Dimension in the context of Geometric modeling

The meaning of spirituality has developed and expanded over time, and various meanings can be found alongside each other. Traditionally, spirituality referred to a religious process of re-formation which "aims to recover the original shape of man", oriented at "the image of God" as exemplified by the founders and sacred texts of the religions of the world. The term was used within early Christianity to refer to a life oriented toward the Holy Spirit and broadened during the Late Middle Ages to include mental aspects of life.

In modern times, the term has spread to other religious traditions. It broadened to refer to a wider range of experiences, including a range of esoteric and religious traditions. Modern usages tend to refer to a subjective experience of a sacred dimension, and the "deepest values and meanings by which people live", often in a context separate from organized religious institutions. This may involve belief in a supernatural realm beyond the ordinarily observable world, personal growth, a quest for an ultimate or sacred meaning, religious experience, or an encounter with one's own "inner dimension" or spirit.

A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimensional spaces are often called planes, or, more generally, surfaces. These include analogs to physical spaces, like flat planes, and curved surfaces like spheres, cylinders, and cones, which can be infinite or finite. Some two-dimensional mathematical spaces are not used to represent physical positions, like an affine plane or complex plane.

In geometry, a three-dimensional space is a mathematical space in which three values (termed coordinates) are required to determine the position of a point. Alternatively, it can be referred to as 3D space, 3-space or, rarely, tri-dimensional space. Most commonly, it means the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may refer colloquially to a subset of space, a three-dimensional region (or 3D domain), a solid figure.

Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space. The set of these n-tuples is commonly denoted ${\displaystyle \mathbb {R} ^{n},}$ and can be identified to the pair formed by a n-dimensional Euclidean space and a Cartesian coordinate system.When n = 3, this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics, it serves as a model of the physical universe, in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-time. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that these directions do not lie in the same plane. Furthermore, if these directions are pairwise perpendicular, the three values are often labeled by the terms width/breadth, height/depth, and length.

Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.

In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat, as in the Euclidean space. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space.

Time is the continuous progression of existence that occurs in an apparently irreversible succession from the past, through the present, and into the future. Time dictates all forms of action, age, and causality, being a component quantity of various measurements used to sequence events, to compare the duration of events (or the intervals between them), and to quantify rates of change of quantities in material reality or in the conscious experience. Time is often referred to as a fourth dimension, along with three spatial dimensions.

Time is primarily measured in linear spans or periods, ordered from shortest to longest. Practical, human-scale measurements of time are performed using clocks and calendars, reflecting a 24-hour day collected into a 365-day year linked to the astronomical motion of the Earth. Scientific measurements of time instead vary from Planck time at the shortest to billions of years at the longest. Measurable time is believed to have effectively begun with the Big Bang 13.8 billion years ago, encompassed by the chronology of the universe. Modern physics understands time to be inextricable from space within the concept of spacetime described by general relativity. Time can therefore be dilated by velocity and matter to pass faster or slower for an external observer, though this is considered negligible outside of extreme conditions, namely relativistic speeds or the gravitational pulls of black holes.

Terrain (from Latin terra 'earth'), alternatively relief or topographical relief, is the dimension and shape of a given surface of a land. In physical geography, terrain is the lay of the land. This is usually expressed in terms of the elevation, slope, and orientation of terrain features. Terrain affects surface water flow and distribution. Over a large area, it can affect weather and climate patterns. Bathymetry is the study of underwater relief, while hypsometry studies terrain relative to sea level.

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.

The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line.

Most commonly, the ambient space is n-dimensional Euclidean space, in which case the hyperplanes are the (n − 1)-dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion (geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces.

The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains the law of gravitation and its relation to the forces of nature. It applies to the cosmological and astrophysical realm, including astronomy.

The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old theory of mechanics created primarily by Isaac Newton. It introduced concepts including 4-dimensional spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves.

Size in general is the magnitude or dimensions of a thing. More specifically, geometrical size (or spatial size) can refer to three geometrical measures: length, area, or volume. Length can be generalized to other linear dimensions (width, height, diameter, perimeter). Size can also be measured in terms of mass, especially when assuming a density range.

In mathematical terms, "size is a concept abstracted from the process of measuring by comparing a longer to a shorter". Size is determined by the process of comparing or measuring objects, which results in the determination of the magnitude of a quantity, such as length or mass, relative to a unit of measurement. Such a magnitude is usually expressed as a numerical value of units on a previously established spatial scale, such as meters or inches.

In geometry, a Cartesian coordinate system (UK: /kɑːrˈtiːzjən/, US: /kɑːrˈtiːʒən/) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system. The point where the axes meet is called the origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame.

Similarly, the position of any point in three-dimensional space can be specified by three Cartesian coordinates, which are the signed distances from the point to three mutually perpendicular planes. More generally, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are the signed distances from the point to n mutually perpendicular fixed hyperplanes.

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its length represents the distance in relation to an arbitrary reference origin O, and its direction represents the angular orientation with respect to given reference axes. Usually denoted x, r, or s, it corresponds to the straight line segment from O to P.In other words, it is the displacement or translation that maps the origin to P:

The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension.

Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. This concept of ordinary space is called Euclidean space because it corresponds to Euclid's geometry, which was originally abstracted from the spatial experiences of everyday life.

Single locations in Euclidean 4D space can be given as vectors or 4-tuples, i.e., as ordered lists of numbers such as (x, y, z, w). For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled x, y, and z). It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of 4D spaces emerge. A hint of that complexity can be seen in the accompanying 2D animation of one of the simplest possible regular 4D objects, the tesseract, which is analogous to the 3D cube.

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve and an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its endpoints).

Euclid's Elements defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry.