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Geometry. In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses. The idealized ruler, known as a straightedge, is assumed ...
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had ...
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number called its ratio, which sends point to a point by the rule [1] X {\displaystyle {\overrightarrow {SX'}}=k {\overrightarrow {SX}}} for a fixed number. ≠ {\displaystyle k\neq 0} .
In geometry, the parallel postulate, also called Euclid 's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right ...
Convexity is a geometric property with a variety of applications in economics. [1] Informally, an economic phenomenon is convex when "intermediates (or combinations) are better than extremes". For example, an economic agent with convex preferences prefers combinations of goods over having a lot of any one sort of good; this represents a kind of ...
He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat ...
Euclidean group. In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space ; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E (n) or ...
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms. [1] These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model ...