 + y 2 − R ) 2 + z 2 = r 2 . {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}=r^{2}.} Algebraically eliminating

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3844-420: The tube and rotation around the torus' axis of revolution, respectively, where the major radius R {\displaystyle R} is the distance from the center of the tube to the center of the torus and the minor radius r {\displaystyle r} is the radius of the tube. The ratio R / r {\displaystyle R/r} is called the aspect ratio of

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