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In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space.
These attitudes are specified with two angles. For a line, these angles are called the trend and the plunge. The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane. [15] For a plane, the two angles are called its strike (angle) and its dip (angle).
A direct proof using classical geometry was developed by James Mercer in 1923. [2] This solution involves drawing one additional line, and then making repeated use of the fact that the internal angles of a triangle add up to 180° to prove that several triangles drawn within the large triangle are all isosceles.
The fourth angle of a Lambert quadrilateral is an obtuse angle in elliptic geometry. The summit angles of a Saccheri quadrilateral are obtuse in elliptic geometry. The sum of the measures of the angles of any triangle is greater than 180° if the geometry is elliptic. That is, the defect of a triangle is negative. [80]
This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined.
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements , it was the three-dimensional space of Euclidean geometry , but in modern mathematics there are Euclidean spaces of any positive integer dimension n , which are called Euclidean n -spaces when one wants to specify their ...