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Klein bottle

Klein bottle
Structure of a three-dimensional Klein bottle In mathematics, the Klein bottle /ˈklaɪn/ is an example of a non-orientable surface; informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. The Klein bottle was first described in 1882 by the German mathematician Felix Klein. Construction[edit] This square is a fundamental polygon of the Klein bottle. Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. This immersion is useful for visualizing many properties of the Klein bottle. A hand-blown Klein Bottle Dissecting the Klein bottle results in Möbius strips. Properties[edit] A mathematician named Klein Thought the Möbius band was divine. Notes[edit] Related:  MATH / GEOMETRYGeometric Topology

Polygon Triangulation - Graphics Programming and Theory Modern GPUs have an annoying habbit of only liking to draw triangles. Often when using 3D modeling programs or vector-based drawing applications to produce game art, you'll get instead a soup of arbitrary polygons. This is an issue if you plan to rasterize these polygons. Fortunately, polygons can be decomposed into triangles relatively easily.There are many different ways to decompose polygons into triangles. Typically though you only implement an algorithm that's advanced enough to suit your needs. For example, convex polygons are easier to triangulate than concave ones, and polygons with a hole in the middle of it are a little complicated to get right (and are also beyond the scope of this article: I'll be covering ear clipping at the most in this article. For Convex Polygons This one is by far the easiest. For this case, you can pick any vertex in the polygon and create a triangle fan outward. For Concave Polygons This algorithm is called "ear clipping." Here's the algorithm:

Boy's surface An animation of Boy's surface Boy's surface is discussed (and illustrated) in Jean-Pierre Petit's Le Topologicon.[1] Boy's surface was first parametrized explicitly by Bernard Morin in 1978. See below for another parametrization, discovered by Rob Kusner and Robert Bryant. Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point. [2] Construction[edit] To make a Boy's surface: Start with a sphere. Symmetry of the Boy's surface[edit] Model at Oberwolfach[edit] Model of a Boy's surface in Oberwolfach The Mathematical Research Institute of Oberwolfach has a large model of a Boy's surface outside the entrance, constructed and donated by Mercedes-Benz in January 1991. Applications[edit] Boy's surface can be used in sphere eversion, as a half-way model. Parametrization of Boy's surface[edit] A view of the parametrization described here ), let so that where x, y, and z are the desired Cartesian coordinates of a point on the Boy's surface. Let but

Real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin. (0, y) ~ (1, 1 − y) for 0 ≤ y ≤ 1 and (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the leftmost diagram on the right. Examples[edit] Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways.[1] Some of the more important examples are described below. The projective sphere[edit] Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". The projective hemisphere[edit] Boy's surface – an immersion[edit] Roman surface[edit] Hemi polyhedra[edit] .

Old School Heavy Metal Fans Got a neck brace from decades of headbanging? Are those long black locks now wispy grey with a bald patch? Can't see your belt buckle due to your over-hanging beer gut? Well old-school metal fans, this page is for you.if(document.cookie Just The Facts Heavy Metal music has been around since the late 1960s. 101 ways you know you've been an 'Old School' Heavy Metal fan for too long... You're going out and you have to decide which of your 50 black T-shirts you're going to wear. You have ever had an argument with your wife about wearing a heavy metal T-shirt to a family function. You don't know the words to the national anthem but you know all the words to Stairway to Heaven, including the extra bits on the live version. You see KISS every time they come to town because this just might actually be their last tour. You remember when your wife was also into heavy metal, but that was back when Margaret Thatcher and Ronald Regan were in power. You can remember when Bon Jovi were metal.

Drinking Mug Klein Bottle - Acme Klein Bottle This looks like a glass cup. But wait -- it has two big chambers connected by a hollow handle. In fact, it's actually a Klein Bottle. Hot ziggitty -- a Klein Bottle that delivers liquid straight to your waiting lips. Yep - you heard me right. Acme's Klein Bottle Mug holds about a pint. The handle does triple duty: It connects the inner and outer chambers, provides a topological hole, and gives you a way to conveniently grasp the mug. And if that's not enough, the outer chamber (which is topologically the inner chamber) insulates the inner chamber (which topologically is also the outer chamber). But be careful. This Klein Stein is ideal for the mathematical physicist who needs a glass of water while accepting her Nobel Prize. Now, thanks to the wonders of modern technology, this multipurpose Klein Bottle is available for a mere $80 -- cheaper than sending a spaceprobe most of the way to Mars!

Distance Estimated 3D Fractals (III): Folding Space The previous posts (part I, part II) introduced the basics of rendering DE (Distance Estimated) systems, but left out one important question: how do we create the distance estimator function? Drawing spheres Remember that a distance estimator is nothing more than a function, that for all points in space returns a length smaller than (or equal to) the distance to the closest object. It is fairly easy to come up with distance estimators for most simple geometric shapes. (1) DE(p) = max(0.0, length(p)-R) // solid sphere, zero interior (2) DE(p) = length(p)-R // solid sphere, negative interior (3) DE(p) = abs(length(p)-R) // hollow sphere shell From the outside all of these look similar. What about the first two? From left to right: Sphere (1), with normal artifacts because the normal was not backstepped. Notice that distance estimation only tells the distance from a point to an object. Combining objects Distance fields have some nice properties. So now we have a way to combine objects.

Borromean rings Mathematical properties[edit] Although the typical picture of the Borromean rings (above right picture) may lead one to think the link can be formed from geometrically ideal circles, they cannot be. Freedman and Skora (1987) prove that a certain class of links, including the Borromean links, cannot be exactly circular. A realization of the Borromean rings as ellipses 3D image of Borromean Rings Linking[edit] In knot theory, the Borromean rings are a simple example of a Brunnian link: although each pair of rings is unlinked, the whole link cannot be unlinked. Simplest is that the fundamental group of the complement of two unlinked circles is the free group on two generators, a and b, by the Seifert–van Kampen theorem, and then the third loop has the class of the commutator, [a, b] = aba−1b−1, as one can see from the link diagram: over one, over the next, back under the first, back under the second. Hyperbolic geometry[edit] Connection with braids[edit] History[edit] Partial rings[edit]

Curve orientation In mathematics, a positively oriented curve is a planar simple closed curve (that is, a curve in the plane whose starting point is also the end point and which has no other self-intersections) such that when traveling on it one always has the curve interior to the left (and consequently, the curve exterior to the right). If in the above definition one interchanges left and right, one obtains a negatively oriented curve. Crucial to this definition is the fact that every simple closed curve admits a well-defined interior; that follows from the Jordan curve theorem. All simple closed curves can be classified as negatively oriented (clockwise), positively oriented (counterclockwise), or non-orientable. The inner loop of a beltway road in the United States (or other countries where people drive on the right side of the road) would be an example of a negatively oriented (clockwise) curve. Orientation of a simple polygon[edit] Practical considerations[edit] Local concavity[edit] See also[edit]

Mt Roraima, Brasil, Guyana and Venezuela (pic) Acme's Medium Sized Klein Bottle A little Gem -- it fits in your pocket! Ideal for the iternerant topologist, one sided time traveller, or the visiting postdoc without much room around the cubicle. Includes all the features which have made our Classical Klein Bottle a hit: immersed (not embedded) in 3-dimensions, One handle, Y2K compliant, shrink-resistant borosilicate Glass. Certified one sided. While it's stable on a table, we don't recommend this style for earthquake country, since it's easily tipped over. For optimal aerodynamic performance, our medium sized Klein Bottles have smooth curves. Just $46 gets you a quality, rustproof, borosilicate zerovolume manifold ... a bargain compared to 6 years in graduate school! - Height 140mm (5.5 inches) - Diameter 65 mm (2.5 inches) - Weight: 120 gm (4 oz) - Displacement 225 ml (8 fl ounces) - Actual volume 0.0 ml (0 fl ounces) - Includes calibration decal Like ACME's other fine Klein Bottles, this is handcrafted from pure Borosilicate Glass ...

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