Circles, squares and lines

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Circles, squares and lines

• A brief look at hyperbolic geometry

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The parallel postulate

• One of Euclids axioms for geometry
• It states that
• Given a line l, and a point P not on l, there is
  a unique line m through P which does not meet l
• Not satisfied in hyperbolic geometry

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Riemannian metrics

• A metric on a set S is a function

• satisfying three conditions
• Positivity
• Symmetry
• Triangle inequality
• A Riemannian metric on a manifold M is an inner
  product lt , gt on the tangent bundle, TM.
• In practice, given a parametrized curve ?(z), we
  can assign a speed to it at every point. We can
  then calculate its length

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• In hyperbolic space we will express this inner
  product as a matrix at each point

(where v and w are tangent vectors at the point x)
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Models of hyperbolic space

• We will use the upper half space and Poincaré
  disc models of hyperbolic space. We will give
  the space, the set of lines and isometries, and
  from this derive the Riemannian metric.

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The hyperbolic metric

• We then ask that inversion in each of these
  hyperbolic lines be isometries.
• Given a circle C with centre O, radius r,
  inversion in C is the map taking any point P to
  the unique point iC(P) on the radial line through
  O and P with

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• Now consider the metric, A(x,y).
• Given two tangent vectors the same point, we can
  map one to the other by an inversion. Hence A
  must have the form
• A(x,y) ?(x,y) .Id
• Since we have the horizontal translations among
  our isometries, this must be independent of x.
• But considering the dilations

We can see the three red segments must have the
same length. Taking the limit as the angled line
approach the vertical, we can see the metric must
be proportional to 1/r. Hence
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Isometries of hyperbolic space

• Let be the set of Mobius maps on
  the Riemann sphere
• and the subset consisting of the maps which
  preserve U2
• is uniquely triply transitive given
  distinct points (z1, z2, z3) and distinct points
  (w1, w2, w3) there is a unique such that
• By considering the images of 0, 1 and ? we can
  show that any Mobius map which preserves the real
  line can be expressed with real coefficients.
• Conversely, any Mobius map with real coefficients
  satisfying
• will preserve the upper half space, so

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• Considering the fixed points, gives us a
  quadratic in z
• Considering the discriminant, ? of this quadratic
  allows us to classify the hyperbolic isometries
• ? gt 0 - Hyperbolic
• Then we have two real solutions. If we assume
  these are 0 and ?, this has the form
• So this is a dilation of hyperbolic space

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• ? 0 - Parabolic
• Now we have one real solution. If we assume this
  solution is ?? then this has the form
• So this is a translation

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• ? lt 0 - Elliptic
• In this case we have a pair of conjugate
  imaginary solutions, one of which is inside U2.
  If we assume these points are i and i, we can
  put this into the form
• So this is a rotation

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Hyperbolic tessellations

• A subgroup of the isometry group is called
  Fuchsian if no sequences of isometries in the
  group approach the identity.
• For such a group, ?, a fundamental region is a
  closed set F whose translates under ? cover
  hyperbolic space, but their interiors never meet.
• Any Fuchsian group has at least one fundamental
  region fix and take
• The translates then give a tessellation of
  hyperbolic space

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You can achieve similar results using two or more
starting polygons
You could also add patterns to your polygons
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• Thanks to Dr. Young Eun Choi, University of
  Warwick.

The end.