POINT LINE VERTEX FROM FINITE TO INFINITE ORTHOGAPHIC PROJECTION IN SPACE A PLACE
The formal definition for the vertex in Maths is a point where two or more edges meet. Vertices are the corner points. A Point is a position in space. A point has no length or width or thickness. A point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. In geometry, a line segment is bounded by two distinct points on a line. Or we can say a line segment is part of the line that connects two points. A line has no endpoints and extends infinitely in both the direction but a line segment has two fixed or definite endpoints. There are more than two points on a line segment. In fact, there are infinitely many points on the line segment. For example, there is a third point between the two endpoints that is also in the line and or a point between the middle point and the start and on and on and on. All triangles have closed shapes. A triangle is a two-dimensional shape in Euclidean geometry, which is seen as three non-collinear points in a unique plane. Hence a triangle is formed by joining three non-collinear points. All triangles have 3 vertices. Three points are vertices of the triangle. The point of intersection of any two sides of a triangle is known as a vertex. A triangle has three vertices. A tetrahedron has 4 vertices. Cubes and cuboids have 8 vertices or 8 points in space. A circle only has one vertex, the centre of the circle. For every two points in space there is a straight line either passing through them, extending from one point to the next named called a segment line which is unique. Two fixed points, with a line through them. In order two dimensional shapes the vertices are the points where the lines of the sides intersect but, given that a circle only has one "side", the circumference, its definition of a vertex is different. Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse.A sphere has 0 vertices. A sphere has 0 edges. A sphere has 1 curved surface.
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint.
Components of an axiomatic system. Primitives (undefined terms) are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while a fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are undefined. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs. His point being that the primitive terms are just empty shells, place holders if you will, and have no intrinsic properties.
Axioms (or postulates) are statements about these primitives; for example, any two points are together incident with just one line (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have.
The laws of logic.Մ, մ (men) Is the twentieth letter of the Armenian alphabet. -մ as a suffix is a pronominal article, a personal ending added to the Verb-stem indicating the First Person Singular Present tense. The syllable մի as an adjective means 'one' classic «one» it is used in compound words. մի՛ is also a negative particle which means no, not, none.EIGHT: Could define the space the Sun travels in 24 hours, four points above and four points below.The eight-pointed star or Octagram is believed to have its origins in early astronomy. Eight lines signified the four corners of space (south, north, east and west) and in time the two equinoxes and two solstices. The eight-pointed star represents as a symbol is surrounded in a circle. The Indians from Native America referred to this as Star Knowledge. Utu's vehicle was a solar chariot,[20] which was pulled by four animals bearing the Sumerian names Uhegalanna ("the abundant light of heaven"), Uhushgalanna ("the terrifying great light of heaven"), Usurmurgalanna ("the dreadful great light of heaven") and Unirgalanna ("the noble light of heaven"). Utu was also the primary god of justice, presumably due to the fact while traveling above through the sky during day and visiting the underworld during our night he was believed to see all at all times, everything that happened in the world day and night with the dead and the living.Gary Beckman notes that the Hittite conception of solar deities does not show any Indo-European influence, and instead was largely similar to that known from Mesopotamia. He points out even the fact that the Sun god of Heaven was believed to travel in a quadriga drawn by horses, similar to Greek Helios, is not necessarily an example of the former, as deities traveling in chariots are already depicted on Mesopotamian seals from the Sargonic period.The symbolic representation of Utu was the sun disc, typically represented as a four-pointed star with wavy lines placed between the points. It is attested as early as in the Sargonic period, and continued to be represented in art through the rest of history of ancient Mesopotamia. It is well known from kudurru (boundary stones), where it is typically depicted in the first row of symbols, next to the eight-pointed star representing Inanna (Ishtar).
The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view.
For clarification in classical Euclidean geometry, a point is a primitive notion that models an exact location in space, and has no length, width, or thickness. In modern mathematics, a point refers more generally to an element of some set called a space.
For clarification in classical Euclidean geometry, a point is a primitive notion that models an exact location in space, and has no length, width, or thickness. In modern mathematics, a point refers more generally to an element of some set called a space.
Components of an axiomatic system. Primitives (undefined terms) are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while a fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are undefined. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs. His point being that the primitive terms are just empty shells, place holders if you will, and have no intrinsic properties.
Axioms (or postulates) are statements about these primitives; for example, any two points are together incident with just one line (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have.
The laws of logic.Մ, մ (men) Is the twentieth letter of the Armenian alphabet. -մ as a suffix is a pronominal article, a personal ending added to the Verb-stem indicating the First Person Singular Present tense. The syllable մի as an adjective means 'one' classic «one» it is used in compound words. մի՛ is also a negative particle which means no, not, none.
Me-ArAr, միարար is a noun for unitive, tending to unite into one. միարկութիւն is the
noun for Unification.
Given a volume, 3 points define a plane within that volume.
Given a hypervolume, 4 points define a volume within that hypervolume.
Representing or pondering upon geometric shapes requires depth awareness. Understanding visualizing projecting and plotting coordinate points in space can make easier the job of the architect, a carpenter or an inventor. If points in space are represented in sets by a real-numbers starting with 1 (the origin) a fixed point and two as the vector to represent the second point and a line and the third point, making a set of three points in space, three lines creating a surface.
Seven: A sphere can be defined by a set of seven points that are perpendicular and equidistant from a given central point. The distance from the center to any of the points is called the radius.
noun for Unification.
միաբան, adjective, adverb for concordant. միացնել verb active To unite, to joint, to link, to connect. միացում, noun Incorporation, junction. եմ, ես եմ, I am. մեզի [first person plural dative case] pronoun to us, for us. մեզմէ [first person plural ablative case], մենէ, pronoun, from us.
This meaning is what lends itself to the idea that Armenian syllable 'Me'alphabetized as Մ, մ (men) in Armenian can represent both 'none' and 'one'. This is reminiscent of the Hindu 'Om' the sacred sound, syllable, mantra, the invocation which represents the supreme primal unity, the vibration that represents consciousness of the cosmic world. The syllable is found at the beginning and the end of chapters in the Vedas, the Upanishads, and other Hindu texts. It is referred to as a "fore-sound", or Om, the primeval sound, also the sound connoting the act of creation, simply a ligature, letters of the alphabet that was in origin a syllable.
The syllable Om is first mentioned in the Upanishads, the mystical texts associated with the Vedanta philosophy. It has variously been associated with concepts like "cosmic sound" or a "mystical syllable" or an "affirmation to something divine", or as a sound that symbolized the essence of the supreme Absolute, an abstracted spiritual concept found in the Upanishads. The syllable for me is linked to knowledge and it stands for the whole the AllOne, the Man-tra, and the Մ, մ (men) in Armenian celebrate all the creative powers of the universe united as one. The syllable can also symbolize the abstract spiritual concepts like the first and last of language, and infinite knowledge.
The syllable Om is first mentioned in the Upanishads, the mystical texts associated with the Vedanta philosophy. It has variously been associated with concepts like "cosmic sound" or a "mystical syllable" or an "affirmation to something divine", or as a sound that symbolized the essence of the supreme Absolute, an abstracted spiritual concept found in the Upanishads. The syllable for me is linked to knowledge and it stands for the whole the AllOne, the Man-tra, and the Մ, մ (men) in Armenian celebrate all the creative powers of the universe united as one. The syllable can also symbolize the abstract spiritual concepts like the first and last of language, and infinite knowledge.
A finite geometry is a geometry with a finite number of points. When confined to a plane, all finite geometries are either projective plane geometries (with no parallel lines) or affine plane geometries (with parallel lines). In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. In synthetic geometry, point and line are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part".[2] In the two-dimensional Euclidean plane, a point is represented by an ordered pair (x, y) of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z) with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, (a1, a2, … , an) where n is the dimension of the space in which the point is located.
Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part".[2] In the two-dimensional Euclidean plane, a point is represented by an ordered pair (x, y) of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z) with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, (a1, a2, … , an) where n is the dimension of the space in which the point is located.
Given a plane, 2 points define a line within that plane.Given a volume, 3 points define a plane within that volume.
Given a hypervolume, 4 points define a volume within that hypervolume.
Given an n-Dimensional space, n points define an (n-1)-Dimensional space within it.
So GBT SAYS 4 points make a tetrahedron (a 3-dimensional solid with 4 triangular faces), 5 points make a pentatope, ( The pentatope is the simplest regular figure in four dimensions, representing the four-dimensional analog of the solid tetrahedron. It is also called the 5-cell, since it consists of five vertices, or pentachoron.) and 6 points make a hexahedron (a 3-dimensional solid with 6 square faces). A hexahedron is a 3-dimensional shape with six faces, straight edges, and sharp corners; the cube is probably the most recognizable hexahedron. There are different kinds of hexahedra: convex and concave. Among the convex hexahedra are seven quadrilaterallly-faced hexahedra, where all six faces have four sides. Beyond that, the mathematical concept you're asking about is called "n-dimensional space," where "n" is the number of points. In general, n points in space determine an n-dimensional space. So for example, 4 points would determine a 4-dimensional space, and 5 points would determine a 5-dimensional space. These concepts are related to a branch of mathematics called geometry, specifically multidimensional geometry.
Seven: A sphere can be defined by a set of seven points that are perpendicular and equidistant from a given central point. The distance from the center to any of the points is called the radius.
4 points in 3-dimensional space define a tetrahedron, which is a solid geometric shape with four triangular faces. However, 4 points in 2-dimensional space do not define a unique geometric shape. It is possible to form a quadrilateral or a rectangle with 4 points.
Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations.In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.In geometry, a coordinate system is a system that 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. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate".
S (Set Theory) is an axiomatic set theory it is the intuitive approach which tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. S, as a first-order theory, is two-sorted because its ontology includes “stages” as well as sets. First-order logic, also known as predicate logic, quantification logic, is a collection of formal systems used in mathematics, philosophy, linguistics.
Set theory in geometry, is branch of mathematics that deals with the properties of well-defined collections of objects, like points in space, which may or may not be of a mathematical nature, such as numbers or functions.
S (Set Theory) is an axiomatic set theory it is the intuitive approach which tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. S, as a first-order theory, is two-sorted because its ontology includes “stages” as well as sets. First-order logic, also known as predicate logic, quantification logic, is a collection of formal systems used in mathematics, philosophy, linguistics.
Ontology is any grouping together of mathematical, abstract, or concrete objects, however formed, is a collection, a synonym for what other set theories refer to as a class. The things that make up a collection are called elements or members. A formal system is essentially an "axiomatic system".A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. Abstraction is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal (real or concrete) signifiers, first principles, or other methods.
"An abstraction" is the outcome of this process—a concept that acts as a common noun for all subordinate concepts and connects any related concepts as a group, field, or category.[
Points and Line
A point is defined as a location in any space and is represented by a dot (.). In geometry, a point is again a location represented by a dot. A point represents position only. It does not have any length, width, shape or size, only a position.
A point is defined as a location in any space and is represented by a dot (.). In geometry, a point is again a location represented by a dot. A point represents position only. It does not have any length, width, shape or size, only a position.
When two distinct points are connected they form a line. The concept of points and lines is important to understand geometry figures. With the help of points, we can label and identify the geometric figures and with the help of lines, we can draw the figures.
A series of points connected by a straight path is defined as a line. A line is usually defined by two points. A line has no thickness and can extend indefinitely in both directions. The length of a line is undefined and it can have infinite numbers of points.
Types of Lines
Just like the different types of points, there are different types of lines which can be distinguished easily on the basis of their unique properties. A horizontal Line is a line that is mapped from left-to-right or right-to-left and is parallel to or is the x-axis in a plane.
A vertical Line is a line that is mapped from up to down or down to up and is parallel to or is the y-axis .
Intersecting Lines are when two lines cross each other and meet at a point, they are known as intersecting lines. The point at which they meet is known as the point of intersection and when two lines intersect exactly at 90°, they are known as perpendicular lines.
Two lines are said to be parallel if they do not intersect at any point and are equidistant.
A point marks the beginning from where and when any figure or shape is formed.
A series of points connected by a straight path is defined as a line. A line is usually defined by two points. A line has no thickness and can extend indefinitely in both directions. The length of a line is undefined and it can have infinite numbers of points.
Types of Lines
Just like the different types of points, there are different types of lines which can be distinguished easily on the basis of their unique properties. A horizontal Line is a line that is mapped from left-to-right or right-to-left and is parallel to or is the x-axis in a plane.
A vertical Line is a line that is mapped from up to down or down to up and is parallel to or is the y-axis .
Intersecting Lines are when two lines cross each other and meet at a point, they are known as intersecting lines. The point at which they meet is known as the point of intersection and when two lines intersect exactly at 90°, they are known as perpendicular lines.
Two lines are said to be parallel if they do not intersect at any point and are equidistant.
Nine: In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line. In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. In geometry, any hyperplane H of a projective space P may be taken as a hyperplane at infinity. Then the set complement P ∖ H is called an affine space. In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point. An affine sphere is called improper if all of the affine normals are constant. In that case, the intersection point mentioned above lies on the hyperplane at infinity.
A sphere (from Ancient Greek σφαῖρα (sphaîra) 'globe, ball') is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere.
Spherical regions or a ball:cap, bounded by one plane
sector, bounded by a conical boundary with apex at the center of the sphere
segment, bounded by a pair of parallel planes
shell, bounded by two concentric spheres of differing radii
wedge, bounded by two planes passing through a sphere center and the surface of the sphere
A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment.
A unit sphere is a sphere with unit radius (r = 1). For convenience, spheres are often taken to have their center at the origin of the coordinate system, and spheres in this article have their center at the origin unless a center is mentioned.
A great circle on the sphere has the same center and radius as the sphere, and divides it into two equal hemispheres.
Although the Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. If a particular point on a sphere is (arbitrarily) designated as its north pole, its antipodal point is called the south pole. The great circle equidistant to each is then the equator. Great circles through the poles are called lines of longitude or meridians. A line connecting the two poles may be called the axis of rotation. Small circles on the sphere that are parallel to the equator are lines of latitude. In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding.[3]
Mathematicians consider a sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space. They draw a distinction a sphere and a ball, which is a three-dimensional manifold with boundary that includes the volume contained by the sphere. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "circle" and "disk" in the plane is similar.
A vertex (plural vertices or vertexes). A Doublet of vortex. From Latin vertex (“whirl, eddy; top, crown, peak, summit”).The highest point, top or apex of something. Synonyms: acme, apex, peak, top(anatomy) The highest surface on the skull; the crown of the head.
(geometry) An angular point of a polygon, polyhedron or higher order polytope.The common point of the two rays that form an angle.
The point at which an axis meets a curve or surface.
(mathematics) A point on the curve with a local minimum or maximum of curvature.
(graph theory) One of the elements of a graph joined or not by edges to other vertices. synonym ▲coordinate term ▼Synonym: node
(computer graphics) A point in 3D space, usually given in terms of its Cartesian coordinates.
(optics) The point where the surface of a lens crosses the optical axis.
(particle physics) An interaction point.
(astrology) The point where the prime vertical meets the ecliptic in the western hemisphere of a natal chart.
(typography) A sharp downward point opposite a crotch, as in the letters "V" and "W" but not "Y". coordinate term ▼
The octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection, as shown above. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Historically, the term line was used in place of the more modern term curve. Hence the terms straight line and right line were used to distinguish what are today called lines from curved lines. For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length" (Def. 2), while a straight line is defined as "a line that lies evenly with the points on itself" (Def. 4). Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points," (Def. 3). Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line[a] is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it.
Three point geometry is categorical. Like many finite geometries, the number of provable theorems in three point geometry is small. One can prove from this collection of axioms that two distinct lines are on exactly one point and that three point geometry contains exactly three lines.Չ, չ, (tcha) The twenty fifth letter of the Armenian alphabet; seven hundred; seven hundredth. A consonant half-occlusive, voiceless, aspirate, sibilant. Չ, is a (negative) as in no and not. A negative prefix forming the negative form of verbs. Է is a verb substantive used to connect the predicate with its subject. Է signifies 'is' as in being. էակ is a noun for being, existence. էաբան is a noun for ontologist. When we add the prefix չ, as չէ, adverb it becomes is not, not a being, non-existent. In the phrases եմ, which means I am the չ, as prefix չ-եմ says I am not, non-existent. As in "no", "not", "nothing", "nothing", "un-", "is not". As in pure awareness, prior to experience or knowledge. This meaning is used especially by the Chan school of Buddhism. A negative. Caused to be nonexistent. Before reason or cause. Nonexistence; non-being. The "original non-being" from which being is produced the all. Nothing, no-thing, or no thing, is the complete absence of anything; the opposite of everything, or its complement. The concept of nothing has been a matter of philosophical debate since at least the 5th century BC.
A sphere (from Ancient Greek σφαῖρα (sphaîra) 'globe, ball') is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere.
Spherical regions or a ball:cap, bounded by one plane
sector, bounded by a conical boundary with apex at the center of the sphere
segment, bounded by a pair of parallel planes
shell, bounded by two concentric spheres of differing radii
wedge, bounded by two planes passing through a sphere center and the surface of the sphere
A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment.
A unit sphere is a sphere with unit radius (r = 1). For convenience, spheres are often taken to have their center at the origin of the coordinate system, and spheres in this article have their center at the origin unless a center is mentioned.
A great circle on the sphere has the same center and radius as the sphere, and divides it into two equal hemispheres.
Although the Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. If a particular point on a sphere is (arbitrarily) designated as its north pole, its antipodal point is called the south pole. The great circle equidistant to each is then the equator. Great circles through the poles are called lines of longitude or meridians. A line connecting the two poles may be called the axis of rotation. Small circles on the sphere that are parallel to the equator are lines of latitude. In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding.[3]
Mathematicians consider a sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space. They draw a distinction a sphere and a ball, which is a three-dimensional manifold with boundary that includes the volume contained by the sphere. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "circle" and "disk" in the plane is similar.
A vertex (plural vertices or vertexes). A Doublet of vortex. From Latin vertex (“whirl, eddy; top, crown, peak, summit”).The highest point, top or apex of something. Synonyms: acme, apex, peak, top(anatomy) The highest surface on the skull; the crown of the head.
(geometry) An angular point of a polygon, polyhedron or higher order polytope.The common point of the two rays that form an angle.
The point at which an axis meets a curve or surface.
(mathematics) A point on the curve with a local minimum or maximum of curvature.
(graph theory) One of the elements of a graph joined or not by edges to other vertices. synonym ▲coordinate term ▼Synonym: node
(computer graphics) A point in 3D space, usually given in terms of its Cartesian coordinates.
(optics) The point where the surface of a lens crosses the optical axis.
(particle physics) An interaction point.
(astrology) The point where the prime vertical meets the ecliptic in the western hemisphere of a natal chart.
(typography) A sharp downward point opposite a crotch, as in the letters "V" and "W" but not "Y". coordinate term ▼
Orthographic projection Stereographic projection
The octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection, as shown above. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Historically, the term line was used in place of the more modern term curve. Hence the terms straight line and right line were used to distinguish what are today called lines from curved lines. For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length" (Def. 2), while a straight line is defined as "a line that lies evenly with the points on itself" (Def. 4). Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points," (Def. 3). Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line[a] is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it.
Three point geometry is categorical. Like many finite geometries, the number of provable theorems in three point geometry is small. One can prove from this collection of axioms that two distinct lines are on exactly one point and that three point geometry contains exactly three lines.
The word չափ, is a noun for measure, measurement, dose, quantity, gauge, portion, moderation, proportion. The word (chapel,) չափել is a verb active it means to measure, to versify, to ponder, and it is this meaning of measure, to ponder that I wish to bring forward to elaborate on to consider, to contemplate,to ruminate. to cogitate, to evaluate and examine in detail, to give thought to all synonymous with measure.
According to the realist interpretation, measurement commits us not just to the logically independent existence of things in space and time, but also to the existence of quantitatively structured properties and relations, and to the existence of real numbers, understood as relations of ratio between specific levels of such attributes. Logical order is when all of the messages and segments within a group are in their logical sequence, next to each other, in the position determined by the physical position of the first item belonging to the group.
A logical sequence is a set of numbers, words, objects etc, following in a sequence with some sort of relation between two consecutive sets.
By a logical measure say (for a given language) we mean a syntactically defined function which associates some value with each well-formed formula of the language. Various such logical measures have played a fundamental role in the development of Logic and the Philosophy of Science.
չորս is the name of the number four. չորք is a noun also four. քառա is also a prefix for four, as in four corners or points or marks, a quaternion, quaternary is չորսնոց, adjective, noun, four, ածական, գոյական չորս. four-cornered, ածական քառանկիւն. four-eyed, ածական քառակնեայ. four-footed, ածական քառոտն. քառաբաժ, adjective divided in four, parts. քառազոյգ adjective quadrigeminal. քառագիծ, adjective for rule in squares. չորսնոց, an adjective, noun stands for four points or marks, quaternion, quaternary.
In semiotics, Khôra (also chora; Ancient Greek: χώρα) is the space that gives a place for being. The term has been used in philosophy by Plato to designate a receptacle (as a "third kind" [triton genos]; Timaeus 48e4), a space, a material substratum, or an interval. In Plato's account, chôra is described as a formless interval, alike to a non-being, in between which the "Forms" were received from the intelligible realm (where they were originally held) and were "copied", shaping into the transitory forms of the sensible realm; it "gives space" and has maternal overtones (a womb, matrix):
“Moreover, a third kind is that of the Khôra (χώρας), everlasting, not admitting destruction, granting an abode to all things having generation, itself to be apprehended with nonsensation, by a sort of bastard reckoning, hardly trustworthy; and looking toward which we dream and affirm that it is necessary that all that is be somewhere in some place and occupy some khôra; and that that which is neither on earth nor anywhere in the heaven is nothing." — Plato, Timaeus, 52a-b
"So likewise it is right that the substance which is to be fitted to receive frequently over its whole extent the copies of all things intelligible and eternal should itself, of its own nature, be void of all the forms. Wherefore, let us not speak of her that is the Mother and Receptacle of this generated world, which is perceptible by sight and all the senses, by the name of earth or air or fire or water, or any aggregates or constituents thereof: rather, if we describe her as a Kind invisible and unshaped, all-receptive, and in some most perplexing and most baffling partaking of the intelligible, we shall describe her truly." — Plato, Timaeus, 51a
Aristotle merged his teacher's concept with his definitions of prima materia (hylé), place (topos) and substratum (hypokeimenon), in the book 4 of Physics: "This is why Plato says in the Timaeus that matter and the khôra are the same; for the receptive and the khÔra are one and the same. Although the manner in which he speaks about the receptive in the Timaeus differs from that in the so-called unwritten teachings, nevertheless he declares that place and the khôra are the same".
KHORA CHORA ARMENIAN VAN?
Key authors addressing khôra include Martin Heidegger, who refers to a "clearing" in which being happens or takes place. Kitaro Nishida stated that he based his concept of basho, Place, on the abyssal nothing mu inspired by his reading of the Plato's notion of khôra.
Jacques Derrida has written a short text with the title Khôra. Jacques Derrida uses khôra to name a radical otherness that "gives place" for being, characterizing khôra is described as a formless interval, alike to a non-being, in between which the "Forms" were received from the intelligible realm (where they were originally held) and were "copied", shaping into the transitory forms of the sensible realm; it "gives space" and has maternal overtones (a womb, matrix):. For Derrida, khôra defies attempts at naming or either/or logic, which he "deconstructs". The project proposed the construction of a garden in the Parc de la Villette in Paris, which included a sieve, or harp-like structure that Derrida envisaged as a physical metaphor for the receptacle-like properties of the khôra. Derrida argues that the subjectile is the space between the sensible and the intelligible, through which everything passes but in which nothing remains. For example, an image needs to be held by something, just as a mirror will hold a reflection.
Following Derrida, John Caputo describes khôra as:
neither present nor absent, active or passive, the good nor evil, living nor nonliving - but rather atheological and nonhuman - khôra is not even a receptacle. Khôra has no meaning or essence, no identity to fall back upon. She/it receives all without becoming anything, which is why she/it can become the subject of neither a philosopheme nor mytheme. In short, the khôra is tout autre [fully other], very.
If, as one contributor concludes, "khôra" means "space", it is an interesting space that "at times appears to be neither this nor that, at times both this and that," wavering "between the logic of exclusion and that of participation." (Derrida, The Name, 89).
Julia Kristeva deploys the "khora" as part of her analysis of the difference between the semiotic and symbolic realms, as the emancipatory employment of semiotic activity as a way of evading the allegedly phallocentric character of symbolic activity (signification through language), which, following Jacques Lacan, is regarded as an inherently limiting and oppressive form of praxis. Kristeva articulates the khôra in terms of a presignifying state: "Although the khôra can be designated and regulated, it can never be definitively posited: as a result, one can situate the khôra and, if necessary, lend it a topology, but one can never give it axiomatic form."
Space is the area between and around objects. In traditional grammar, all numerals, including ordinal numerals, are grouped into a separate part of speech (Latin: nomen numerale, hence, "noun numeral" in older English grammar books). However, in modern interpretations of English grammar, ordinal numerals are usually conflated with adjectives. Ordinal numbers may be written in English with numerals and letter suffixes: 1st, 2nd, 3rd etc., with the suffix acting as an ordinal indicator.
In semiotics, Khôra (also chora; Ancient Greek: χώρα) is the space that gives a place for being. The term has been used in philosophy by Plato to designate a receptacle (as a "third kind" [triton genos]; Timaeus 48e4), a space, a material substratum, or an interval. In Plato's account, chôra is described as a formless interval, alike to a non-being, in between which the "Forms" were received from the intelligible realm (where they were originally held) and were "copied", shaping into the transitory forms of the sensible realm; it "gives space" and has maternal overtones (a womb, matrix):
“Moreover, a third kind is that of the Khôra (χώρας), everlasting, not admitting destruction, granting an abode to all things having generation, itself to be apprehended with nonsensation, by a sort of bastard reckoning, hardly trustworthy; and looking toward which we dream and affirm that it is necessary that all that is be somewhere in some place and occupy some khôra; and that that which is neither on earth nor anywhere in the heaven is nothing." — Plato, Timaeus, 52a-b
"So likewise it is right that the substance which is to be fitted to receive frequently over its whole extent the copies of all things intelligible and eternal should itself, of its own nature, be void of all the forms. Wherefore, let us not speak of her that is the Mother and Receptacle of this generated world, which is perceptible by sight and all the senses, by the name of earth or air or fire or water, or any aggregates or constituents thereof: rather, if we describe her as a Kind invisible and unshaped, all-receptive, and in some most perplexing and most baffling partaking of the intelligible, we shall describe her truly." — Plato, Timaeus, 51a
In Ancient Greek: χώρα, romanized: khṓrā is the space where something is, or any generic place.
Aristotle merged his teacher's concept with his definitions of prima materia (hylé), place (topos) and substratum (hypokeimenon), in the book 4 of Physics: "This is why Plato says in the Timaeus that matter and the khôra are the same; for the receptive and the khÔra are one and the same. Although the manner in which he speaks about the receptive in the Timaeus differs from that in the so-called unwritten teachings, nevertheless he declares that place and the khôra are the same".
KHORA CHORA ARMENIAN VAN?
Key authors addressing khôra include Martin Heidegger, who refers to a "clearing" in which being happens or takes place. Kitaro Nishida stated that he based his concept of basho, Place, on the abyssal nothing mu inspired by his reading of the Plato's notion of khôra.
Jacques Derrida has written a short text with the title Khôra. Jacques Derrida uses khôra to name a radical otherness that "gives place" for being, characterizing khôra is described as a formless interval, alike to a non-being, in between which the "Forms" were received from the intelligible realm (where they were originally held) and were "copied", shaping into the transitory forms of the sensible realm; it "gives space" and has maternal overtones (a womb, matrix):. For Derrida, khôra defies attempts at naming or either/or logic, which he "deconstructs". The project proposed the construction of a garden in the Parc de la Villette in Paris, which included a sieve, or harp-like structure that Derrida envisaged as a physical metaphor for the receptacle-like properties of the khôra. Derrida argues that the subjectile is the space between the sensible and the intelligible, through which everything passes but in which nothing remains. For example, an image needs to be held by something, just as a mirror will hold a reflection.
Following Derrida, John Caputo describes khôra as:
neither present nor absent, active or passive, the good nor evil, living nor nonliving - but rather atheological and nonhuman - khôra is not even a receptacle. Khôra has no meaning or essence, no identity to fall back upon. She/it receives all without becoming anything, which is why she/it can become the subject of neither a philosopheme nor mytheme. In short, the khôra is tout autre [fully other], very.
If, as one contributor concludes, "khôra" means "space", it is an interesting space that "at times appears to be neither this nor that, at times both this and that," wavering "between the logic of exclusion and that of participation." (Derrida, The Name, 89).
Julia Kristeva deploys the "khora" as part of her analysis of the difference between the semiotic and symbolic realms, as the emancipatory employment of semiotic activity as a way of evading the allegedly phallocentric character of symbolic activity (signification through language), which, following Jacques Lacan, is regarded as an inherently limiting and oppressive form of praxis. Kristeva articulates the khôra in terms of a presignifying state: "Although the khôra can be designated and regulated, it can never be definitively posited: as a result, one can situate the khôra and, if necessary, lend it a topology, but one can never give it axiomatic form."
Natural numbers are a part of the number system, including all the positive numbers from 1 to infinity. Natural numbers are also known as counting numbers because they do not include zero or negative numbers.
A spatial reference system (SRS) or coordinate reference system (CRS) is a framework used to precisely measure locations on the surface of Earth as coordinates. It is thus the application of the abstract mathematics of coordinate systems and analytic geometry to geographic space. Thousands of coordinate systems have been specified for use around the world or in specific regions and for various purposes, necessitating transformations between different SRS.
To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. For example, although a two-dimensional map is a useful tool for navigating from one place to another, in some cases the topography of the land is important. If for example the route go through the mountains. To appreciate fully the impact of these geographic features, you must use three dimensions. A natural extension of the two-dimensional Cartesian coordinate plane into three dimensions.
In a three-dimensional space, a plane can and is defined by three points, as long as those points are not on the same line. Our motion in three dimensional space incorporates all plus and minus X, Y and Z axes. Motion of birds flying in the sky is also an example of three dimensional motion.
In mathematics today, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighborhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.
A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.
Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called point-set topology or general topology.
To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. For example, although a two-dimensional map is a useful tool for navigating from one place to another, in some cases the topography of the land is important. If for example the route go through the mountains. To appreciate fully the impact of these geographic features, you must use three dimensions. A natural extension of the two-dimensional Cartesian coordinate plane into three dimensions.
In a three-dimensional space, a plane can and is defined by three points, as long as those points are not on the same line. Our motion in three dimensional space incorporates all plus and minus X, Y and Z axes. Motion of birds flying in the sky is also an example of three dimensional motion.
In mathematics today, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighborhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.
A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.
Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called point-set topology or general topology.
Space is the area between and around objects. In traditional grammar, all numerals, including ordinal numerals, are grouped into a separate part of speech (Latin: nomen numerale, hence, "noun numeral" in older English grammar books). However, in modern interpretations of English grammar, ordinal numerals are usually conflated with adjectives. Ordinal numbers may be written in English with numerals and letter suffixes: 1st, 2nd, 3rd etc., with the suffix acting as an ordinal indicator.
Written dates often omit the suffix, although it is nevertheless pronounced. In other languages, different ordinal indicators are used to write ordinal numbers.
In American Sign Language, the ordinal numbers first through ninth are formed with hand-shapes similar to those for the corresponding cardinal numbers with the addition of a small twist of the wrist.
The Babylonian numeral system, base 60, was the first positional system to be developed, and its influence is present today in the way time and angles are counted in tallies related to 60, such as 60 minutes in an hour and 360 degrees in a circle.
There are 3 axis that we use to orient ourselves in space forward-backwards, left-right and up-down. That technically gives 6 directions.In linguistics, ordinal numerals or ordinal number words are words representing position or rank in a sequential order; the order may be of size, importance, chronology, and so on.
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two or three dimensional plane that are both perpendicular to a third line (in the same plane or not). As for carnality, say if X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), and generalizing this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.
In classical Euclidean geometry, a point is a primitive notion that models an exact location in space, but has no length, width, or thickness. In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. Being a primitive notion means that a point is defined only by some properties, called axioms, that it must satisfy, like for example "there is exactly one line that passes through two different points".
In classical philosophy an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a starting point for reasoning. The axioms provide the foundational statements of truth, from which all else in the theory derives and primitive notions provide the language of the theory, The "primitive notions" and "axioms" are building blocks for a mathematical theory, using the tools of logical reasoning to prove propositions/theorems.
In the philosophy of language, as well as linguistics, logic, mathematics, semantics, semiotics an intention is any property or quality connected by a syllable, word, phrase, or another symbol. In the case of a word, the definition almost always implies an intention.
Swiss linguist Ferdinand de Saussure(1857–1913) contrasts three concepts as follows, one the signifier – the "sound image" or the string of letters on a page that one recognizes as the form of a sign, the signified, Second the meaning, the concept or idea that a sign expresses or evokes, the referent. third the actual thing or set of things a sign refers to. Accumulation point cluster point or limit point of a sequence?
Without an intention of some sort, a word has no meaning.
An intentional definition gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used. In the case of nouns, this is equivalent to specifying the properties that an object needs to have in order to be counted as a referent of the term.
In classical Euclidean geometry, a point is a primitive notion that models an exact location in space, but has no length, width, or thickness. In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. Being a primitive notion means that a point is defined only by some properties, called axioms, that it must satisfy, like for example "there is exactly one line that passes through two different points".
In classical philosophy an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a starting point for reasoning. The axioms provide the foundational statements of truth, from which all else in the theory derives and primitive notions provide the language of the theory, The "primitive notions" and "axioms" are building blocks for a mathematical theory, using the tools of logical reasoning to prove propositions/theorems.
In the philosophy of language, as well as linguistics, logic, mathematics, semantics, semiotics an intention is any property or quality connected by a syllable, word, phrase, or another symbol. In the case of a word, the definition almost always implies an intention.
Swiss linguist Ferdinand de Saussure(1857–1913) contrasts three concepts as follows, one the signifier – the "sound image" or the string of letters on a page that one recognizes as the form of a sign, the signified, Second the meaning, the concept or idea that a sign expresses or evokes, the referent. third the actual thing or set of things a sign refers to. Accumulation point cluster point or limit point of a sequence?
Without an intention of some sort, a word has no meaning.
An intentional definition gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used. In the case of nouns, this is equivalent to specifying the properties that an object needs to have in order to be counted as a referent of the term.
In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
Defining a set's intension or defining sets by properties is also known as set comprehension, set abstraction.
Defining a set's intension or defining sets by properties is also known as set comprehension, set abstraction.
NINE has the quality of being infinite, for the state or quality of being infinite for example as in “the infinity of space”, or as the synonyms of infinite offer, endlessness, boundlessness, limitlessness.
In Mathematics the infinity, symbol /∞/ is recent and is defined as a number greater than any assignable quantity or countable number it just happens to be a version of the symbol representing the number 8. If infinity of old was indeed defined by the number 9 it makes sense, since no matter what the number 9 is added to, the result is equal to the same number it was/is added to and also no matter what 9 is multiplied by it gives the result of itself. This phenomenon happens infinitely as well.
So if the last number in a base 10 system could be consider as infinite, this could bring a new definition to infinity, instead of being defined as limitless, infinity could be defined as forever returning to base. A little like the earth revolves around the sun and itself.
Earth revolving around itself would be multiplying number 9, while revolving around the sun would be making additions to number 9 but yet always going back to the base.
If we decide to divide 9, we get the best of both worlds. Meaning the results we were getting while doing additions for some numbers as well as the ones we get while multiplying number 9.
In Mathematics the infinity, symbol /∞/ is recent and is defined as a number greater than any assignable quantity or countable number it just happens to be a version of the symbol representing the number 8. If infinity of old was indeed defined by the number 9 it makes sense, since no matter what the number 9 is added to, the result is equal to the same number it was/is added to and also no matter what 9 is multiplied by it gives the result of itself. This phenomenon happens infinitely as well.
So if the last number in a base 10 system could be consider as infinite, this could bring a new definition to infinity, instead of being defined as limitless, infinity could be defined as forever returning to base. A little like the earth revolves around the sun and itself.
Earth revolving around itself would be multiplying number 9, while revolving around the sun would be making additions to number 9 but yet always going back to the base.
If we decide to divide 9, we get the best of both worlds. Meaning the results we were getting while doing additions for some numbers as well as the ones we get while multiplying number 9.
So if infinity is defined as coming back on itself or to whatever is added to it, that means that it had to be an infinite beginning. It had to be a number 9. It also means that everything either always goes back to its original value or if multiplied by infinity becomes infinity.
So one can say 9 = ∞.
So one can say 9 = ∞.
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