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Topic: Hyperbolic plane

Related Topics

Hyperbolic space

Ultraparallel theorem

Riemann sphere

Felix Klein

	Upper half-plane - Wikipedia, the free encyclopedia
		Other names are hyperbolic plane, Poincaré plane and Lobachevsky plane, particularly in texts by Russian authors.
		The open unit disk D is equivalent by a conformal mapping, meaning that it is usually possible to pass between H and D.
		It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions.
	en.wikipedia.org /wiki/Upper_half_plane (273 words)

	The Institute For Figuring // An Interview with David Henderson and Daina Taimina
		A hyperbolic plane is a surface in which the space curves away from itself at every point.
		The hyperbolic plane is sometimes described as a surface in which the space expands.
		DH: The discovery of the hyperbolic plane came from the attempt to prove Euclid’s fifth postulate, which is also known as the parallel postulate.
	www.theiff.org /lectures/05a.html (2350 words)

	Semi-Regular Tilings of the Plane Part 3: General Theorems
		A reflection tiling (green) of the hyperbolic plane using 'kites' with angles Pi/6, Pi/2, 2*Pi/5, and Pi/2.
		The green p^q = 5^4 tiling of the hyperbolic plane and its associated tiling by rhombii.
		Using Theorem 4 and the existence of the regular tilings of the Euclidean plane and the sphere in Theorem 1, we obtain most of the semi-regular tilings in the Euclidean plane and on the sphere listed in Theorems 2 and 3.
	people.hws.edu /mitchell/tilings/Part3.html (2038 words)

	math lessons - Hyperbolic geometry
		One remarkable property of the hyperbolic plane is that there is a unique common perpendicular for each pair of ultraparallel lines (see Ultraparallel theorem).
		There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disc model, the Poincaré half-plane model and the Lorentz model.
		It is also possible to see quite plainly the negative curvature of the hyperbolic plane, via its effect on the sum of angles in triangles and squares.
	www.mathdaily.com /lessons/Hyperbolic_geometry (587 words)

	Dynamic, Interactive Geometry with PoincaréDraw
		While this is an excellent means for students to understand how hyperbolic geometry "sits inside" Euclidean geometry, the Euclidean constructions are too cumbersome for all but the most elementary hyperbolic constructions, and the dynamic aspect of Geometer's Sketchpad is significantly slowed by the numerous dependencies among the objects of the constructions.
		The hyperbolic plane is represented by the open unit disk in the complex plane.
		Hyperbolic points and lines are represented by certain 2x2 matrices in the group of hyperbolic motions and its Lie algebra.
	persweb.wabash.edu /facstaff/footer/Pdraw/paper.htm (1001 words)

	MAM2003 Essay: Hyperbolic Art and the Poster Pattern (Site not responding. Last check: 2007-11-06)
		But it was only recently that the hyperbolic plane has been utilized for artistic purposes, though mathematicians have been drawing hyperbolic patterns for more than 100 years (see [Ma1] for examples).
		For example, all the triangles in Figure 1 are the same hyperbolic size, as are all the fl fish (or white fish) of Figure 2.
		Points on such arcs are an equal hyperbolic distance from the hyperbolic line with the same endpoints on the bounding circle.
	www.mathaware.org /mam/03/essay1.html (2110 words)

	Hyperbolic Tessellations (Site not responding. Last check: 2007-11-06)
		For this representation, a straight line in the hyperbolic plane is represented as the part (in the disk) of a circle that meets the boundary of the disk at right angles.
		For instance, here is a representation of the tessellation of the hyperbolic plane by pentagons where four pentagons meet at each vertex, that is, the {5,4}-tessellation.
		The dual tessellation {4,5} of the hyperbolic plane
	aleph0.clarku.edu /~djoyce/poincare/poincare.html (784 words)

	The Geometry Junkyard: Hyperbolic Geometry
		Embedding the hyperbolic plane in higher dimensional Euclidean spaces.
		Packing circles in the hyperbolic plane, Java animation by Kevin Pilgrim illustrating the effects of changing radii in the hyperbolic plane.
		The tractrix and the pseudosphere, hyperbolic surfaces modeled in Cabri.
	www.ics.uci.edu /~eppstein/junkyard/hyper.html (392 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Because of this, drawing a hyperbolic circle through a given point with a given center is different from drawing a Euclidean circle through a given point with a given center, and it is not clear (to me) that the former construction can be reduced to the latter (or to any other Euclidean construction).
		One might also consider scaling down a putative hyperbolic trisection to the infinitesimal scale to obtain a Euclidean trisection (which we know to be impossible), but the hyperbolic plane admits no scaling transformations (unlike the Euclidean plane), and anyway the limit of a valid construction may not be a valid construction (degeneracies may arise).
		Since this interpretation fails for other geometries such as the hyperbolic plane or the sphere, it would be nice to understand the problem in a way that would extend naturally to other geometries.
	www.math.niu.edu /~rusin/known-math/98/hyperbolic_tris (457 words)

	Body (Site not responding. Last check: 2007-11-06)
		the ratio determines the radius (the r in the annular hyperbolic plane) of the hyperbolic plane.
		This is the usual upper half plane model of the hyperbolic plane thought of as a map of the hyperbolic plane in the same way that we use planar maps of the spherical surface of the earth.
		Thus, we have established that the annular hyperbolic plane is the same as the usual upper half plane model of the hyperbolic plane.
	www.math.cornell.edu /~dwh/papers/crochet/crochet.html (3801 words)

	Java Gallery: Hyperbolic Triangles (Site not responding. Last check: 2007-11-06)
		One of the most surprising facts in hyperbolic geometry is that there is an upper limit to the possible area a triangle can have, even though there is not an upper limit to the lengths of the sides of the triangle.
		In hyperbolic geometry, the sum of the angles of a triangle is always less than 180 degrees (PI radians).
		The hyperbolic plane is embedded inside of a disk; the edge of the disk represents infinity.
	www.geom.uiuc.edu /java/triangle-area (275 words)

	Semi-Regular Tilings of the Plane Part 4: Hyperbolic Results
		Recall that because triangles in the hyperbolic plane have angle sum less than PI, it follows that the measure of the angle at a vertex of a regular p-gon in the hyperbolic plane is less than PI-(2*PI/p).
		There are regular tilings of the hyperbolic plane of vertex type p^q for all positive integers p and q such that 1/p + 1/q < 1/2.
		Discussion: An elementary proof of the fact that the images of triangle T tile the hyperbolic plane is given in [Ca, Chapter 3].
	people.hws.edu /mitchell/tilings/Part4.html (1887 words)

	Xah: Special Plane Curves: Hyperbolic Trig Functions (Site not responding. Last check: 2007-11-06)
		They are called hyperbolic trig functions because they bear strong similarities to the trig functions.
		Trig functions relate to a circle, while hyperbolic trig functions relate to a rectangular hyperbola x^-y^==1 in a similar way.
		The hyperbolic cosine Cosh is famously known as catenary.
	xahlee.org /SpecialPlaneCurves_dir/Sinh_dir/sinh.html (158 words)

	Hyperbolic Geometry - Circles (Site not responding. Last check: 2007-11-06)
		A circle in the hyperbolic plane is the locus of all points a fixed distance from the center, just as in the Euclidean plane.
		The only difference is that since distances are larger the nearer you are to the edge, the center of the hyperbolic circle is not the same as the Eucidean center, but is offset toward the edge of the half-plane.
		The points are connected by a (hyperbolic) line segment, the radius, in red, and the (hyperbolic) circle itself is drawn in blue.
	www.math.ksu.edu /math572/circ.html (379 words)

	Models of the Hyperbolic Plane
		In the Klein model of the hyperbolic plane, the "plane" is the unit disk; in other words, the interior of the Euclidean unit circle.
		The upper half plane model takes the Euclidean upper half plane as the "plane." Now the "lines" are portions of circles with their center on the boundary, as shown in Figure 1.
		In this case the "space" is the unit sphere, "lines" are portions of circles intersecting the boundary of the unit sphere at right angles, and "planes" are portions of spheres which meet the unit sphere at right angles.
	www.geom.uiuc.edu /docs/forum/hype/model.html (519 words)

	ipedia.com: Hyperbolic geometry Article (Site not responding. Last check: 2007-11-06)
		Hyperbolic geometry, also called saddle geometry or Lobachevskian geometry, is the non-Euclidean geometry obtained by replacing the parallel postulate with the hyperbolic postulate, which states: "Giv...
		The Klein model uses the interior of a circle for the hyperbolic plane, and chordss of the circle as lines.
		The Poincaré disc model also employs the interior of a circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle.
	www.ipedia.com /hyperbolic_geometry.html (306 words)

	Cabinet Magazine Online - Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
		The discovery of hyperbolic space in the 1820s and 1830s by the Hungarian mathematician Janos Bolyai and the Russian mathematician Nicholay Lobatchevsky marked a turning point in mathematics and initiated the formal field of non-Euclidean geometry.
		In the 1970s the American geometer William Thurston had described a model of hyperbolic space that could be made by taping together a series of paper annuli, or thin circular strips.
		On a Euclidean plane the internal angles of a triangle sum to 180 degrees, but on a hyperbolic plane they always sum to less than 180 degrees.
	www.cabinetmagazine.org /issues/16/crocheting.php (2537 words)

	Ivars Peterson's MathTrek - Hyperbolic Five
		Just as a flat surface is a piece of the infinite mathematical surface known as the Euclidean plane, so a saddle-shaped surface can be thought of as a small piece of the hyperbolic plane.
		To get a feel for what the hyperbolic plane is like, you could try to sew together pieces of cloth in the shape of pentagons.
		The hyperbolic measure of an angle is equal to that measured in the disk representation of the hyperbolic plane.
	www.maa.org /mathland/mathtrek_09_01_03.html (773 words)

	Intro to hyperbolic plane & H_n graphs (Site not responding. Last check: 2007-11-06)
		The hyperbolic plane is a (up to isomorphism, the) connected 2-dimensional manifold with constant negative curvature.
		In the hyperbolic plane, as on the sphere, there is no such thing as similar figures at different sizes.
		On the hyperbolic plane, with constant negative curvature, A is minus the area.
	web.mat.bham.ac.uk /marijke/F2/pix/Hn.html (1236 words)

	Hyperbolic Geometry
		It is this bumpy sheet with angular excesses all over the place that you might think of when you try to visualize the hyperbolic plane.
		Since we know that angular excess corresponds to negative curvature, we see that the hyperbolic plane is a negatively curved space.
		There are a number of different models for the hyperbolic plane.
	geom.math.uiuc.edu /docs/education/institute91/handouts/node37.html (760 words)

	[No title]
		But there is another way to get a sturdier model of the hyperbolic plane, which you can work and play with as much as you wish.
		You will get a hyperbolic plane ONLY if you will be increasing the number of stitches in the same ratio all the time.
		Thus, the plane can be called a sphere (or hyperbolic plane) with infinite radius.
	www.mathsci.appstate.edu /~sjg/class/1010/wc/geom/crochet.html (780 words)

	Chapter 5 (Site not responding. Last check: 2007-11-06)
		A paper model of the hyperbolic plane may be constructed as follows: Cut out many identical annular ("annulus" is the region between two concentric circles) strips as in Figure 5.2.
		The hyperbolic plane can be approximately constructed by using heptagons (7-sided) surrounded by seven hexagons and two hexagons and one heptagon together around each vertex.
		The hyperbolic soccer ball construction is related to the {3,7} construction in the sense that if a neighborhood of each vertex in the {3,7} construction is replaced by a heptagon then the remaining portion of each triangle is a hexagon
	www.math.cornell.edu /~dwh/books/eg00/supplements/AHPmodel (1566 words)

	NPR : Mathematicians Get Crafty with Geometry
		A model of the hyperbolic plane crocheted by Daina Taimina.
		Those curves -- an example of a high-level geometry concept called the hyperbolic plane -- were not even defined by geometry theorists until the 19th century.
		In 1997, Taimina, of Cornell University, found a way to crochet her way into "hyperbolic space." Her woolen creations, which resemble crenulated flowers and hair scrunchies, became the first physical models of the hyperbolic plane.
	www.npr.org /templates/story/story.php?storyId=4531695 (294 words)

	Foundations of Geometry (Site not responding. Last check: 2007-11-06)
		Hyperbolic geometry is the geometry where we accept the first four axioms of Euclidean geometry but negate the fifth postulate, i.e.
		One of the standard models of flattening out the hyperbolic plane is due to the French mathematician Henri Poincare.
		In this model, the hyperbolic plane is squashed onto a Euclidean half-plane.
	www.math.ksu.edu /math572/hyp.html (201 words)

	Amazon.com: Books: Hyperbolic Geometry (Springer Undergraduate Mathematics Series) (Site not responding. Last check: 2007-11-06)
		The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries.
		Topics covered include the upper half-space model of the hyperbolic plane, Möbius transformations, the general Möbius group and the subgroup preserving path length in the upper half-space model, arc-length and distance, the Poincaré; disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications.
		Provides a self-contained introduction to the subject of hyperbolic geometry, taking the approach that the subject consists of the study of those quantities invariant under the action of a natural group of transformations.
	www.amazon.com /exec/obidos/tg/detail/-/1852331569?v=glance (609 words)

	The Geometer's Sketchpad® - The Half-Plane Model
		In the half-plane model of hyperbolic geometry, we consider points on one side of a horizontal boundary line.
		This Half-Plane Model of Hyperbolic Geometry sketch depicts the hyperbolic plane and contains Custom Tools to create constructions in the upper half-plane.
		The Hyperbolic Triangles sketch depicts the same hyperbolic plane and contains Custom Tools for creating various centers of triangles constructed in the half-plane.
	www.keypress.com /sketchpad/general_resources/advanced_sketch_gallery/half_plane_model/index.php (198 words)

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