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### Hyperbolic Geometry And Amplituhedra In 1+ 2 Dimensions

Recently, the existence of an Amplituhedron for tree level amplitudes in the
bi-adjoint scalar field theory has been proved by Arkhani-Hamed et al. We argue
that hyperbolic geometry constitutes a natural framework to address the study
of positive geometries in moduli spaces of Riemann surfaces, and thus to try to
extend this achievement beyond tree level. In this paper we begin an
exploration of these ideas starting from the simplest example of hyperbolic
geometry, the hyperbolic plane. The hyperboloid model naturally guides us to
re-discover the moduli space Associahedron, and a new version of its
kinematical avatar. As a by-product we obtain a solution to the scattering
equations which can be interpreted as a special case of the two well known
solutions in terms of spinor-helicity formalism. The construction is done in
$1+2$ dimensions and this makes harder to understand how to extract the
amplitude from the dlog of the space time Associahedron. Nevertheless, we
continue the investigation accommodating a lo Mar 15, 2018 ... We argue that hyperbolic geometry constitutes a natural framework to address the study of positive geometries in moduli spaces of Riemann ...
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### Hyperbolic Geometry - Wikipedia

Hyperbolic plane geometry is also the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. A modern ...
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### Hyperbolic Geometry

of non-Euclidean geometry called hyperbolic geometry. Recall that one ... We will lead into hyperbolic geometry by considering the Saccheri quadrilateral:.
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### Hyperbolic Geometry Of Complex Networks

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting Sep 9, 2010 ... We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these ...
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### Hyperbolic Geometry, Section 5

Hyperbolic geometry is the geometry you get by assuming all the postulates of Euclid, except the fifth one, which is replaced by its negation.
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### String Theory And The History Of Non-Euclidean Geometry - Dummies

Before string theory introduced the concept of extra dimensions, the fascination with strange warping of space in the 1800s was perhaps nowhere as clear as in the creation of non-Euclidean geometry, where mathematicians began to explore new types of geometry that weren’t based on the rules laid out 2,000 years earlier by Euclid. One version of non-Euclidean geometry is Riemannian geometry, but there are others, such as projective geometry.
The reason for the creation of non-Euclidean geometry is based in Euclid’s Elements itself, in his “fifth postulate,” which was much more complex than the first four postulates. The fifth postulate is sometimes called the parallel postulate and, though it’s worded fairly technically, one consequence is important for string theory’s purposes: A pair of parallel lines never intersects.
Well, that’s all well and good on a flat surface, but on a sphere, for example, two parallel lines can and do intersect. Lines of long Mar 26, 2016 ... One version of non-Euclidean geometry is Riemannian geometry, but there are others, such as projective geometry. The reason for the creation ...
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### DIY Hyperbolic Geometry

Which are duals of each other? Which shapes tile the hyperbolic plane? i) Triangles and regular tilings. Take a geodesic triangle with angles π/ ...
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### Hyperbolic Geometry | Mathematics | Britannica

hyperbolic geometry, also called Lobachevskian Geometry, a non-Euclidean geometry that rejects the validity of Euclid’s fifth, the “parallel,” postulate. Simply stated, this Euclidean postulate is: through a point not on a given line there is exactly one line parallel to the given line. In hyperbolic geometry, through a point not on a given line there are at least two lines parallel to the given line. The tenets of hyperbolic geometry, however, admit the other four Euclidean postulates. Although many of the theorems of hyperbolic geometry are identical to those of Euclidean, others differ. In Euclidean geometry, for example, two parallel hyperbolic geometry, also called Lobachevskian Geometry, a non-Euclidean geometry that rejects the validity of Euclid's fifth, the “parallel,” postulate.
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### The Geometric Viewpoint | History Of Hyperbolic Geometry

Dec 8, 2016 ... History of Hyperbolic Geometry · 1. A straight line segment can be drawn joining any two points. · 2. Any straight line segment can be extended ...
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### Hyperbolic Geometry -- From Wolfram MathWorld

A non-Euclidean geometry, also called Lobachevsky-Bolyai-Gauss geometry, having constant sectional curvature -1. This geometry satisfies all of Euclid's postulates except the parallel postulate, which is modified to read: For any infinite straight line L and any point P not on it, there are many other infinitely extending straight lines that pass through P and which do not intersect L. In hyperbolic geometry, the sum of angles of a triangle is less than 180 degrees, and triangles with the... There are no similar triangles in hyperbolic geometry. The best-known example of a hyperbolic space are spheres in Lorentzian four-space.
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### Hyperbolic Geometry - Simple English Wikipedia, The Free ...

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate that defines Euclidean geometry isn't true.
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