1.
Trefoil knot
–
In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of knot theory. The trefoil knot is named after the three-leaf clover plant, specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the image of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of a parametric equation. In algebraic geometry, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere S3 with the plane curve of zeroes of the complex polynomial z2 + w3. If one end of a tape or belt is turned over three times and then pasted to the other, the forms a trefoil knot. The trefoil knot is chiral, in the sense that a knot can be distinguished from its own mirror image. The two resulting variants are known as the trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, though the trefoil knot is chiral, it is also invertible, meaning that there is no distinction between a counterclockwise-oriented trefoil and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, the trefoil knot is nontrivial, meaning that it is not possible to untie a trefoil knot in three dimensions without cutting it. From a mathematical point of view, this means that a knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie a trefoil, proving this requires the construction of a knot invariant that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability, the trefoil is tricolorable, in addition, virtually every major knot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants. In knot theory, the trefoil is the first nontrivial knot and it is a prime knot, and is listed as 31 in the Alexander-Briggs notation. The Dowker notation for the trefoil is 462, the trefoil can be described as the -torus knot. It is also the knot obtained by closing the braid σ13, the trefoil is an alternating knot
2.
3D computer graphics
–
Such images may be stored for viewing later or displayed in real-time. 3D computer graphics rely on many of the same algorithms as 2D computer vector graphics in the wire-frame model, 3D computer graphics are often referred to as 3D models. Apart from the graphic, the model is contained within the graphical data file. However, there are differences, a 3D model is the representation of any three-dimensional object. A model is not technically a graphic until it is displayed, a model can be displayed visually as a two-dimensional image through a process called 3D rendering or used in non-graphical computer simulations and calculations. With 3D printing, 3D models are rendered into a 3D physical representation of the model. William Fetter was credited with coining the term computer graphics in 1961 to describe his work at Boeing, 3D computer graphics software began appearing for home computers in the late 1970s. The earliest known example is 3D Art Graphics, a set of 3D computer graphics effects, written by Kazumasa Mitazawa, models can also be produced procedurally or via physical simulation. Basically, a 3D model is formed from points called vertices that define the shape, a polygon is an area formed from at least three vertexes. A polygon of n points is an n-gon, the overall integrity of the model and its suitability to use in animation depend on the structure of the polygons. Before rendering into an image, objects must be out in a scene. This defines spatial relationships between objects, including location and size, Animation refers to the temporal description of an object. These techniques are used in combination. As with animation, physical simulation also specifies motion, rendering converts a model into an image either by simulating light transport to get photo-realistic images, or by applying an art style as in non-photorealistic rendering. The two basic operations in realistic rendering are transport and scattering and this step is usually performed using 3D computer graphics software or a 3D graphics API. Altering the scene into a form for rendering also involves 3D projection. There are a multitude of websites designed to help, educate, some are managed by software developers and content providers, but there are standalone sites as well. These communities allow for members to seek advice, post tutorials, not all computer graphics that appear 3D are based on a wireframe model
3.
Knot theory
–
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, in mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Knots can be described in various ways, given a method of description, however, there may be more than one description that represents the same knot. For example, a method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in different ways using a knot diagram. Therefore, a problem in knot theory is determining when two descriptions represent the same knot. A complete algorithmic solution to this exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, important invariants include knot polynomials, knot groups, and hyperbolic invariants. The original motivation for the founders of theory was to create a table of knots and links. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century, to gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other spaces and objects other than circles can be used. Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space, archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics, Knots appear in various forms of Chinese artwork dating from several centuries BC. The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, the Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork. Mathematical studies of knots began in the 19th century with Gauss, in the 1860s, Lord Kelvins theory that atoms were knots in the aether led to Peter Guthrie Taits creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings and this record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology. This would be the approach to knot theory until a series of breakthroughs transformed the subject. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem, many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants
4.
Knot (mathematics)
–
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, the term knot is also applied to embeddings of S j in S n, especially in the case j = n −2. The branch of mathematics that studies knots is known as knot theory, a knot is an embedding of the circle into three-dimensional Euclidean space. Or the 3-sphere, S3, since the 3-sphere is compact, two knots are defined to be equivalent if there is an ambient isotopy between them. A knot in R3, can be projected onto a plane R2, in this case, by choosing a projection side, one can completely encode the isotopy class of the knot by its regular projection by recording a simple over/under information at these crossings. In graph theory terms, a projection of a knot. The local modifications of this graph which allow to go from one diagram to any other diagram of the knot are called Reidemeister moves. The simplest knot, called the unknot or trivial knot, is a circle embedded in R3. In the ordinary sense of the word, the unknot is not knotted at all, the simplest nontrivial knots are the trefoil knot, the figure-eight knot and the cinquefoil knot. Several knots, linked or tangled together, are called links, Knots are links with a single component. A polygonal knot is a knot whose image in R3 is the union of a set of line segments. A tame knot is any knot equivalent to a polygonal knot, Knots which are not tame are called wild, and can have pathological behavior. In knot theory and 3-manifold theory, often the adjective tame is omitted, smooth knots, for example, are always tame. A framed knot is the extension of a knot to an embedding of the solid torus D2 × S1 in S3. The framing of the knot is the number of the image of the ribbon I × S1 with the knot. A framed knot can be seen as the ribbon and the framing is the number of twists. This definition generalizes to a one for framed links
5.
Rotational symmetry
–
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
6.
Chirality (mathematics)
–
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral, in 3 dimensions, not all achiral objects have a mirror plane. For example, a 3-dimensional object with inversion centre as its only nontrivial symmetry operation is achiral but has no mirror plane, a chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ, the hand, the most familiar chiral object, 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, many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. A right shoe is different from a left shoe only for being mirror images of each other, in contrast thin gloves may not be considered chiral if you can wear them inside-out. The J, L, S and Z-shaped tetrominoes of the video game Tetris also exhibit chirality. Individually they contain no mirror symmetry in the plane, a figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. In three dimensions, every figure that possesses a plane of symmetry S1, an inversion center of symmetry S2. Note, however, that there are achiral figures lacking both plane and center of symmetry, an example is the figure F0 = which is invariant under the orientation reversing isometry ↦ and thus achiral, but it has neither plane nor center of symmetry. The figure F1 = also is achiral as the origin is a center of symmetry, note also that achiral figures can have a center axis. In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry and its symmetry group is a frieze group generated by a single glide reflection. A knot is called if it can be continuously deformed into its mirror image. For example the unknot and the knot are achiral, whereas the trefoil knot is chiral. Chirality in Metric Spaces, Symmetry, Culture and Science 21, pp. 27–36 Chiral Polyhedra by Eric W. Weisstein, when Topology Meets Chemistry by Erica Flapan. Chiral manifold at the Manifold Atlas
7.
Link (knot theory)
–
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component, links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a reference link, usually called the unlink. For example, a co-dimension two link in 3-dimensional space is a subspace of 3-dimensional Euclidean space whose connected components are homeomorphic to circles, the simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked, the Borromean rings thus form a Brunnian link and in fact constitute the simplest such link. The notion of a link can be generalized in a number of ways, frequently the word link is used to describe any submanifold of the sphere S n diffeomorphic to a disjoint union of a finite number of spheres, S j. If M is disconnected, the embedding is called a link, if M is connected, it is called a knot. While links are defined as embeddings of circles, it is interesting and especially technically useful to consider embedded intervals. The type of a tangle is the manifold X, together with an embedding of ∂ X. The condition that the boundary of X lies in R × says that intervals either connect two lines or connect two points on one of the lines, but imposes no conditions on the circles. Tangles include links, braids, and others besides – for example, in this context, a braid is defined as a tangle which is always going down – whose derivative always has a non-zero component in the vertical direction. In particular, it must consist solely of intervals, and not double back on itself, however, a string link is a tangle consisting of only intervals, with the ends of each strand required to lie at. – i. e. connecting the integers, and ending in the order that they began, if this has ℓ components. A string link need not be a braid – it may double back on itself, a braid that is also a string link is called a pure braid, and corresponds with the usual such notion. The key technical value of tangles and string links is that they have algebraic structure, the tensor structure is given by juxtaposition of tangles – putting one tangle to the right of the other. For a fixed ℓ, isotopy classes of ℓ-component string links form a monoid, however, concordance classes of string links do have inverses, where inverse is given by flipping the string link upside down, and thus form a group. Linking number Hyperbolic link Unlink Link group
8.
Braid theory
–
In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the operation is do the first braid on a set of strings. Such groups may be described by explicit presentations, as was shown by Emil Artin, for an elementary treatment along these lines, see the article on braid groups. Braid groups are also understood by a deeper mathematical interpretation, as the group of certain configuration spaces. To explain how to reduce a braid group in the sense of Artin to a fundamental group and that is, an ordered n-tuple is in the same orbit as any other that is a re-ordered version of it. A path in the symmetric product is the abstract way of discussing n points of X, considered as an unordered n-tuple. Since we must require that the strings never pass through other, it is necessary that we pass to the subspace Y of the symmetric product. That is, we remove all the subspaces of Xn defined by conditions xi = xj and this is invariant under the symmetric group, and Y is the quotient by the symmetric group of the non-excluded n-tuples. Under the dimension condition Y will be connected, with this definition, then, we can call the braid group of X with n strings the fundamental group of Y. The case where X is the Euclidean plane is the one of Artin. In some cases it can be shown that the homotopy groups of Y are trivial. When X is the plane, the braid can be closed, i. e. corresponding ends can be connected in pairs, to form a link, i. e. a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to n, a theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the closure of a braid. Different braids can give rise to the link, just as different crossing diagrams can give rise to the same knot. Markov describes two moves on braid diagrams that yield equivalence in the corresponding closed braids, a single-move version of Markovs theorem, was published by Lambropoulou & Rourke. Vaughan Jones originally defined his polynomial as an invariant and then showed that it depended only on the class of the closed braid. The braid index is the least number of strings needed to make a closed braid representation of a link and it is equal to the least number of Seifert circles in any projection of a knot. Additionally, the length is the longest dimension of a braid
9.
Cinquefoil knot
–
In knot theory, the cinquefoil knot, also known as Solomons seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the 51 knot in the Alexander-Briggs notation, the cinquefoil is the closed version of the double overhand knot. The cinquefoil is a prime knot and its writhe is 5, and it is invertible but not amphichiral. These are the same as the Alexander, Conway, and Jones polynomials of the knot 10132, however, the Kauffman polynomial can be used to distinguish between these two knots. The name “cinquefoil” comes from the flowers of plants in the genus Potentilla. Pentagram Trefoil knot 7₁ knot Skein relation A Pentafoil Knot at the Wayback Machine
10.
Mikhail Leonidovich Gromov
–
Mikhail Leonidovich Gromov, is a French-Russian mathematician known for important contributions in many different areas of mathematics, including geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University, Gromov has won several prizes, including the Abel Prize in 2009 for his revolutionary contributions to geometry. Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union and his father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. Gromov was born during World War II, and his mother, when Gromov was nine years old, his mother gave him the book The Enjoyment of Mathematics by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him. Gromov studied mathematics at Leningrad State University where he obtained a degree in 1965. His thesis advisor was Vladimir Rokhlin, in 1970, invited to give a presentation at the International Congress of Mathematicians in France, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings, disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14. In the early 1970s he ceased publication, hoping that this would help his application to move to Israel and he changed his last name to that of his mother. When the request was granted in 1974, he moved directly to New York where a position had been arranged for him at Stony Brook. In 1981 he left Stony Brook to join the faculty of University of Paris VI, at the same time, he has held professorships at the University of Maryland, College Park from 1991 to 1996, and at the Courant Institute of Mathematical Sciences since 1996. He adopted French citizenship in 1992, Gromovs style of geometry often features a coarse or soft viewpoint, analyzing asymptotic or large-scale properties. In the 1980s, Gromov introduced the Gromov–Hausdorff metric, a measure of the difference between two metric spaces. The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ c, Gromov was also the first to study the space of all possible Riemannian structures on a given manifold. Gromov introduced geometric group theory, the study of infinite groups via the geometry of their Cayley graphs, in 1981 he proved Gromovs theorem on groups of polynomial growth, a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The proof uses the Gromov–Hausdorff metric mentioned above, along with Eliyahu Rips he introduced the notion of hyperbolic groups. Gromov founded the field of symplectic topology by introducing the theory of pseudoholomorphic curves and this led to Gromov–Witten invariants which are used in string theory and to his non-squeezing theorem. Gromov is also interested in biology, the structure of the brain and the thinking process. Member of the French Academy of Sciences Gromov, M. Hyperbolic manifolds, groups, riemann surfaces and related topics, Proceedings of the 1978 Stony Brook Conference, pp. 183–213, Ann. of Math
11.
Torus
–
In geometry, a torus 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 shape and is called a torus of revolution. Real-world examples of objects include inner tubes, swim rings, and the surface of a doughnut. A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, a solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, vadai or vada, many lifebuoys, in topology, a ring torus is homeomorphic to the Cartesian product of two circles, S1 × S1, and the latter is taken to be 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 three-dimensional Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane. This produces an object called the Clifford torus, a surface in 4-space. In the field of topology, a torus is any space that is topologically equivalent to a torus. R is known as the radius and r is known as the minor radius. The ratio R divided by r is known as the aspect ratio, a doughnut has an aspect ratio of about 2 to 3. An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is 2 + z 2 = r 2, or the solution of f =0, algebraically eliminating the square root gives a quartic equation,2 =4 R2. The three different classes of standard tori correspond to the three aspect ratios between R and r, When R > r, the surface will be the familiar ring torus. R = r corresponds to the torus, which in effect is a torus with no hole. R < r describes the self-intersecting spindle torus, when R =0, the torus degenerates to the sphere. When R ≥ r, the interior 2 + z 2 < r 2 of this torus is diffeomorphic to a product of an Euclidean open disc, the losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. In traditional spherical coordinates there are three measures, R, the distance from the center of the system, and θ and φ. As a torus has, effectively, two points, the centerpoints of the angles are moved, φ measures the same angle as it does in the spherical system. The center point of θ is moved to the center of r and these terms were first used in a discussion of the Earths magnetic field, where poloidal was used to denote the direction toward the poles
12.
3-sphere
–
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a central point in 4-dimensional Euclidean space. A 3-sphere is an example of a 3-manifold, in coordinates, a 3-sphere with center and radius r is the set of all points in real, 4-dimensional space such that ∑ i =032 =2 +2 +2 +2 = r 2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S3 and it is often convenient to regard R4 as the space with 2 complex dimensions or the quaternions. The unit 3-sphere is then given by S3 = or S3 = and this description as the quaternions of norm one, identifies the 3-sphere with the versors in the quaternion division ring. Just as the circle is important for planar polar coordinates. See polar decomposition of a quaternion for details of development of the three-sphere. This view of the 3-sphere is the basis for the study of space as developed by Georges Lemaître. The 3-dimensional cubic hyperarea of a 3-sphere of radius r is 2 π2 r 3 while the 4-dimensional quartic hypervolume is 12 π2 r 4, every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere. Then the 2-sphere shrinks again down to a point as the 3-sphere leaves the hyperplane. A 3-sphere is a compact, connected, 3-dimensional manifold without boundary, what this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold with these properties, the 3-sphere is homeomorphic to the one-point compactification of R3. In general, any space that is homeomorphic to the 3-sphere is called a topological 3-sphere. The homology groups of the 3-sphere are as follows, H0, any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to S3, infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1/n on any knot in the 3-sphere gives a homology sphere, as to the homotopy groups, we have π1 = π2 = and π3 is infinite cyclic. The higher-homotopy groups are all finite abelian but otherwise follow no discernible pattern, for more discussion see homotopy groups of spheres. The 3-sphere is naturally a smooth manifold, in fact, an embedded submanifold of R4