1.
String theory
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In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how strings propagate through space and interact with each other. On distance scales larger than the scale, a string looks just like an ordinary particle, with its mass, charge. In string theory, one of the vibrational states of the string corresponds to the graviton. Thus string theory is a theory of quantum gravity, String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. Despite much work on problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows to choose the details. String theory was first studied in the late 1960s as a theory of the nuclear force. Subsequently, it was realized that the properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest version of string theory, bosonic string theory, incorporated only the class of known as bosons. It later developed into superstring theory, which posits a connection called supersymmetry between bosons and the class of particles called fermions. In late 1997, theorists discovered an important relationship called the AdS/CFT correspondence, one of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, and these issues have led some in the community to criticize these approaches to physics and question the value of continued research on string theory unification. In the twentieth century, two theoretical frameworks emerged for formulating the laws of physics, one of these frameworks was Albert Einsteins general theory of relativity, a theory that explains the force of gravity and the structure of space and time. The other was quantum mechanics, a different formalism for describing physical phenomena using probability. In spite of successes, there are still many problems that remain to be solved. One of the deepest problems in physics is the problem of quantum gravity. The general theory of relativity is formulated within the framework of classical physics, in addition to the problem of developing a consistent theory of quantum gravity, there are many other fundamental problems in the physics of atomic nuclei, black holes, and the early universe. String theory is a framework that attempts to address these questions

2.
Fourth dimension in art
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New possibilities opened up by the concept of four-dimensional space helped inspire many modern artists in the first half of the twentieth century. Early Cubists, Surrealists, Futurists, and abstract artists took ideas from higher-dimensional mathematics, french mathematician Maurice Princet was known as le mathématicien du cubisme. Picassos Portrait of Daniel-Henry Kahnweiler in 1910 was an important work for the artist, early cubist Max Weber wrote an article entitled In The Fourth Dimension from a Plastic Point of View, for Alfred Stieglitzs July 1910 issue of Camera Work. Another influence on the School of Paris was that of Jean Metzinger and Albert Gleizes, in 1936 in Paris, Charles Tamkó Sirató published his Manifeste Dimensioniste, which described how the Dimensionist tendency has led to, Literature leaving the line and entering the plane. Painting leaving the plane and entering space, sculpture stepping out of closed, immobile forms. …The artistic conquest of space, which to date has been completely art-free. The manifesto was signed by prominent modern artists worldwide. In 1953, the surrealist Salvador Dalí proclaimed his intention to paint an explosive, nuclear and he said that, This picture will be the great metaphysical work of my summer. Completed the next year, Crucifixion depicts Jesus Christ upon the net of a hypercube, the unfolding of a tesseract into eight cubes is analogous to unfolding the sides of a cube into six squares. The Metropolitan Museum of Art describes the painting as a new interpretation of an oft-depicted subject, christs spiritual triumph over corporeal harm. Some of Piet Mondrians abstractions and his practice of Neoplasticism are said to be rooted in his view of a utopian universe, the fourth dimension has been the subject of numerous fictional stories. De Stijl Five-dimensional space Four-dimensional space Duration Philosophy of space and time Octacube Clair, spirits, Art, and the Fourth Dimension. Dalí, Salvador, Gómez de la Serna, Ramón, Maurice Princet, Le Mathématicien du Cubisme. Overview of The Fourth Dimension And Non-Euclidean Geometry In Modern Art, traité élémentaire de géométrie à quatre dimensions et introduction à la géométrie à n dimensions. Art in the Fourth Dimension, Giving Form to Form – The Abstract Paintings of Piet Mondrian, spaces of Utopia, An Electronic Journal, 23–35. Einstein, Picasso, space, time, and beauty that causes havoc, making Music Modern, New York in the 1920s. Shadows of Reality, The Fourth Dimension in Relativity, Cubism, in The Fourth Dimension from a Plastic Point of View. The Fourth Dimension And Non-Euclidean Geometry In Modern Art, duchamp in Context, Science and Technology in the Large Glass and Related Works

3.
Flatland
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Flatland, A Romance of Many Dimensions is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Several films have made from the story, including the feature film Flatland. Other efforts have been short or experimental films, including one narrated by Dudley Moore, the story describes a two-dimensional world occupied by geometric figures, whereof women are simple line-segments, while men are polygons with various numbers of sides. The narrator is a square named A Square, a member of the caste of gentlemen and professionals, who guides the readers through some of the implications of life in two dimensions. The first half of the story goes through the practicalities of existing in a two-dimensional universe as well as a history leading up to the year 1999 on the eve of the 3rd Millennium. In the end, the monarch of Lineland tries to kill A Square rather than tolerate his nonsense any further, following this vision, he is himself visited by a three-dimensional sphere named A Sphere, which he cannot comprehend until he sees Spaceland for himself. This Sphere visits Flatland at the turn of each millennium to introduce a new apostle to the idea of a dimension in the hopes of eventually educating the population of Flatland. After this proclamation is made, many witnesses are massacred or imprisoned, including A Squares brother, let us leave this God of Pointland to the ignorant fruition of his omnipresence and omniscience, nothing that you or I can do can rescue him from his self-satisfaction. The Square recognizes the identity of the ignorance of the monarchs of Pointland and Lineland with his own ignorance of the existence of higher dimensions. Eventually the Square himself is imprisoned for just this reason, with only occasional contact with his brother who is imprisoned in the same facility and he does not manage to convince his brother, even after all they have both seen. Men are portrayed as polygons whose social status is determined by their regularity, on the other hand, females consist only of lines and are required by law to sound a peace-cry as they walk, lest they be mistaken face-to-face for a point. The Square evinces accounts of cases where women have accidentally or deliberately stabbed men to death, in the world of Flatland, classes are distinguished by the Art of Hearing, the Art of Feeling, and the Art of Sight Recognition. Classes can be distinguished by the sound of voice, but the lower classes have more developed vocal organs. Feeling, practised by the classes and women, determines the configuration of a person by feeling one of its angles. The Art of Sight Recognition, practised by the classes, is aided by Fog. With this, polygons with sharp angles relative to the observer will fade more rapidly than polygons with more gradual angles, colour of any kind is banned in Flatland after Isosceles workers painted themselves to impersonate noble Polygons. The Square describes these events, and the class war at length. Thus the son of a Square is a Pentagon, the son of a Pentagon, a Hexagon and this rule is not the case when dealing with Isosceles Triangles with only two congruent sides

4.
Four-dimensional space
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For example, the volume of a rectangular box is found by measuring its length, width, and depth. More than two millennia ago Greek philosophers explored in detail the implications of this uniformity, culminating in Euclids Elements. However, it was not until recent times that a handful of insightful mathematical innovators generalized the concept of dimensions to more than three. The idea of adding a fourth dimension began with Joseph-Louis Lagrange in the mid 1700s, in 1880 Charles Howard Hinton popularized these insights in an essay titled What is the Fourth Dimension. Which was notable for explaining the concept of a cube by going through a step-by-step generalization of the properties of lines, squares. The simplest form of Hintons method is to draw two ordinary cubes separated by a distance, and then draw lines between their equivalent vertices. This form can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube, the eight lines connecting the vertices of the two cubes in that case represent a single direction in the unseen fourth dimension. Higher dimensional spaces have become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without the use of such spaces, calendar entries for example are usually 4D locations, such as a meeting at time t at the intersection of two streets on some building floor. In list form such a meeting place at the 4D location. Einsteins concept of spacetime uses such a 4D space, though it has a Minkowski structure that is a bit more complicated than Euclidean 4D space, when dimensional locations are given as ordered lists of numbers such as they are called vectors or n-tuples. It is only when such locations are linked together into more complicated shapes that the richness and geometric complexity of 4D. A hint of that complexity can be seen in the animation of one of simplest possible 4D objects. Lagrange wrote in his Mécanique analytique that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, and one of time, the possibility of geometry in higher dimensions, including four dimensions in particular, was thus established. An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843 and this associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R, one of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension. Published in the Dublin University magazine and he coined the terms tesseract, ana and kata in his book A New Era of Thought, and introduced a method for visualising the fourth dimension using cubes in the book Fourth Dimension. Hintons ideas inspired a fantasy about a Church of the Fourth Dimension featured by Martin Gardner in his January 1962 Mathematical Games column in Scientific American, in 1886 Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams

5.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension

6.
Six-dimensional space
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Six-dimensional space is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are a number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes, six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature. Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6-vectors in this space, as such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined, and can be used to calculate the metric,6 ×6 matrices can be used to describe transformations such as rotations that keep the origin fixed. More generally, any space that can be described locally with six coordinates, one example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional Euclidean space ℝ7 that are equidistant from the origin and this constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions. Such non-Euclidean spaces are far more common than Euclidean spaces, a polytope in six dimensions is called a 6-polytope. The most studied are the regular polytopes, of which there are three in six dimensions, the 6-simplex, 6-cube, and 6-orthoplex. A wider family are the uniform 6-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 6-demicube is a unique polytope from the D6 family, and 221 and 122 polytopes from the E6 family. The 5-sphere, or hypersphere in six dimensions, is the five dimensional surface equidistant from a point and it has symbol S5, and the equation for the 5-sphere, radius r, centre the origin is S5 =. The volume of space bounded by this 5-sphere is V6 = π3 r 66 which is 5.16771 × r6. The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point and it has symbol S6, and the equation for the 6-sphere, radius r, centre the origin is S6 =. The volume of the bounded by this 6-sphere is V7 =16 π3 r 7105 which is 4.72477 × r7. In three dimensional space a transformation has six degrees of freedom, three translations along the three coordinate axes and three from the rotation group SO. Often these transformations are handled separately as they have different geometrical structures. In screw theory angular and linear velocity are combined into one six-dimensional object, a similar object called a wrench combines forces and torques in six dimensions

7.
Degrees of freedom (physics and chemistry)
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In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all dimensions of a system is known as a phase space, a degree of freedom of a physical system is an independent parameter that is necessary to characterize the state of a physical system. In general, a degree of freedom may be any property that is not dependent on other variables. The location of a particle in space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. On the other hand, a system with an object that can rotate or vibrate can have more than six degrees of freedom. In statistical mechanics, a degree of freedom is a scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the phase space. In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer and it is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a function to the energy of the system. In three-dimensional space, three degrees of freedom are associated with the movement of a particle, a diatomic gas molecule has 7 degrees of freedom. This set may be decomposed in terms of translations, rotations, the center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two degrees of motion and two vibrational modes. The rotations occur around the two axes perpendicular to the line between the two atoms, the rotation around the atom–atom bond is not a physical rotation. This yields, for a molecule, a decomposition of. In special cases, such as adsorbed large molecules, the degrees of freedom can be limited to only one. As defined above one can also count degrees of freedom using the number of coordinates required to specify a position. This is done as follows, For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space, thus its degree of freedom in a 3-D space is 3

8.
Eight-dimensional space
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In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n =8, the set of all locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance, eight-dimensional Euclidean space is eight-dimensional space equipped with a Euclidean metric, which is defined by the dot product. More generally the term may refer to a vector space over any field, such as an eight-dimensional complex vector space. It may also refer to a manifold such as an 8-sphere. A polytope in eight dimensions is called an 8-polytope, the most studied are the regular polytopes, of which there are only three in eight dimensions, the 8-simplex, 8-cube, and 8-orthoplex. A broader family are the uniform 8-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 8-demicube is a unique polytope from the D8 family, and 421,241, and 142 polytopes from the E8 family. The 7-sphere or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point and it has symbol S7, with formal definition for the 7-sphere with radius r of S7 =. The volume of the bounded by this 7-sphere is V8 = π424 R8 which is 4.05871 × r8. The kissing number problem has been solved in eight dimensions, thanks to the existence of the 421 polytope, the kissing number in eight dimensions is 240. The octonions are a division algebra over the real numbers. Mathematically they can be specified by 8-tuplets of real numbers, so form an 8-dimensional vector space over the reals, a normed algebra is one with a product that satisfies ∥ x y ∥ ≤ ∥ x ∥ ∥ y ∥ for all x and y in the algebra. A normed division algebra additionally must be finite-dimensional, and have the property that every non-zero vector has a multiplicative inverse. Hurwitzs theorem prohibits such a structure from existing in other than 1,2,4. The complexified quaternions C ⊗ H, or biquaternions, are an eight-dimensional algebra dating to William Rowan Hamiltons work in the 1850s and this algebra is equivalent to the Clifford algebra C ℓ2 and the Pauli algebra. It has also proposed as a practical or pedagogical tool for doing calculations in special relativity. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C

9.
Five-dimensional space
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A five-dimensional space is a space with five dimensions. If interpreted physically, that is one more than the three spatial dimensions and the fourth dimension of time used in relativitistic physics. It is an abstraction which occurs frequently in mathematics, where it is a legitimate construct, in physics and mathematics, a sequence of N numbers can be understood to represent a location in an N-dimensional space. Whether or not the universe is five-dimensional is a topic of debate, although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century. To explain why this dimension would not be observable, Klein suggested that the fifth dimension would be rolled up into a tiny. While not detectable, it would imply a connection between seemingly unrelated forces. Superstring theory then evolved into a generalized approach known as M-theory. M-theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings, the other 10 dimensions are compacted, or rolled up, to a size below the subatomic level. Kaluza–Klein theory today is seen as essentially a gauge theory, with the gauge being the circle group, the fifth dimension is difficult to directly observe, though the Large Hadron Collider provides an opportunity to record indirect evidence of its existence. Mathematical approaches were developed in the early 20th century that viewed the fifth dimension as a theoretical construct and these theories make reference to Hilbert space, a concept that postulates an infinite number of mathematical dimensions to allow for a limitless number of quantum states. They suggested that electromagnetism resulted from a field that is “polarized” in the fifth dimension. The main novelty of Einstein and Bergmann was to consider the fifth dimension as a physical entity, rather than an excuse to combine the metric tensor. But they then reneged, modifying the theory to break its five-dimensional symmetry, minkowski space and Maxwells equations in vacuum can be embedded in a five-dimensional Riemann curvature tensor. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal particle. T Hooft has speculated that the dimension is really the spacetime fabric. According to Klein’s definition, a geometry is the study of the invariant properties of a spacetime, therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations. In five or more dimensions, only three regular polytopes exist, in five dimensions, they are, The 5-simplex of the simplex family, with 6 vertices,15 edges,20 faces,15 cells, and 6 hypercells. The 5-cube of the family, with 32 vertices,80 edges,80 faces,40 cells

10.
Nine-dimensional space
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In mathematics, a sequence of n real numbers can be understood as a point in n-dimensional space. When n =9, the set of all locations is called 9-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance, nine-dimensional Euclidean space is nine-dimensional space equipped with a Euclidean metric, which is defined by the dot product. More generally, the term may refer to a vector space over any field, such as a nine-dimensional complex vector space. It may also refer to a manifold such as a 9-sphere. A polytope in nine dimensions is called an 9-polytope, the most studied are the regular polytopes, of which there are only three in nine dimensions, the 9-simplex, 9-cube, and 9-orthoplex. A broader family are the uniform 9-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 9-demicube is a unique polytope from the D9 family. H. S. M. Coxeter, H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 Wiley, Kaleidoscopes, Selected Writings of H. S. M

11.
Seven-dimensional space
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In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n =7, the set of all locations is called 7-dimensional space. Often such a space is studied as a space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean metric, more generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions. It may also refer to a manifold such as a 7-sphere. Seven-dimensional spaces have a number of properties, many of them related to the octonions. An especially distinctive property is that a product can be defined only in three or seven dimensions. This is related to Hurwitzs theorem, which prohibits the existence of structures like the quaternions and octonions in dimensions other than 2,4. The first exotic spheres ever discovered were seven-dimensional, a polytope in seven dimensions is called a 7-polytope. The most studied are the regular polytopes, of which there are three in seven dimensions, the 7-simplex, 7-cube, and 7-orthoplex. A wider family are the uniform 7-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 7-demicube is a unique polytope from the D7 family, and 321,231, and 132 polytopes from the E7 family. The 6-sphere or hypersphere in seven-dimensional Euclidean space is the six-dimensional surface equidistant from a point and it has symbol S6, with formal definition for the 6-sphere with radius r of S6 =. The volume of the bounded by this 6-sphere is V7 =16 π3105 r 7 which is 4.72477 × r7. A cross product, that is a valued, bilinear, anticommutative. Along with the usual cross product in three dimensions it is the only such product, except for trivial products. In 1956, John Milnor constructed an exotic sphere in 7 dimensions, in 1963 he showed that the exact number of such structures is 28. Euclidean geometry List of geometry topics List of regular polytopes H. S. M, dover,1973 J. W. Milnor, On manifolds homeomorphic to the 7-sphere

12.
VC dimension
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In statistical learning theory and computational learning theory, the VC dimension is a measure of the capacity of a space of functions that can be learned by a statistical classification algorithm. It is defined as the cardinality of the largest set of points that the algorithm can shatter and it is a core concept in Vapnik–Chervonenkis theory, and was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Formally, the capacity of a model is related to how complicated it can be. For example, consider the thresholding of a polynomial, if the polynomial evaluates above zero. A high-degree polynomial can be wiggly, so it can fit a set of training points well. But one can expect that the classifier will make errors on other points, such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function and this function may not fit the training set well, because it has a low capacity. This notion of capacity is made rigorous below, let H be a set family and C a set. The VC dimension of a model f is the number of points that can be arranged so that f shatters them. More formally, it is the maximum integer D such that data point set of cardinality D can be shattered by f.1. Its VC-dimension is 0 since it cannot shatter even a single point, in general, the VC dimension of a finite classification model, which can return at most 2 d different classifiers, is at most d.2. F is a single-parametric threshold classifier on real numbers, i. e, for a certain threshold θ, the VC dimension of f is 1 because, It can shatter a single point. For every point x, a classifier f θ labels it as 0 if θ > x and it cannot shatter any set of two points. For every set of two numbers, if the smaller is labeled 1, then the larger must also be labeled 1, so not all labelings are possible. F is a single-parametric interval classifier on real numbers, i. e, for a certain parameter θ, the VC dimension of f is 2 because, It can shatter some sets of two points. E. g, for every set, a classifier f θ labels it as if θ < x −8, as if θ ∈, as if θ ∈ and it cannot shatter any set of three points. For every set of three numbers, if the smallest and the largest are labeled 1, then the one must also be labeled 1. F is a line as a classification model on points in a two-dimensional plane