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- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/00%3A_Front_Matter/04%3A_AcknowledgementsI would like to thank Rob Beezer, David Farmer, and all my colleagues at the UTMOST \(2017\) Textbook Workshop and in the PreTeXt Community for helping me convert my dusty tex code to a workable PreTe...I would like to thank Rob Beezer, David Farmer, and all my colleagues at the UTMOST \(2017\) Textbook Workshop and in the PreTeXt Community for helping me convert my dusty tex code to a workable PreTeXt document in order to make the text freely available online, both as a webpage and as a printable document.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/03%3A_Transformations/3.03%3A_The_Extended_PlaneConsider again inversion about the circle C given by |z−z_0|=r, and observe that points close to z_0 get mapped to points in the plane far away from z_0. In fact, a sequence of points in compl...Consider again inversion about the circle C given by |z−z_0|=r, and observe that points close to z_0 get mapped to points in the plane far away from z_0. In fact, a sequence of points in complex numbers whose limit is z_0 will be inverted to a sequence of points whose magnitudes go to ∞. Conversely, any sequence of points in complex numbers having magnitudes marching off to ∞ will be inverted to a sequence of points whose limit is z_0.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/03%3A_Transformations/3.02%3A_InversionInversion offers a way to reflect points across a circle. This transformation plays a central role in visualizing the transformations of non-Euclidean geometry, and this section is the foundation of m...Inversion offers a way to reflect points across a circle. This transformation plays a central role in visualizing the transformations of non-Euclidean geometry, and this section is the foundation of much of what follows.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/04%3A_GeometryWhereas Euclid's approach to geometry was additive (he started with basic definitions and axioms and proceeded to build a sequence of results depending on previous ones), Klein's approach was subtract...Whereas Euclid's approach to geometry was additive (he started with basic definitions and axioms and proceeded to build a sequence of results depending on previous ones), Klein's approach was subtractive. Klein's approach to geometry, called the Erlangen Program after the university at which he worked at the time, has the benefit that all three geometries (Euclidean, hyperbolic and elliptic) emerge as special cases from a general space and a general set of transformations.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/00%3A_Front_Matter
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/06%3A_Elliptic_Geometry/6.02%3A_Elliptic_GeometryAs was the case in hyperbolic geometry, the space in elliptic geometry is derived from C+, and the group of transformations consists of certain Möbius transformations.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/00%3A_Front_Matter/03%3A_PrefaceFinally, the essays in Chapter 8 on cosmic topology and our understanding of the universe have been updated to include research done since the original publication of this text, some of which is due t...Finally, the essays in Chapter 8 on cosmic topology and our understanding of the universe have been updated to include research done since the original publication of this text, some of which is due to sharper measurements of the temperature of the cosmic microwave background radiation obtained with the launch of the Planck satellite in \(2009\).
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/08%3A_Cosmic_TopologyCosmic topology can be described as the effort to determine the shape of our universe through observational techniques. In this chapter we discuss two programs of research in cosmic topology: the cosm...Cosmic topology can be described as the effort to determine the shape of our universe through observational techniques. In this chapter we discuss two programs of research in cosmic topology: the cosmic crystallography method and the circles-in-the-sky method. The chapter begins with a discussion of three-dimensional geometry and some \(3\)-manifolds that have been given consideration as models for the shape of our universe.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/02%3A_The_Complex_Plane/2.01%3A_Basic_NotionsThe set of complex numbers is obtained algebraically by adjoining the number i to the set R of real numbers, where i is defined by the property that i^2=−1. We will take a geometric approach and defin...The set of complex numbers is obtained algebraically by adjoining the number i to the set R of real numbers, where i is defined by the property that i^2=−1. We will take a geometric approach and define a complex number to be an ordered pair (x,y) of real numbers.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/02%3A_The_Complex_Plane/2.04%3A_Complex_ExpressionsIn this section we look at some equations and inequalities that will come up throughout the text.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/07%3A_Geometry_on_Surfaces/7.06%3A_Geometry_of_SurfacesIf you've got a surface in your hand, you can find a homeomorphic version of the surface on which to construct hyperbolic geometry, elliptic geometry, or Euclidean geometry. And the choice of geometry...If you've got a surface in your hand, you can find a homeomorphic version of the surface on which to construct hyperbolic geometry, elliptic geometry, or Euclidean geometry. And the choice of geometry is unique: No surface admits more than one of these geometries. As we shall see, of the infinitely many surfaces, all but four admit hyperbolic geometry (two admit Euclidean geometry and two admit elliptic geometry).
