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Given a point A 0 in a Euclidean space and a translation S, define the point A i to be the point obtained from i applications of the translation S to A 0, so A i = S i (A 0).The set of vertices A i with i any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. S. M. Coxeter.
A blue horocycle in the Poincaré disk model and some red normals. The normals converge asymptotically to the upper central ideal point.. In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant curvature where all the perpendicular geodesics through a point on a horocycle are limiting parallel, and ...
For example, in Circle Limit III every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this, we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth.
All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon . Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint.
That the algebra of the real numbers can be employed to yield results about the ... The equation x 2 + y 2 = r 2 is the equation for any circle centered at the origin ...
Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary.
The most famous of these problems, squaring the circle, otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass. Squaring the circle has been proved impossible, as it involves generating a transcendental number, that is, √ π.
Free algebra. Clifford algebra • Geometric algebra Operator algebra. In mathematics, a direct limit is a way to construct a (typically large) object from many ...