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A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection f from the space onto itself that multiplies all distances by the same positive real number r, so that for any two points x and y we have ((), ()) = (,), where d(x,y) is the Euclidean distance from x to y. [16]
In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency. [1]
Figure 1: The point O is an external homothetic center for the two triangles. The size of each figure is proportional to its distance from the homothetic center. In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another.
A spiral similarity taking triangle ABC to triangle A'B'C'. Spiral similarity is a plane transformation in mathematics composed of a rotation and a dilation. [1] It is used widely in Euclidean geometry to facilitate the proofs of many theorems and other results in geometry, especially in mathematical competitions and olympiads.
Usually, a text based on natural numbers (for example, the article "prime number") does not specify the used definition of natural numbers. Likewise, a text based on topological spaces (for example, the article "homotopy", or "inductive dimension") does not specify the used definition of a topological space. Thus, it is possible (and rather ...
A similarity system of triangles is a specific configuration involving a set of triangles. [1] A set of triangles is considered a configuration when all of the triangles share a minimum of one incidence relation with one of the other triangles present in the set. [1]
Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. [2] Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to ...
The same term can also be used more informally to refer to something "standard" or "classic". For example, one might say that Euclid's proof is the "canonical proof" of the infinitude of primes. There are two canonical proofs that are always used to show non-mathematicians what a mathematical proof is like:
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