Visualizing Hyperbolic Geometry

... These alternatives to Playfair’s Axiom might seem bizarre, but if you’ve ever flown on a plane, you’re familiar with one of them already. ...

... These alternatives to Playfair’s Axiom might seem bizarre, but if you’ve ever flown on a plane, you’re familiar with one of them already. ...

What is Hyperbolic Geometry? - School of Mathematics, TIFR

... Given a straight line L in a plane P and a point x on the plane P lying outside the line L, there exists a unique straight line L0 lying on P passing through x and parallel to L. Problem Prove the Parallel Postulate from the other axioms of Euclidean geometry. ...

... Given a straight line L in a plane P and a point x on the plane P lying outside the line L, there exists a unique straight line L0 lying on P passing through x and parallel to L. Problem Prove the Parallel Postulate from the other axioms of Euclidean geometry. ...

What is Hyperbolic Geometry?

... Given a straight line L in a plane P and a point x on the plane P lying outside the line L, there exists a unique straight line L0 lying on P passing through x and parallel to L. Problem Prove the Parallel Postulate from the other axioms of Euclidean geometry. ...

... Given a straight line L in a plane P and a point x on the plane P lying outside the line L, there exists a unique straight line L0 lying on P passing through x and parallel to L. Problem Prove the Parallel Postulate from the other axioms of Euclidean geometry. ...

Pants decompositions of random surfaces

... The proof of Theorem 2 is closely parallel to the proof of Theorem 1. In some ways, the proof is not as clean, but on the other hand the proof is completely elementary and self-contained. In a similar spirit, Brooks and Makover suggested that for large N , the counting measure on the combinatorial m ...

... The proof of Theorem 2 is closely parallel to the proof of Theorem 1. In some ways, the proof is not as clean, but on the other hand the proof is completely elementary and self-contained. In a similar spirit, Brooks and Makover suggested that for large N , the counting measure on the combinatorial m ...

Tangent circles in the hyperbolic disk - Rose

... Among the axioms underlying Euclidean geometry, there are five that get special attention. Other geometries use some of these same axioms, but they may not use all of them. Axioms are statements that are universally accepted as true without requiring a proof. Hyperbolic geometry follows the first fo ...

... Among the axioms underlying Euclidean geometry, there are five that get special attention. Other geometries use some of these same axioms, but they may not use all of them. Axioms are statements that are universally accepted as true without requiring a proof. Hyperbolic geometry follows the first fo ...

On Klein`s So-called Non

... For a beginner, projective geometry is, compared to the Euclidean, a mysterious geometry. There are several reasons for that. First of all, the nonnecessity of any notion of distance or of length may be misleading (What do we measure in this geo-metry?) Secondly, the Euclidean coordinates are replac ...

... For a beginner, projective geometry is, compared to the Euclidean, a mysterious geometry. There are several reasons for that. First of all, the nonnecessity of any notion of distance or of length may be misleading (What do we measure in this geo-metry?) Secondly, the Euclidean coordinates are replac ...

THE GEOMETRIES OF 3

... lines of these foliations are permuted by the isometries of S2 x U and H2 x U. It follows that any 3-manifold with a geometric structure modelled on S2 x U or H2 x U is foliated by lines or circles. If the manifold is compact, it is usually true that the foliation is by circles. For some of the othe ...

... lines of these foliations are permuted by the isometries of S2 x U and H2 x U. It follows that any 3-manifold with a geometric structure modelled on S2 x U or H2 x U is foliated by lines or circles. If the manifold is compact, it is usually true that the foliation is by circles. For some of the othe ...

kucukarslan et al.

... fuzzification of different classical mathematical structures and to study properties of these fuzzy objects. Many scientists have studied fuzzy sets theory by different aspects for many years. Also, the concepts of fuzzy set, fuzzy logic, fuzzy number, fuzzy topology and fuzzy geometry were studied. ...

... fuzzification of different classical mathematical structures and to study properties of these fuzzy objects. Many scientists have studied fuzzy sets theory by different aspects for many years. Also, the concepts of fuzzy set, fuzzy logic, fuzzy number, fuzzy topology and fuzzy geometry were studied. ...

fractal geometry : an introduction

... self-similar, meaning thereby that a fractal is exactly or approximately similar to a part of itself. Self-similarity fractal is based on some equation which undergoes iteration. The length of a coastline measured with different length rulers may be an example of fractal. The shorter the ruler, the ...

... self-similar, meaning thereby that a fractal is exactly or approximately similar to a part of itself. Self-similarity fractal is based on some equation which undergoes iteration. The length of a coastline measured with different length rulers may be an example of fractal. The shorter the ruler, the ...

Slides for Nov. 12, 2014, lecture

... imposes additional structure; unboundedness is an extension relation, because it is qualitative and not quantitative, as opposed to infinitude, which is a measure relation because it is quantitative; (for cognoscenti: the extension relations are the differential structure and topology of a manifold) ...

... imposes additional structure; unboundedness is an extension relation, because it is qualitative and not quantitative, as opposed to infinitude, which is a measure relation because it is quantitative; (for cognoscenti: the extension relations are the differential structure and topology of a manifold) ...

Compact hyperbolic tetrahedra with non

... Lemma 2 Suppose that three planes Pv1 , Pv2 , Pv3 intersect pairwise in H3 with non-obtuse dihedral angles α, β, and γ. Then, Pv1 , Pv2 , Pv3 intersect at a vertex in H3 if and only if α + β + γ ≥ π. The planes intersect in H3 if and only if the inequality is strict. Proof: The planes intersect in ...

... Lemma 2 Suppose that three planes Pv1 , Pv2 , Pv3 intersect pairwise in H3 with non-obtuse dihedral angles α, β, and γ. Then, Pv1 , Pv2 , Pv3 intersect at a vertex in H3 if and only if α + β + γ ≥ π. The planes intersect in H3 if and only if the inequality is strict. Proof: The planes intersect in ...

[edit] Construction of the Lebesgue measure

... The intuitive dimension of a geometric object is the number of independent parameters you need to pick out a unique point inside. But you can easily take a single real number, one parameter, and split its digits to make two real numbers. The example of a space-filling curve shows that you can even t ...

... The intuitive dimension of a geometric object is the number of independent parameters you need to pick out a unique point inside. But you can easily take a single real number, one parameter, and split its digits to make two real numbers. The example of a space-filling curve shows that you can even t ...

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

... 2.4 We will recall the results obtained in [10] concerning the change of xi coordinates by 2. Suppose (p1 ∪. . .∪p3g+r−3 , b, col) is a marking on an oriented surface F , and DT is the associated Dehn–Thurston coordinate. Let [a] and [b] be two isotopy classes of curve systems so that their twisting ...

... 2.4 We will recall the results obtained in [10] concerning the change of xi coordinates by 2. Suppose (p1 ∪. . .∪p3g+r−3 , b, col) is a marking on an oriented surface F , and DT is the associated Dehn–Thurston coordinate. Let [a] and [b] be two isotopy classes of curve systems so that their twisting ...

On Lobachevsky`s trigonometric formulae

... details, and highlighting applications of his work outside the world of pure geometry,6 with the hope of attracting the attention of his colleagues. His efforts in that direction were vain, and his work was acknowledged only ten years after his death. These memoirs that he left, despite the fact tha ...

... details, and highlighting applications of his work outside the world of pure geometry,6 with the hope of attracting the attention of his colleagues. His efforts in that direction were vain, and his work was acknowledged only ten years after his death. These memoirs that he left, despite the fact tha ...

... details, and highlighting applications of his work outside the world of pure geometry,6 with the hope of attracting the attention of his colleagues. His efforts in that direction were vain, and his work was acknowledged only ten years after his death. These memoirs that he left, despite the fact tha ...

Metric Spaces - UGA Math Department

... By the usual abuse of notation, when only one metric on X is under discussion we will typically refer to “the metric space X.” Example 1.1. (Discrete Metric) Let X be a set, and for any x, y ∈ X, put ...

... By the usual abuse of notation, when only one metric on X is under discussion we will typically refer to “the metric space X.” Example 1.1. (Discrete Metric) Let X be a set, and for any x, y ∈ X, put ...

Introduction to Conjugate Plateau Constructions

... where conjugate minimal pieces have been obtained by solving degree arguments, for example in [KPS], [Po], a trivial perturbation argument shows that constant mean curvature deformations of the constructed minimal surfaces also exist. Constant mean curvature one surfaces in H3 , also called Bryant ...

... where conjugate minimal pieces have been obtained by solving degree arguments, for example in [KPS], [Po], a trivial perturbation argument shows that constant mean curvature deformations of the constructed minimal surfaces also exist. Constant mean curvature one surfaces in H3 , also called Bryant ...

Non Euclidean Geometry

... by cutting the polygon into n–2 triangles. A spherical polygon with n sides can be cut in the same way into n–2 spherical triangles, each of which has angle sum more than 180°, and so the angle sum of a spherical n-gon is more than (n – 2)180°. Put another way, the angle sum of a spherical polygon ...

... by cutting the polygon into n–2 triangles. A spherical polygon with n sides can be cut in the same way into n–2 spherical triangles, each of which has angle sum more than 180°, and so the angle sum of a spherical n-gon is more than (n – 2)180°. Put another way, the angle sum of a spherical polygon ...

Ab-initio construction of some crystalline 3D Euclidean networks

... We describe a technique for construction of 3D Euclidean (E3 ) networks with partially-prescribed rings. The algorithm starts with 2D hyperbolic (H2 ) tilings, whose symmetries are commensurate with the intrinsic 2D symmetries of triply periodic minimal surfaces (or infinite periodic minimal surface ...

... We describe a technique for construction of 3D Euclidean (E3 ) networks with partially-prescribed rings. The algorithm starts with 2D hyperbolic (H2 ) tilings, whose symmetries are commensurate with the intrinsic 2D symmetries of triply periodic minimal surfaces (or infinite periodic minimal surface ...

Ā - Non-Aristotelian Evaluating

... arithmetic (Greek arthmētikē techne, act of counting), apart of Mathematics instead concerned with numbers (computation of numbers, etc), hence discrete magnitudes. Since it remains with the Greeks that any sophisticated geometry has survived, who tended to view it as a 'deductively' (a priori, Lati ...

... arithmetic (Greek arthmētikē techne, act of counting), apart of Mathematics instead concerned with numbers (computation of numbers, etc), hence discrete magnitudes. Since it remains with the Greeks that any sophisticated geometry has survived, who tended to view it as a 'deductively' (a priori, Lati ...

Euclidean Geometry and History of Non

... 5. Through a point not on a given straight line, infinitely many lines can be drawn that never meet the given line. Hyperbolic geometry, (think of a balloon with a stamp on it) also called saddle geometry or Lobachevskian geometry, is the non-Euclidean geometry obtained by replacing the parallel pos ...

... 5. Through a point not on a given straight line, infinitely many lines can be drawn that never meet the given line. Hyperbolic geometry, (think of a balloon with a stamp on it) also called saddle geometry or Lobachevskian geometry, is the non-Euclidean geometry obtained by replacing the parallel pos ...

Mathematical Tools for 3D Shape Analysis and - imati-cnr

... X the collection T is called a topology on X ...

... X the collection T is called a topology on X ...

AN INTRODUCTION TO THE MEAN CURVATURE FLOW Contents

... of ε : A simply connected Riemannian manifold with sectional curvatures in the interval (1/4, 1] is homeomorphic to a sphere. If the value 1/4 is allowed, there are counterexamples. Actually, any compact symmetric space of rank 1 admits a metric whose sectional curvatures lie in the interval [1, 4]. ...

... of ε : A simply connected Riemannian manifold with sectional curvatures in the interval (1/4, 1] is homeomorphic to a sphere. If the value 1/4 is allowed, there are counterexamples. Actually, any compact symmetric space of rank 1 admits a metric whose sectional curvatures lie in the interval [1, 4]. ...

Pdf slides - Daniel Mathews

... the right of the door a little knot had formed round a small table, the center of which was the mathematics student, who was eagerly talking. He had made the assertion that one could draw through a given point more than one parallel to a straight line; Frau Hagenström had cried out that this was imp ...

... the right of the door a little knot had formed round a small table, the center of which was the mathematics student, who was eagerly talking. He had made the assertion that one could draw through a given point more than one parallel to a straight line; Frau Hagenström had cried out that this was imp ...