Non-Euclidean Geometry
These traits give hyperbolic geometry a wide variety of real world applications. It is used as a framework to study complex networks (a branch of network theory, network theory being a branch of graph theory, where nodes and edges have attributes and names). 12 The heterogeneous (diverse in terms of dimension or degrees) nature of complex networks is a reflection of the negative curvature of hyperbolic space. This correlation allows hyperbolic geometry to be used inmodels of complex networks such as national power grids, airline networks and protein-protein interactions. 13 Another important application of hyperbolic geometry is in Riemann’s uniformization theorem. 14 The uniformization theorem states: ‘ every simply connected Riemann surface is conformally equivalent to either the unit disk, the complex plane or the Riemann sphere ’ . 15 Riemann surfaces are defined as one-
dimensional complex manifolds, where manifolds are collections of points which make up a certain kind of set. 16 As these manifolds are complex, they chart to the open unit disk whichmakes up the set of points used in Poincaré ’s diskmodel of hyperbolic geometry. The functions that map points to the disk are described as ‘ holomorphic ’ . This means that it is a function of complex variables that is complex differentiable (meaning that it satisfies the Cauchy-Riemann series of differential equations) in the neighbourhood of each point. The last two conditions have the important implication that holomorphic equations are equal to their own taylor series (an expression of functions as the summation of infinitely long polynomial equations). 17
Objects of varying genus (g)
All Riemann Surfaces of genus greater than one are equivalent to the unit disk and therefore abide by the axioms of hyperbolic geometry. The genus of a surface correlates to how many holes it has. This makes hyperbolic geometry an essential tool in the analysis of Riemann surfaces. 18
So far, this essay has only grappled with one branch of non-Euclidean geometry: hyperbolic geometry. However there is another, vastly older, branch called spherical geometry (sometimes called elliptic geometry despite there being a subtle difference between the two; in spherical geometry lines (great circles) intersect in two points whereas in elliptic geometry these two points are opposite and therefore represent the same point). True to its name this geometry occurs on the flat surface of a sphere and as such has a constant positive curvature (in contrast to the negative curvature of hyperbolic geometry) and in it Euclid’s fifth postulate also doesn’t hold, albeit for different reasons. 19
Spherical geometry has been referenced many times throughout antiquity by mathematicians such as Autolycus of Pitane (whose work, ‘On the moving sphere’ is considered the oldest surviving Greek
12 Vincent 2011, NKS 2011, Retvari 2011, McLaury 2015; see also Krioukov 2010. 13 Small, Hou and Zhang 2014. 14 Vincent 2011, NKS 2011, Retvari 2011, McLaury 2015. 15 Uniformization theorem (Wikipedia). 16 Manifold (Wikipedia ). 17 Kailasa, Lai and Khim (unknown). 18 Vincent 2011, NKS 2011, Retvari 2011, McLaury 2015; see also Lee 2013. 19 Gowers 2002.
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