Abstract algebra | Dimension | Mathematical concepts | Geometric measurement

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. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) 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 four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), 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 Euclidean spaces or more general parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space in which we live. (Wikipedia).

Dimensions (1 of 3: The Traditional Definition - Directions)

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From playlist Exploring Mathematics: Fractals

Chapter 5 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

Chapter 2 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

Chapter 1 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

Chapter 6 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

Chapter 4 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

From playlist Dimensions Arabe/Arabic / العربية

From playlist Dimensions Arabe/Arabic / العربية

From playlist Dimensions

Commutative algebra 53: Dimension Introductory survey

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We give an introductory survey of many different ways of defining dimension. Reading: Section Exercises:

From playlist Commutative algebra

Kenneth Falconer: Intermediate dimensions, capacities and projections

The talk will review recent work on intermediate dimensions which interpolate between Hausdorff and box dimensions. We relate these dimensions to capacities which leading to ‘Marstrand-type’ theorems on the intermediate dimensions of projections of a set in $\mathbb{R}^{n}$ onto almost all

From playlist Analysis and its Applications

algebraic geometry 14 Dimension

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the dimension of a topological space, algebraic set, or ring.

From playlist Algebraic geometry I: Varieties

Juan M. Maldacena - Are there Extra Dimensions?

Free access to Closer to Truth's library of 5,000 videos: http://bit.ly/2UufzC7 Extra dimensions—beyond length, width, height—seem like the stuff of science fiction. What would extra dimensions be like? Is time the fourth dimension? Does string theory require ten or eleven dimensions? Cou

From playlist Exploring the Multiverse - Closer To Truth - Core Topic

Dave Richeson - A Romance of Many (and Fractional) Dimensions - CoM Oct 2021

Dimension seems like an intuitive idea. We are all familiar with zero-dimensional points, one-dimensional curves, two-dimensional surfaces, and three-dimensional solids. Yet dimension is a slippery idea that took mathematicians many years to understand. We will discuss the history of dimen

From playlist Celebration of Mind 2021

Are there Extra Dimensions? | Episode 406 | Closer To Truth

Extra dimensions -beyond length, width, height- seem the stuff of science fiction. What would extra dimensions be like? Is time the fourth dimension? Could deep reality be so strange? And, anyway, why would we care? Featuring interviews with Lawrence Krauss, Michio Kaku, David Gross, Nima

From playlist Closer To Truth | Season 4

Benjamin Schweinhart (4/3/18): Persistent homology and the upper box dimension

We prove the first results relating persistent homology to a classically defined fractal dimension. Several previous studies have demonstrated an empirical relationship between persistent homology and fractal dimension; our results are the first rigorous analogue of those comparisons. Spe

From playlist AATRN 2018

Commutative algebra 55: Dimension of local rings

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We give 4 definitions of the dimension of a Noetherian local ring: Brouwer-Menger-Urysohn dimension, Krull dimension, degree o

From playlist Commutative algebra

Data Warehouse Concepts | Data Warehouse Tutorial | Data Warehouse Architecture | Edureka

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From playlist Data Warehousing Tutorial Videos

From playlist Dimensions Arabe/Arabic / العربية

SHM - 16/01/15 - Constructivismes en mathématiques - Marie Françoise Roy

Marie-Françoise Roy (IRMAR, Université Rennes 1), « Méthodes constructives en algèbre abstraite : l'exemple de la dimension »

From playlist Les constructivismes mathématiques - Séminaire d'Histoire des Mathématiques