Worksheet on Hyperbolic Geometry

... The program KaleidoTile can be found at the website _______________________. Since triangles are ‘thin’ in the hyperbolic geometry setting the symmetry to _____ will put you into the tiling. For a binary tree (part of the hyperbolic plane) each parent has ____ daughters. This means the number of des ...

... The program KaleidoTile can be found at the website _______________________. Since triangles are ‘thin’ in the hyperbolic geometry setting the symmetry to _____ will put you into the tiling. For a binary tree (part of the hyperbolic plane) each parent has ____ daughters. This means the number of des ...

Introduction to Teichmüller Spaces

... Definition 3.1. A marked Riemann surface (X, f ) is a Riemann surface X together with a homemorphism f : S → X. Two marked surfaces (X, f ) ∼ (Y, g) are equivalent if gf −1 : X → Y is isotopic to an isomorphism. Definition 3.2. We define the Teichmüler Space Tg = {(X, f )}/ ∼ For g ≥ 2, Tg is also ...

... Definition 3.1. A marked Riemann surface (X, f ) is a Riemann surface X together with a homemorphism f : S → X. Two marked surfaces (X, f ) ∼ (Y, g) are equivalent if gf −1 : X → Y is isotopic to an isomorphism. Definition 3.2. We define the Teichmüler Space Tg = {(X, f )}/ ∼ For g ≥ 2, Tg is also ...

5 The hyperbolic plane

... As we see above, the analogy between Euclidean geometry and its theorems and the geometry of the hyperbolic plane is very close, so long as we replace lines by geodesics, and Euclidean isometries (translations, rotations and reflections) by the isometries of H or D. In fact it played an important hi ...

... As we see above, the analogy between Euclidean geometry and its theorems and the geometry of the hyperbolic plane is very close, so long as we replace lines by geodesics, and Euclidean isometries (translations, rotations and reflections) by the isometries of H or D. In fact it played an important hi ...

Exercises - Durham University

... isometries fix two points of the absolute? Classify the isometries fixing 0 and ∞ in the upper half-plane model. 14.11. (a) Show that the group of isometries of hyperbolic plane is generated by reflections. (b) How many reflections do you need to map a triangle ABC to a congruent triangle A0 B 0 C 0 ...

... isometries fix two points of the absolute? Classify the isometries fixing 0 and ∞ in the upper half-plane model. 14.11. (a) Show that the group of isometries of hyperbolic plane is generated by reflections. (b) How many reflections do you need to map a triangle ABC to a congruent triangle A0 B 0 C 0 ...

WHAT IS HYPERBOLIC GEOMETRY? - School of Mathematics, TIFR

... It is the fifth postulate that is equivalent to the parallel postulate stated above. The reason why people tried to prove it from the rest of the axioms is that they thought it was not ‘sufficiently self-evident’ to be given the status of an axiom, and an ‘axiom’ in Euclid’s times was a ‘self-eviden ...

... It is the fifth postulate that is equivalent to the parallel postulate stated above. The reason why people tried to prove it from the rest of the axioms is that they thought it was not ‘sufficiently self-evident’ to be given the status of an axiom, and an ‘axiom’ in Euclid’s times was a ‘self-eviden ...

A Formula for the Intersection Angle of Backbone Arcs with the

... Outline of the Calculation 1. Calculate the Weierstrass coordinates of the points P and Q. 2. Find the coordinates of ℓ from those of P and Q. 3. Use the coordinates of ℓ to compute the matrix of the reflection across ℓ. 4. Reflect O across ℓ to obtain the Weierstrass coordinates of R, and thus the ...

... Outline of the Calculation 1. Calculate the Weierstrass coordinates of the points P and Q. 2. Find the coordinates of ℓ from those of P and Q. 3. Use the coordinates of ℓ to compute the matrix of the reflection across ℓ. 4. Reflect O across ℓ to obtain the Weierstrass coordinates of R, and thus the ...

Pairs of Pants and Congruence Laws of Geometry - Rose

... What are geometric surfaces and why study them? ...

... What are geometric surfaces and why study them? ...

Lengths of simple loops on surfaces with hyperbolic metrics Geometry & Topology G

... 1.1 Given a compact orientable surface of negative Euler characteristic, there exists a natural length pairing between the Teichmüller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or ...

... 1.1 Given a compact orientable surface of negative Euler characteristic, there exists a natural length pairing between the Teichmüller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or ...

Hyperbolic Geometry and 3-Manifold Topology

... surface, such that E is identiﬁed with i(S × (0, ∞)). Exercise 1.16. Let M be an open irreducible 3-manifold with ﬁnitely generated fundamental group. The end E of M is tame if and only if there exists a core C of M such that the closure of the component Z of M \C containing E is homeomorphic to S × ...

... surface, such that E is identiﬁed with i(S × (0, ∞)). Exercise 1.16. Let M be an open irreducible 3-manifold with ﬁnitely generated fundamental group. The end E of M is tame if and only if there exists a core C of M such that the closure of the component Z of M \C containing E is homeomorphic to S × ...

Pants decompositions of random surfaces

... construct it by gluing some hyperbolic pairs of pants with total boundary length ≤ L. A hyperbolic pair of pants is determined by its boundary lengths, so the number (really, Weil-Petersson volume) of possible surfaces with total pants length ≤ L is governed by the number of possible ways of choosin ...

... construct it by gluing some hyperbolic pairs of pants with total boundary length ≤ L. A hyperbolic pair of pants is determined by its boundary lengths, so the number (really, Weil-Petersson volume) of possible surfaces with total pants length ≤ L is governed by the number of possible ways of choosin ...