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# Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. [1]

## "—And He Built a Crooked House—"

'—And He Built a Crooked House—' is a science fiction short story by Robert A. Heinlein first published in Astounding Science Fiction in February 1941.

## A Wrinkle in Time

A Wrinkle in Time is a science fantasy novel written by American writer Madeleine L'Engle, first published in 1962.

## Absolute space and time

Absolute space and time is a concept in physics and philosophy about the properties of the universe.

## Alan E. Nourse

Alan Edward Nourse (August 11, 1928 – July 19, 1992) was an American science fiction (SF) writer and physician.

## Albert Einstein

Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).

## Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

## Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.

## Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry.

## Allegory of the Cave

The Allegory of the Cave, or Plato's Cave, was presented by the Greek philosopher Plato in his work Republic (514a–520a) to compare "the effect of education (παιδεία) and the lack of it on our nature".

## Angle

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

## Arrow of time

The Arrow of Time, or Time's Arrow, is a concept developed in 1927 by the British astronomer Arthur Eddington involving the "one-way direction" or "asymmetry" of time.

## Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

## Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

## Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

## Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

## Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.

Brady John Haran (born 18 June 1976) is an Australian-born British independent filmmaker and video journalist who is known for his educational videos and documentary films produced for BBC News and his YouTube channels, the most notable being Periodic Videos and Numberphile.

## Brane

In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions.

## Brane cosmology

Brane cosmology refers to several theories in particle physics and cosmology related to string theory, superstring theory and M-theory.

## C-symmetry

Charge conjugation is a transformation that switches all particles with their corresponding antiparticles, and thus changes the sign of all charges: not only electric charge but also the charges relevant to other forces.

## Calabi–Yau manifold

In algebraic geometry, a Calabi&ndash;Yau manifold, also known as a Calabi&ndash;Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics.

## Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Charles Howard Hinton

Charles Howard Hinton (1853, United Kingdom &ndash; 30 April 1907, Washington D.C., United States) was a British mathematician and writer of science fiction works titled Scientific Romances.

## Circa

Circa, usually abbreviated c., ca. or ca (also circ. or cca.), means "approximately" in several European languages (and as a loanword in English), usually in reference to a date.

## Classical mechanics

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

## Clifford A. Pickover

Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity and is employed at the IBM Thomas J. Watson Research Center in Yorktown, New York.

## Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

## Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Configuration space (mathematics)

T^3/S_3, is the above orbifold. --> In mathematics, a configuration space (also known as Fadell's configuration space) is a construction closely related to state spaces or phase spaces in physics.

## Connectedness

In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece".

## Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

## Correlation dimension

In chaos theory, the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension.

## Cover (topology)

In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset.

## Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

New!!: Dimension and Cube · See more »

## Curse of dimensionality

The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces (often with hundreds or thousands of dimensions) that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience.

## Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

## CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory.

## Cylinder

A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.

## Cylindrical coordinate system

A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.

## D-brane

In string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named.

## Degrees of freedom (mechanics)

In physics, the degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration.

## Degrees of freedom (physics and chemistry)

In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system.

## Degrees of freedom (statistics)

In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.

## Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

## Dimension (data warehouse)

A dimension is a structure that categorizes facts and measures in order to enable users to answer business questions.

## Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

## Dimension of an algebraic variety

In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways.

## Dimensional analysis

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed.

## Edwin Abbott Abbott

Edwin Abbott Abbott (20 December 1838 – 12 October 1926) was an English schoolmaster and theologian, best known as the author of the novella Flatland (1884).

## Electromagnetism

Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.

## Entropy

In statistical mechanics, entropy is an extensive property of a thermodynamic system.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Event (relativity)

In physics, and in particular relativity, an event is the instantaneous physical situation or occurrence associated with a point in spacetime (that is, a specific place and time).

## Experimental psychology

Experimental psychology refers to work done by those who apply experimental methods to psychological study and the processes that underlie it.

## Exterior dimension

In geometry, exterior dimension is a type of dimension that can be used to characterize the scaling behavior of "fat fractals".

## Extra dimensions

In physics, extra dimensions are proposed additional space or time dimensions beyond the (3 + 1) typical of observed spacetime, such as the first attempts based on the Kaluza–Klein theory.

## Flatland

Flatland: A Romance of Many Dimensions is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co.

## Four-dimensional space

A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space.

## Fourth dimension in art

New possibilities opened up by the concept of four-dimensional space (and difficulties involved in trying to visualize it) helped inspire many modern artists in the first half of the twentieth century.

## Fourth dimension in literature

The idea of a fourth dimension has been a factor in the evolution of modern art, but use of concepts relating to higher dimensions has been little discussed by academics in the literary world.

## Fractal

In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects.

## Fractal dimension

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured.

## Free module

In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements.

## Function space

In mathematics, a function space is a set of functions between two fixed sets.

## Fundamental interaction

In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions.

## Gauge theory

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.

## General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

## Geographic coordinate system

A geographic coordinate system is a coordinate system used in geography that enables every location on Earth to be specified by a set of numbers, letters or symbols.

## Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

## Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

## Gravity

Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another.

## Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

## Gustav Fechner

Gustav Theodor Fechner (19 April 1801 – 18 November 1887), was a German philosopher, physicist and experimental psychologist.

## Habilitation

Habilitation defines the qualification to conduct self-contained university teaching and is the key for access to a professorship in many European countries.

## Hamiltonian mechanics

Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.

## Hausdorff dimension

Hausdorff dimension is a measure of roughness in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a space, taking into account the distance between its points.

## Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

## Hiding in the Mirror

Hiding in the Mirror is a popular science book by the theoretical physicist Lawrence M. Krauss.

## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

## Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

## Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

## Hurst exponent

The Hurst exponent is used as a measure of long-term memory of time series.

## Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

## Hyperspace (book)

Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension (1994) is a book by Michio Kaku, a theoretical physicist from the City College of New York.

## Hyperspace (disambiguation)

Hyperspace is a faster-than-light method of traveling used in science fiction.

## Immanuel Kant

Immanuel Kant (22 April 1724 – 12 February 1804) was a German philosopher who is a central figure in modern philosophy.

## Inductive dimension

In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X).

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Intrinsic dimension

In signal processing of multidimensional signals, for example in computer vision, the intrinsic dimension of the signal describes how many variables are needed to represent the signal.

## Isolated point

In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S but there exists a neighborhood of x which does not contain any other points of S. This is equivalent to saying that the singleton is an open set in the topological space S (considered as a subspace of X).

## Isoperimetric dimension

In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large-scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space).

## Kaluza–Klein theory

In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the usual four of space and time and considered an important precursor to string theory.

## Knot (mathematics)

In mathematics, a knot is an embedding of a circle S^1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies).

## Krull dimension

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals.

## Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

## Large extra dimension

In particle physics, the ADD model, also known as the model with large extra dimensions (LED), is a model framework that attempts to solve the hierarchy problem by explaining the weakness of gravity relative to the other forces.

The Large Hadron Collider (LHC) is the world's largest and most powerful particle collider, the most complex experimental facility ever built and the largest single machine in the world.

## Latitude

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface.

## Laws of thermodynamics

The four laws of thermodynamics define fundamental physical quantities (temperature, energy, and entropy) that characterize thermodynamic systems at thermal equilibrium.

## Lebesgue covering dimension

In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.

## Length of a module

In abstract algebra, the length of a module is a measure of the module's "size".

## Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

## Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

## List of convex uniform tilings

This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.

## Local property

In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points.

## Longitude

Longitude, is a geographic coordinate that specifies the east-west position of a point on the Earth's surface.

## Ludwig Schläfli

Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces.

## M-theory

M-theory is a theory in physics that unifies all consistent versions of superstring theory.

Madeleine L'Engle Camp (November 29, 1918 – September 6, 2007) was an American writer who wrote young adult fiction, including A Wrinkle in Time and its sequels: A Wind in the Door, A Swiftly Tilting Planet, Many Waters, and An Acceptable Time.

## Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mean dimension

In mathematics, the mean (topological) dimension of a topological dynamical system is a non-negative extended real number that is a measure of the complexity of the system.

## Metaphysics

Metaphysics is a branch of philosophy that explores the nature of being, existence, and reality.

## Metric dimension (graph theory)

In graph theory, the metric dimension of a graph G is the minimum cardinality of a subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given value, is NP-complete.

## Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

## Miles J. Breuer

Miles John Breuer (January 3, 1889 – October 14, 1945) was an American physician and science fiction writer.

## Minkowski space

In mathematical physics, Minkowski space (or Minkowski spacetime) is a combining of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.

## Minkowski–Bouligand dimension

Estimating the box-counting dimension of the coast of Great Britain In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rn, or more generally in a metric space (X, d).

## Multidimensional analysis

In statistics, econometrics, and related fields, multidimensional analysis (MDA) is a data analysis process that groups data into two categories: data dimensions and measurements.

## Murray Leinster

Murray Leinster (June 16, 1896 &ndash; June 8, 1975) was a nom de plume of William Fitzgerald Jenkins, an American writer of science fiction and alternate history literature.

## N-skeleton

In mathematics, particularly in algebraic topology, the of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the.

## Network science

Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors represented by nodes (or vertices) and the connections between the elements or actors as links (or edges).

## Normal space

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods.

## Number line

In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by \mathbb.

## Observer (special relativity)

In special relativity, an observer is a frame of reference from which a set of objects or events are being measured.

## Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are three lower-dimensional normed division algebras over the reals: the real numbers R themselves, the complex numbers C, and the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.

## One-dimensional space

In physics and mathematics, a sequence of n numbers can specify a location in n-dimensional space.

## Order dimension

In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.

## Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

## P. D. Ouspensky

Pyotr Demianovich Ouspenskii (known in English as Peter D. Ouspensky, Пётр Демья́нович Успе́нский; 5 March 1878 – 2 October 1947), was a Russian esotericist known for his expositions of the early work of the Greek-Armenian teacher of esoteric doctrine George Gurdjieff, whom he met in Moscow in 1915.

## Parallel universes in fiction

A parallel universe is a hypothetical self-contained reality co-existing with one's own.

## Parameter space

In science, a parameter space is the set of all possible combinations of values for all the different parameters contained in a particular mathematical model.

## Parity (physics)

In quantum mechanics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate.

## Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

## Plane (esotericism)

In esoteric cosmology, a plane is conceived as a subtle state, level, or region of reality, each plane corresponding to some type, kind, or category of being.

## Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

## Plato

Plato (Πλάτων Plátōn, in Classical Attic; 428/427 or 424/423 – 348/347 BC) was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world.

## Platonic solid

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

## Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

## Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

## Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

## Prime ideal

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.

## Project Gutenberg

Project Gutenberg (PG) is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks".

## Prolegomena to Any Future Metaphysics

Prolegomena to Any Future Metaphysics That Will Be Able to Present Itself as a Science (Prolegomena zu einer jeden künftigen Metaphysik, die als Wissenschaft wird auftreten können) is a book by the German philosopher Immanuel Kant, published in 1783, two years after the first edition of his Critique of Pure Reason.

## Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite, but need only be a non-degenerate bilinear form, which is a weaker condition.

## Pseudonym

A pseudonym or alias is a name that a person or group assumes for a particular purpose, which can differ from their first or true name (orthonym).

## Quantum field theory

In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.

## Quantum gravity

Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics, and where quantum effects cannot be ignored, such as near compact astrophysical objects where the effects of gravity are strong.

## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

## Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers.

## Quotient stack

In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Regular 4-polytope

In mathematics, a regular 4-polytope is a regular four-dimensional polytope.

## René Descartes

René Descartes (Latinized: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist.

## Republic (Plato)

The Republic (Πολιτεία, Politeia; Latin: Res Publica) is a Socratic dialogue, written by Plato around 380 BC, concerning justice (δικαιοσύνη), the order and character of the just, city-state, and the just man.

## Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

## Robert A. Heinlein

Robert Anson Heinlein (See also the biography at the end of For Us, the Living, 2004 edition, p. 261. July 7, 1907 – May 8, 1988) was an American science-fiction writer.

## Rudy Rucker

Rudolf von Bitter Rucker (born March 22, 1946) is an American mathematician, computer scientist, science fiction author, and one of the founders of the cyberpunk literary movement.

## Science fiction

Science fiction (often shortened to Sci-Fi or SF) is a genre of speculative fiction, typically dealing with imaginative concepts such as advanced science and technology, spaceflight, time travel, and extraterrestrial life.

## Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

## Singular point of an algebraic variety

In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined.

## Size

Size is the magnitude or dimensions of a thing.

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## Space

Space is the boundless three-dimensional extent in which objects and events have relative position and direction.

## Space (mathematics)

In mathematics, a space is a set (sometimes called a universe) with some added structure.

## Space-filling curve

In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an n-dimensional unit hypercube).

## Spacetime

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

## Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

## Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

## Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

## Stack (mathematics)

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets.

## Stereoscopy

Stereoscopy (also called stereoscopics, or stereo imaging) is a technique for creating or enhancing the illusion of depth in an image by means of stereopsis for binocular vision.

## String theory

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.

## Supergravity

In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity where supersymmetry obeys locality; in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model.

## Superstring theory

Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.

## Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.

## T-symmetry

T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal: T-symmetry can be shown to be equivalent to the conservation of entropy, by Noether's Theorem.

## Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

## Tesseract

In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.

## The Boy Who Reversed Himself

The Boy Who Reversed Himself (1986) is a science fiction novel by William Sleator.

## The Fourth Dimension (book)

The Fourth Dimension is a non-fiction work written by Rudy Rucker, the Silicon Valley professor of mathematics and computer science, and was published in 1984 by Houghton Mifflin.

## Thomas Banchoff

Thomas Francis Banchoff (born 1938) is an American mathematician specializing in geometry.

## Three-dimensional space

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).

## Time

Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future.

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## Two-dimensional space

Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).

## Tychonoff space

In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.

## Unit circle

In mathematics, a unit circle is a circle with a radius of one.

## Universal extra dimension

In particle physics, models with universal extra dimensions include one or more spatial dimensions beyond the three spatial and one temporal dimensions that are observed.

## University of Nottingham

The University of Nottingham is a public research university in Nottingham, United Kingdom.

## UV completion

In theoretical physics, ultraviolet completion, or UV completion, of a quantum field theory is the passing from a lower energy quantum field theory to a more general quantum field theory above a threshold value known as the cutoff.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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## Western esotericism

Western esotericism (also called esotericism and esoterism), also known as the Western mystery tradition, is a term under which scholars have categorised a wide range of loosely related ideas and movements which have developed within Western society.

## William Rowan Hamilton

Sir William Rowan Hamilton MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician who made important contributions to classical mechanics, optics, and algebra.

## William Sleator

William Warner Sleator III (February 13, 1945 – August 3, 2011), known as William Sleator, was an American science fiction author who wrote primarily young adult novels but also wrote for younger readers.

## Zero-dimensional space

In mathematics, a zero-dimensional topological space (or nildimensional) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.

## 3-manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.

## 4-manifold

In mathematics, a 4-manifold is a 4-dimensional topological manifold.

## References

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