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Topic: Klein bottle

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Mobius Band

Klein quartic

Felix Klein

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Mean curvature

Topological space

Riemann surface

Hyperplane

Ruled surface

List of mathematical shapes

Minimal surface

Erlangen programme

Curve

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Algebraic topology

	Klein Bottle
		A side view of the knitted Klein Bottle, and also a bottom view of the same work, so you can see how the sidedness changes and the neck widens.
		This essentially makes a Klein bottle minus a disc, which is still pretty cool.
		I knitted a figure-eight cylinder: just a cylinder with a line of self-intersection down the middle, so it has a figure-eight cross-section (here's a picture of it, when it was short).
	web.meson.org /topology/klein.php (1628 words)

	UR Mathematics: About the Klein Bottle (Site not responding. Last check: 2007-11-02)
		The Klein bottle is the non-orientable surface with Euler characteristic equal to 0.
		A Klein bottle can be made from a rectangular piece of the plane by identifying the top and bottom edges using the same orientation, but identifying the left and right edges with opposite orientation (as in the formation of a Möbius band).
		Finally, the Klein bottle is the connected sum of two real projective planes, since the projective plane minus a disk is just a Möbius band.
	www.math.rochester.edu /misc/klein-bottle.html (177 words)

	PlanetMath: Klein bottle
		Where a Möbius strip is a two dimensional object with only one surface and one edge, a Klein bottle is a two dimensional object with a single surface, and no edges.
		To construct a pseudo-Klein bottle in 3-dimensional space, you would first take a cylinder and cut a hole at one point on the side.
		This is version 9 of Klein bottle, born on 2003-05-07, modified 2006-07-17.
	planetmath.org /encyclopedia/KleinBottle.html (242 words)

	Klein biography
		Klein received his doctorate, which was supervised by Plücker, from the University of Bonn in 1868, with a dissertation Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form on line geometry and its applications to mechanics.
		Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872), profoundly influenced mathematical development.
		Klein initiated a correspondence with Poincaré, and soon a friendly rivalry ensued as both sought to formulate and prove a grand uniformization theorem that would serve as a capstone to this theory.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Klein.html (2301 words)

	Klein
		Klein received his doctorate from the University of Bonn in 1868, where he had studied mathematics and physics.
		Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programme, profoundly influenced mathematical development.
		Klein was elected a member of the Royal Society in 1885 and received the Copley medal of the Society in 1912.
	library.wolfram.com /examples/quintic/people/Klein.html (335 words)

	Klein bottle Summary
		Although the Klein bottle cannot be constructed in space without self-intersection, mathematicians are still able to study the shape of the Klein bottle by looking at it intrinsically, considering what life would be like for a two-dimensional creature living on the Klein bottle.
		Looking at the Klein bottle's intrinsic structure, it becomes clear that the Klein bottle is an example of a non-orientable surface, that is, a surface on which it is impossible to distinguish between left-handedness and right-handedness.
		The Klein bottle is also closely related to another non-orientable surface, the projective plane, which is formed from a round sheet of paper by gluing together opposite points on the boundary circle (like the Klein bottle, this surface cannot actually be constructed without the sheet of paper passing through itself).
	www.bookrags.com /Klein_bottle (1495 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-11-02)
		In mathematics, the Klein bottle is a certain non-orientable surface, i.e.
		This square is known as the fundamental polygon of the Klein bottle.
		The generalization of the Klein bottle to higher genus is given in the article on the fundamental polygon.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Klein_bottle (870 words)

	Klein bottle
		It was discovered in 1882 by Felix Klein when he imagined, as in the limerick, joining two Möbius bands together to create a single-sided bottle with no boundary.
		A three-dimensional glass model of a Klein bottle can be made by stretching the neck of a bottle through its side and joining its end to a hole in the base.
		Just as a photo of such a bottle is of a three-dimensional Klein bottle immersion, so the immersion in real life is like a photo of the true four-dimensional bottle.
	www.daviddarling.info /encyclopedia/K/Klein_bottle.html (445 words)

	Imaging maths - Inside the Klein bottle (Site not responding. Last check: 2007-11-02)
		The Klein bottle was discovered in 1882 by Felix Klein [1] and since then has joined the gallery of popular mathematical shapes known to the general public outside the "ivory tower".
		In his original work [1], Klein introduced the bottle as a "certain unbounded double surface" which "can be visualized by inverting a piece of a rubber tube and by letting it pass through itself so that outside and inside meet".
		The Möbius band and the Klein bottle were discovered in the 19th century during the search for a classification of surfaces and shapes.
	plus.maths.org /issue26/features/mathart/index-gifd.html (2870 words)

	Klein Bottle (Site not responding. Last check: 2007-11-02)
		Klein bottle - named after its creator, Edgar Bottle, the Klein bottle is essentially a four-dimensional mobius strip - a bottle with no inside or outside.
		It can be simulated in three dimensions, by gluing together both pairs of opposite edges of a rectangle and giving one set a half-twist, but that requires cutting a hole that should not have to be there.
		Mathematical basis for the Klein bottle (and a nifty 3d-model).
	community.middlebury.edu /~dwalker/class/klein.html (83 words)

	Impossible Bottle Built
		A Klein bottle, named after the late-nineteenth-century mathematician Felix Klein, is a surface made by gluing together the ends of a tube in a prescribed way.
		Gassner said Parker Brothers, the famous manufacturer of games, had already approached him about marketing the Klein bottle as a toy, to be called the "Impossibottle." But because Gassner was not paying attention to what he was doing when he made the Klein bottle, he doesn't know how to make another.
		Those who believe the Klein bottle is evidence of a higher deity and those who think instead that it means the universe is non-orientable agree about one thing: the discovery of the Klein bottle is a monumental step forward for mathematics.
	nasw.org /users/klarreich/kleinbottle.htm (739 words)

	Impossible Bottle Built
		A Klein bottle, named after the late-nineteenth-century mathematician Felix Klein, is a surface made by gluing together the ends of a tube in a prescribed way.
		Gassner said Parker Brothers, the famous manufacturer of games, had already approached him about marketing the Klein bottle as a toy, to be called the "Impossibottle." But because Gassner was not paying attention to what he was doing when he made the Klein bottle, he doesn't know how to make another.
		Those who believe the Klein bottle is evidence of a higher deity and those who think instead that it means the universe is non-orientable agree about one thing: the discovery of the Klein bottle is a monumental step forward for mathematics.
	www.nasw.org /users/nasw/klarreich/kleinbottle.htm (739 words)

	Gravity: Klein Bottle (Site not responding. Last check: 2007-11-02)
		But suvw are periodic in the manner of a Klein bottle.
		A connection component (force) that increases the x and u components will force the trajectory around the bottle, but the force will be less effective there because it has to be continuous through the flipped periodicity, and hence (probably) has a cos(th) dependence on the u "angle".
		The loop of the bottle is so oriented (toward u=pi/2) that a line with u==0 starting at (0,0) returns to (0,0) without crossing s==0 elsewhere.
	www.math.ucla.edu /~jimc/klein_h/bottle.html (571 words)

	Imaging maths - Inside the Klein bottle (Site not responding. Last check: 2007-11-02)
		The Klein bottle was discovered in 1882 by Felix Klein [1] and since then has joined the gallery of popular mathematical shapes known to the general public outside the "ivory tower".
		In his original work [1], Klein introduced the bottle as a "certain unbounded double surface" which "can be visualized by inverting a piece of a rubber tube and by letting it pass through itself so that outside and inside meet".
		The Klein bottle is topologically equivalent to the connected sum of two crosscaps.
	pass.maths.org.uk /issue26/features/mathart/index-gifd.html (2870 words)

	Mathematical Recreations
		A few years ago he became intrigued by the mysterious shapes that arise in topology - Möbius bands, Klein bottles and the like-and came across a curious puzzle.
		Unlike an ordinary bottle, the "spout" or "neck" has been bent around, passed through the bottle's surface and joined to the main bottle from the inside.
		What appear to be the inside and outside of a Klein bottle connect together seamlessly: it is indeed one-sided.
	www.fortunecity.com /emachines/e11/86/klein1.html (1216 words)

	Grand Illusions - Toy Shop - The Wine Klein Bottle
		A Klein bottle is a non-orientable surface, where there is no distinction between inside and outside.
		It was first described by the German mathematician Felix Klein in 1882, and it is related to the Möbius strip.
		Klein bottles are sometimes made by the glassblowing department in a university, to show off their skills, and maybe to make a leaving present for a retiring Maths professor!
	www.grand-illusions.com /toyshop/wine_klein/index.shtml (185 words)

	Math Trek: Immersed in Klein Bottles, Science News Online, Feb. 17, 2001 (Site not responding. Last check: 2007-11-02)
		The intriguing subject of these cryptic entreaties is a bizarre mathematical object known as a Klein bottle, discovered in 1882 by German mathematician Felix Klein (1849-1925).
		One way to describe a Klein bottle is in terms of instructions for making one from a rectangular sheet of paper.
		If one were to make a model of a Klein bottle out of glass, there would have to be a circular hole where the stretched-out, bent end intersects the tube's side.
	www.sciencenews.org /20010217/mathtrek.asp (667 words)

	What is a Klein Bottle?
		In this sense, a Klein Bottle is a 2-dimensional manifold, and its inside is the same as its outside.
		A photograph of a stapler is a 2-dimensional immersion of a 3-dimensional stapler.
		We represent a Klein Bottle in glass by stretching the neck of a bottle through its side and joining its end to a hole in the base.
	www.kleinbottle.com /whats_a_klein_bottle.htm (899 words)

	[No title]
		I'm still working out detailed stitch-by-stitch instructions for the blind-follower types; but this is the hat ideas that I have at the moment given the beginning of the Klein bottle pattern that Mary gave me...
		I got the basic Klein bottle shape pretty good, but the hat shaping part I'm still working on.
		My gauge with the worsted was 18 sts for 4 inches, which made about a 20 inch outside circumferance hat which fit my roughly 18 inch head pretty well with the addition of the lining and the ribbing.
	www.woolworks.org /patterns/klein.txt (1353 words)

	Klein bottle (Site not responding. Last check: 2007-11-02)
		Most containers have an inside and an outside, a klein bottle is a closed surface with no interior and only one surface.
		This source code creates the classical klein bottle shape as descibed at the top of this document.
		If you would like to create a klein bottle for a CAD package, this source code creates a DXF file of the classical klein bottle.
	local.wasp.uwa.edu.au /~pbourke/surfaces/klein/index.html (278 words)

	Strange Surfaces: New Ideas (Site not responding. Last check: 2007-11-02)
		Felix Klein, a German mathematician, introduced one of the best known surfaces, the Klein bottle, in 1882.
		In 1995 Alan Bennett, a retired glass-blower, became interested in Klein bottles and was in a unique position to satisfy his curiosity.
		This is one of a series of glass Klein bottles made by Alan Bennett for the Science Museum.
	www.sciencemuseum.org.uk /on-line/surfaces/new.asp (138 words)

	Klein Bottle Checker Board Paradox (Site not responding. Last check: 2007-11-02)
		A Klein bottle is a particular shape of a 2-dimensional space.
		Since a Klein bottle space is a closed finite space that we can shift or rotate our checkered pattern anyway we want like a oil slick, this seems to prove that it is impossible to color a Klein bottle with 8 x 8 checkered pattern, due to some inherent parity issue.
		If you connect the right and left edges with a flip, indeed the whole Klein Bottle space is patterned by alternating colored squares.
	xahlee.org /Periodic_dosage_dir/20031224_klein_checker.html (538 words)

	Gigantic Klein botttle - Boing Boing
		Klein bottles are basically Moebius strips with one extra dimension - bottles that have one continous volume without any "inside" or "outside." This Klein bottle was made by Cliff Stoll (who wrote the classic true-cybercrime thriller The Cuckoo's Egg) who runs the Acme Klein studio in the East Bay.
		I've bought one of Cliff's Klein bottles as a gift for my mathematician father, and it was very appreciated.
		Klein bottles are basically Moebius strips with one extra dimension -- bottles that have one continous volume without any "inside" or "outside." This Klein bottle was made by Cliff Stoll (who wrote the classic true-cybercrime thriller The Cuckoo's Egg) who runs the Acme Klein studio in the East Bay.
	www.boingboing.net /2005/08/23/gigantic_klein_bottt.html (297 words)

	Yikes! A Drinking Mug Klein Bottle
		There's ~230 mL in the outer chamber (which topologists will recognize as equivalent to the inner chamber) and ~250 mL in the inner chamber (which topologists claim to be the same as the outer chamber).
		This Klein Stein is ideal for the mathematical physicist who needs a glass of water while accepting her Nobel Prize.
		Indeed, think of all the seminars, colloquia, interviews, and funerals that would be jazzed up with an Acme Klein Bottle Mug at your side.
	www.kleinbottle.com /drinking_mug_klein_bottle.htm (728 words)

	The Klein Bottle (Site not responding. Last check: 2007-11-02)
		The tube that goes through the side of the bottle does so in much the same way as a television character from the 1960s, or at least that is one way of looking at it.
		The Klein bottle is simply a set on which we have placed a certain mathematical "structure." The Klein bottle is an example of a two-dimensional differentiable manifold.
		Now in the picture of the Klein bottle, the big problem is that the tube collides with the side of the bottle.
	www.und.nodak.edu /instruct/lapeters/res/kb (602 words)

	Klein Bottle & Projective plane (Site not responding. Last check: 2007-11-02)
		That is to say, Klein bottle is made of two Mobius strips glued.
		The picture of this Klein Bottle is not like a bottle and seems nothing to do with inside or outside, not similar to the standard Klein Bottle.
		Thus it is confirmed that the tubular Klein Bottle is the same as the standard Klein Bottle.
	www1.kcn.ne.jp /~iittoo/us27i_klpr.htm (3773 words)

	Epinions.com - Klein bottles: the way to a math geek's heart
		A Klein bottle is like that, only in three dimensions: its inside and outside are one and the same and it has zero volume .
		Now take off the cap, grab the top of the bottle and streeeettttch it (this is a thought experiment, ok?) and stick it into the side of the bottle.
		Acme Klein Bottles are available in several shapes and variations.
	www.epinions.com /gift-review-5DD4-AC98C8C-39E28528-prod1 (589 words)

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