"the vibration theory of smell."" While Turin comments that Block used ""cells in a dish rather than within whole organisms"" and that ""expressing an olfactory receptor in human embryonic kidney cells doesn't adequately reconstitute the complex nature of olfaction..."" Vosshall responds ""Embryonic kidney cells are not identical to the cells in the nose ... but if you are looking at receptors, it's the best system in the world."" Shape theory of olfaction The Shape theory of smell proposes that a molecule's smell character is due to its molecular shape, molecular size and functional groups. It has also been described by"
"Vermiform Vermiform describes something shaped like a worm. The expression is often employed in biology and anatomy to describe more or less tubular or cylindrical, usually soft body parts or animals. The word root is Latin, ""vermes""-worms and ""formes""-shaped. A well known example is the vermiform appendix, a small, blind section of the gut in humans and a number of other mammals. A number of soft-bodied animal phyla are typically described as vermiform. The more well known one are undoubtedly the annelids (earthworm and relatives) and the roundworms (a very common, mainly parasitic group), but a number of less well"
"be false. Space form In mathematics, a space form is a complete Riemannian manifold ""M"" of constant sectional curvature ""K"". The three obvious examples are Euclidean ""n""-space, the ""n""-dimensional sphere, and hyperbolic space, although a space form need not be simply connected. The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an ""n""-dimensional space form formula_1 with curvature formula_2 is isometric to formula_3, hyperbolic space, with curvature formula_4 is isometric to formula_5, Euclidean ""n""-space, and with curvature formula_6 is isometric to formula_7, the n-dimensional sphere of points distance 1 from the origin in formula_8. By rescaling the"
"shoe is different from a left shoe, and clockwise is different from anticlockwise. See for a full mathematical definition. A chiral object and its mirror image are said to be enantiomorphs. The word ""enantiomorph"" stems from the Greek (enantios) 'opposite' + (morphe) 'form'. A non-chiral figure is called achiral or amphichiral. The helix (and by extension a spun string, a screw, a propeller, etc.) and Möbius strip are chiral two-dimensional objects in three-dimensional ambient space. The J, L, S and Z-shaped ""tetrominoes"" of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space. Many other familiar"
"and spheres. A wide range of other shape representations have also been investigated. There also seems to be something about shape that can guide human attention. Shape A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material composition. Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can"
"Space form In mathematics, a space form is a complete Riemannian manifold ""M"" of constant sectional curvature ""K"". The three obvious examples are Euclidean ""n""-space, the ""n""-dimensional sphere, and hyperbolic space, although a space form need not be simply connected. The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an ""n""-dimensional space form formula_1 with curvature formula_2 is isometric to formula_3, hyperbolic space, with curvature formula_4 is isometric to formula_5, Euclidean ""n""-space, and with curvature formula_6 is isometric to formula_7, the n-dimensional sphere of points distance 1 from the origin in formula_8. By rescaling the Riemannian metric"
"Shape A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material composition. Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares, etc. Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, and parabolas. Among the most common"
"a Riemannian manifold is the largest radius of a ball which is a ""convex"" subset. Ray is a one side infinite geodesic which is minimizing on each interval Riemann curvature tensor Riemannian manifold Riemannian submersion is a map between Riemannian manifolds which is submersion and ""submetry"" at the same time. Second fundamental form is a quadratic form on the tangent space of hypersurface, usually denoted by II, an equivalent way to describe the ""shape operator"" of a hypersurface, It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space."
"the geodesic symmetry at ""p"" (less ""p"" itself). See . It has sectional curvature ranging from 1/4 to 1, and is the roundest manifold that is not a sphere (or covered by a sphere): by the 1/4-pinched sphere theorem, any complete, simply connected Riemannian manifold with curvature strictly between 1/4 and 1 is diffeomorphic to the sphere. Complex projective space shows that 1/4 is sharp. Conversely, if complete simply connected Riemannian manifold has sectional curvatures in the closed interval [1/4,1], then it is either diffeomorphic to the sphere, or isometric to the complex projective space, the quaternionic projective space, or"
"bilinear form on the tangent space of the manifold . In relativity theory, the Ricci tensor is the part of the curvature of spacetime that determines the degree to which matter will tend to converge or diverge in time (via the Raychaudhuri equation). It is related to the matter content of the universe by means of the Einstein field equation. In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. If the Ricci tensor"
"while halibut are right-eyed. Chirality Chirality is a property of asymmetry important in several branches of science. The word ""chirality"" is derived from the Greek (""kheir""), ""hand,"" a familiar chiral object. An object or a system is ""chiral"" if it is distinguishable from its mirror image; that is, it cannot be superposed onto it. Conversely, a mirror image of an ""achiral"" object, such as a sphere, cannot be distinguished from the object. A chiral object and its mirror image are called ""enantiomorphs"" (Greek, ""opposite forms"") or, when referring to molecules, ""enantiomers"". A non-chiral object is called ""achiral"" (sometimes also ""amphichiral"")"
"particularly difficult to find: in the monograph on the subject by the pseudonymous author Arthur Besse, readers are offered a meal in a starred restaurant in exchange for a new example. Einstein manifold In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not"
"3-dimensional shapes are polyhedra, which are shapes with flat faces; ellipsoids, which are egg-shaped or sphere-shaped objects; cylinders; and cones. If an object falls into one of these categories exactly or even approximately, we can use it to describe the shape of the object. Thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk. There are several ways to compare the shapes of two objects: Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required"
"express the contribution that the Ricci curvature (or rather, the Schouten tensor) and the Weyl tensor each makes to the curvature of a Riemannian manifold. This so-called Ricci decomposition is useful in differential geometry. When there is a metric tensor ""g"", the Kulkarni–Nomizu product of ""g"" with itself is the identity endomorphism of the space of 2-forms, Ω(""M""), under the identification (using the metric) of the endomorphism ring End(Ω(""M"")) with the tensor product Ω(""M"") ⊗ Ω(""M""). A Riemannian manifold has constant sectional curvature ""k"" if and only if the Riemann tensor has the form where ""g"" is the metric tensor."
"Einstein manifold In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity. If ""M"" is the underlying ""n""-dimensional manifold and ""g"" is its metric tensor the Einstein condition means that for"
"It is usually assumed that the manifold is globally hyperbolic, all hypersurfaces are spacelike, and the foliation covers the whole manifold. Each hypersurface is called a leaf or a slice of the foliation. In order to give a more precise definition of foliation, it is necessary to define some auxiliary elements. A ""rectangular neighborhood"" in R is an open subset of the form ""B"" = ""J"" × ⋅⋅⋅ × ""J"", where ""J"" is a (possibly unbounded) relatively open interval in the ""i""th coordinate axis. If ""J"" is of the form (""a"",0], it is said that ""B"" has boundary In the"
"Rostriform. In the form of a rostrum. Rudimentary. Not fully formed; imperfect. Rugose. Rough or wrinkled, as parts of some shells. Sacculated. Somewhat like a sac, or composed of sac-like parts. Scalar. Resembling a ladder. Secreted. Produced or deposited from the blood or glands, as the shell material in mollusks. Semicircular. Half round or circular, as the aperture in some snails. Semidentate. Half toothed, as the parietal wall in some land snails. Semielliptic. Half elliptical. Semiglobose. Half, or not quite globose. Semilunate. Half lunate. Semioval. Half, or not quite oval. Serrated. Notched, like the teeth on a saw. Serriform. In"
"the Greek (cheir), the hand, the most familiar chiral object; the word ""enantiomorph"" stems from the Greek (enantios) 'opposite' + (morphe) 'form'. A non-chiral figure is called achiral or amphichiral. Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule. Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out. The J,"
"into paraboloids that are accurate enough to be used as primary reflectors in telescopes. No correction of the shape is necessary. Spin-cast glass mirrors need correction because of distortions that arise during and after solidification. Rotating furnace A rotating furnace is a device for making solid objects which have concave surfaces that are segments of axially symmetrical paraboloids. Usually, the objects are made of glass. The furnace makes use of the fact, which was known already to Newton, that the centrifugal-force-induced shape of the top surface of a spinning liquid is a concave paraboloid, identical to the shape of a"
"SL(2, R) is denoted formula_1. It fibers over H. The group ""G"" has 2 components. Its identity component has the structure formula_2. The point stabilizer is O(2,R). Examples of these manifolds include: the manifold of unit vectors of the tangent bundle of a hyperbolic surface, and more generally the Brieskorn homology spheres (excepting the 3-sphere and the Poincare dodecahedral space). This geometry can be modeled as a left invariant metric on the Bianchi group of type VIII. Finite volume manifolds with this geometry are orientable and have the structure of a Seifert fiber space. The classification of such manifolds is"
"the 2-plane σ, and is denoted ""K""(σ). Riemannian manifolds with constant sectional curvature are the simplest. These are called space forms. By rescaling the metric there are three possible cases The model manifolds for the three geometries are hyperbolic space, Euclidean space and a unit sphere. They are the only connected, complete, simply connected Riemannian manifolds of given sectional curvature. All other connected complete constant curvature manifolds are quotients of those by some group of isometries. If for each point in a connected Riemannian manifold (of dimension three or greater) the sectional curvature is independent of the tangent 2-plane, then"
"and the sphere. A shape is said to be convex if all of the points on a line segment between any two of its points are also part of the shape. Geometric shape A geometric shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object. That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape. Objects that have the same shape as each other are said to"
"the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions. One way of modeling non-rigid movements is by homeomorphisms. Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated mathematical joke is that topologists can't tell their coffee cup from their donut, since a sufficiently pliable donut could be reshaped to the form of a"
"Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting"
"of ""n"" = 0, not covered by the above formulas), are as follows: Torus In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a ""torus of revolution"". Real-world examples of toroidal objects include inner tubes. A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, around an axis. A solid torus is a"
"known phyla answer to the same description. Examples range from the minute parasitic mesozoans to the larger bodied free-living phyla like ribbon worms, peanut worms and priapulids. Vermiform Vermiform describes something shaped like a worm. The expression is often employed in biology and anatomy to describe more or less tubular or cylindrical, usually soft body parts or animals. The word root is Latin, ""vermes""-worms and ""formes""-shaped. A well known example is the vermiform appendix, a small, blind section of the gut in humans and a number of other mammals. A number of soft-bodied animal phyla are typically described as vermiform."
"pseudo-Riemannian manifold. Nondegenerate form In linear algebra, a nondegenerate form or nonsingular form is a bilinear form that is not degenerate, meaning that formula_1 is an isomorphism, or equivalently in finite dimensions, if and only if The most important examples of nondegenerate forms are inner products and symplectic forms. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map formula_3 be an isomorphism, not positivity. For example, a manifold with an inner product structure on its tangent spaces is a Riemannian manifold, while relaxing this to a symmetric nondegenerate form"
"curve, a three-dimensional curve formed by intersecting a sphere with a cylinder, also has a figure eight shape, and has the lemniscate of Gerono as its planar projection. Other figure-eight shaped algebraic curves include Lemniscate In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin ""lēmniscātus"" meaning ""decorated with ribbons"", from the Greek λημνίσκος meaning ribbons, or alternatively may refer to the wool from which the ribbons were made. Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli,"
"Conifold In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base. In physics, in particular in flux compactifications of string theory, the base is usually a five-dimensional real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces. Conifolds are important objects in string theory: Brian Greene explains the physics of conifolds in Chapter 13 of his book ""The Elegant Universe""—including the fact that the space can tear near the cone, and its topology can change. This"
"Lentoid Lentoid is a geometric shape of a three-dimensional body, best described as a circle viewed from one direction and a convex lens viewed from every orthogonal direction. The term is most often used in describing jewelry and cellular phenomena in microbiology. Since ancient times, the lentoid shape has been used to fashion jewelry and seals for identification made from a variety of gemstones and metals. In Minoan Crete, for example, seals have been found with complex carving on lentoid stones. The lentoid shape was one of the most commonly recovered seal shapes from Minoan Knossos on Crete dating to"
"Andre Lichnerowicz states that: Given a compact ""n""-dimensional Riemannian manifold with no boundary with formula_29. Assume the Ricci curvature satisfies the lower bound: where formula_31 is the metric tensor and formula_32 is any tangent vector on the manifold formula_22. Then the first positive eigenvalue formula_34 of the eigenvalue equation satisfies the lower bound: This lower bound is sharp and achieved on the sphere formula_36. In fact on formula_37 the eigenspace for formula_34 is three dimensional and spanned by the restriction of the coordinate functions formula_39 from formula_40 to formula_37. Using spherical coordinates formula_42, on formula_37 the two dimensional sphere, set"
"""small resolution""). It is believed that nearly all Calabi–Yau manifolds can be connected via these ""critical transitions"", resonating with Reid's conjecture. Conifold In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base. In physics, in particular in flux compactifications of string theory, the base is usually a five-dimensional real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces. Conifolds are important objects in string theory: Brian Greene explains the physics of conifolds in Chapter 13 of"
"of ""manifold"" grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere: A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of R: The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of R. Consider the northern hemisphere, which is the part with positive ""z"" coordinate (coloured red in the picture on the right). The function χ defined by maps"