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This equivalence can be extended to the negative real numbers by noting () (+) = and taking the limit as n goes to infinity. Characterization 1 ⇔ characterization 3 [ edit ]
Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment , with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question.
Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum.
The apeirogonal hosohedron is the arithmetic limit of the family of hosohedra {2,p}, as p tends to infinity, thereby turning the hosohedron into a Euclidean tiling.All the vertices have then receded to infinity and the digonal faces are no longer defined by closed circuits of finite edges.
The apeirogonal tiling is the arithmetic limit of the family of prisms t{2, p} or p.4.4, as p tends to infinity, thereby turning the prism into a Euclidean tiling.. An alternation operation can create an apeirogonal antiprism composed of three triangles and one apeirogon at each vertex.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram, progressing to infinity.
The apeirogonal antiprism is the arithmetic limit of the family of antiprisms sr{2, p} or p.3.3.3, as p tends to infinity, thereby turning the antiprism into a Euclidean tiling. The apeirogonal antiprism can be constructed by applying an alternation operation to an apeirogonal prism .