The Klein bottle which has no distinction between inside and outside surfaces can be made by gluing two of what similar objects?
"The three embeddings above cannot be smoothly transformed into each other in three dimensions. If you cut the traditional Klein bottle lengthwise it deconstructs into two, oppositely chiral Möbius strips. If you cut a left handed figure-8 Klein bottle it deconstructs into two left handed Möbius strips. And similarly for the right handed figure-8 Klein bottle. If the traditional Klein bottle is two color painted, this induces chirality on it, creating four homotopy classes. The generalization of the Klein bottle to higher genus is given in the article on the fundamental polygon. In another order of ideas, constructing 3-manifolds, it"
"a self-intersecting Klein bottle. To construct the Klein bottle, glue the red arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends of the cylinder together so that the arrows on the circles match, you must pass one end through the side of the cylinder. This creates a circle of self-intersection – this is an immersion of the Klein bottle in three dimensions. This immersion is useful for visualizing many properties of the Klein bottle. For example, the Klein bottle has no ""boundary"", where the surface stops abruptly, and it is non-orientable, as"
"The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It may have been originally named the ""Kleinsche Fläche"" (""Klein surface"") and then misinterpreted as ""Kleinsche Flasche"" (""Klein bottle""), which ultimately may have led to the adoption of this term in the German language as well. The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding coloured edges with the arrows matching, as in the diagrams below. Note that this is an ""abstract"" gluing in the sense that trying to realize this in three dimensions results in"
"it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space R, the Klein bottle cannot. It can be embedded in R, however. The Klein bottle can be seen as a fiber bundle over the circle ""S"", with fibre ""S"", as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be ""E"", the total space, while the base space ""B"" is given by the unit interval in ""y"", modulo ""1~0"". The projection π:""E""→""B"" is then given by π([""x"", ""y""]) = [""y""]. The Klein bottle can be constructed (in"
"is known that a solid Klein bottle is homeomorphic to the Cartesian product of a Möbius strip and a closed interval. The ""solid Klein bottle"" is the non-orientable version of the solid torus, equivalent to formula_8 A Klein surface is, as for Riemann surfaces, a surface with an atlas allowing the transition maps to be composed using complex conjugation. One can obtain the so-called dianalytic structure of the space. Klein bottle In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal"
"Klein bottle In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary (for comparison, a sphere is an orientable surface with no boundary)."
"""a"" is thought of as a cut, then −""a"" can be thought of as a gluing operation. Making a cut and then re-gluing it does not change the surface, so ""a"" + (−""a"") = 0. But now consider two ""a""-cycles. Since the Klein bottle is nonorientable, you can transport one of them all the way round the bottle (along the ""b""-cycle), and it will come back as −""a"". This is because the Klein bottle is made from a cylinder, whose ""a""-cycle ends are glued together with opposite orientations. Hence 2""a"" = ""a"" + ""a"" = ""a"" + (−""a"") = 0."
"then be glued back together in different ways. The square can be twisted to allow edges to meet in the opposite direction, as shown by the arrows in the diagram. Up to symmetry, there are four distinct ways of gluing the sides, each creating a different surface: formula_4 is the Klein bottle, which is a torus with a twist in it (The twist can be seen in the square diagram as the reversal of the bottom arrow). It is a theorem that the re-glued surface must self-intersect (when immersed in Euclidean 3-space). Like the torus, cycles ""a"" and ""b"" cannot"
"to construct a true glass Klein bottle, a one-sided figure formally described as “an enclosure continuous with its outer surface constructed by twisting a tube through an opening in the side of the tube and joining it to the other end"". Klein bottles had previously been made by other glassblowers, but their versions were sealed and did not have an open hole where the tube went through the side of the bottle. After several days of failing to make such a Klein bottle with an opening, he declared it impossible. However, the solution to the problem was revealed in a"
"reflected in the one-sidedness of the immersion. The common physical model of a Klein bottle is a similar construction. The Science Museum in London has on display a collection of hand-blown glass Klein bottles, exhibiting many variations on this topological theme. The bottles date from 1995 and were made for the museum by Alan Bennett. The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. Gently push a piece of the tube containing"
"on the surface of a Klein bottle; this is the only exception to the Heawood conjecture, a generalization of the four color theorem, which would require seven. A Klein bottle is homeomorphic to the connected sum of two projective planes. It is also homeomorphic to a sphere plus two cross caps. When embedded in Euclidean space, the Klein bottle is one-sided. However, there are other topological 3-spaces, and in some of the non-orientable examples a Klein bottle can be embedded such that it is two-sided, though due to the nature of the space it remains non-orientable. Dissecting a Klein bottle"
"Solid Klein bottle In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder formula_1 to the bottom disk by a reflection across a diameter of the disk. Alternatively, one can visualize the solid Klein bottle as the trivial product formula_2, of the möbius strip and an interval formula_3. In this model one can see that the core central curve at 1/2 has a regular neighborhood which is again a trivial cartesian product: formula_4 and whose"
"that in three-dimensional space, a Klein bottle's surface must pass through itself. Building a Klein bottle which is not self-intersecting requires four or more dimensions of space. Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called ""antipodes"". Although there is no way to do so physically, it is possible (by considering a quotient space) to mathematically merge each antipode pair into a single point. The closed surface so produced is the real projective plane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but"
"boundary is a Klein bottle. Solid Klein bottle In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder formula_1 to the bottom disk by a reflection across a diameter of the disk. Alternatively, one can visualize the solid Klein bottle as the trivial product formula_2, of the möbius strip and an interval formula_3. In this model one can see that the core central curve at 1/2 has a regular neighborhood which is again a trivial"
"no gravity, as did also a letter to an editor by Clément. This special case is often referred to as the ""Ellis wormhole"". When the full-blown drainhole is considered in its role as the prototypical traversable wormhole, the name of Bronnikov is attached to it alongside that of Ellis. Imagine two euclidean planes, one above the other. Pick two circles of the same radius, one above the other, and remove their interiors. Now glue the exteriors together at the circles, bending the exteriors smoothly so that there is no sharp edge at the gluing. If done with care the result"
"Cat but leaving its ever-expanding smile behind. By the time the growth front gets to where the bud had been, there’s nothing there to intersect and the growth completes without piercing existing structure. The 4-figure as defined cannot exist in 3-space but is easily understood in 4-space. More formally, the Klein bottle is the quotient space described as the square [0,1] × [0,1] with sides identified by the relations for and for . Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. Unlike the Möbius strip, the Klein bottle is a ""closed"" manifold, meaning"
"Klein surfaces: there is a one-parameter family of inequivalent Klein surfaces arising in this way from annuli. By removing a number of open disks from the Riemann sphere, we obtain another class of Klein surfaces (compact, with boundary). The real projective plane can be turned into a Klein surface (compact, without boundary), in essentially only one way. The Klein bottle can be turned into a Klein surface (compact, without boundary); there is a one-parameter family of inequivalent Klein surfaces structures defined on the Klein bottle. Similarly, there is a one-parameter family of inequivalent Klein surface structures (compact, with boundary) defined"
"be etc.). A common technique is blow molding. Like any object made by extrusion, it is possible to produce multilayer plastic bottles. This is called coextrusion. The combination of different materials or colours can be used to produce flacons with different properties, such as: impermeability to light, oxygen, recycled inner layer. The glass bottles are manufactured by blow molding. Charles Baudelaire wrote a poem about the flacon, entitled ""Le Flacon"" (The Perfume Flask). Other words derived from flasco Flacon A flacon (from Late Latin flasco, meaning ""bottle""; cf. ""flagon"") is a small, often decorative, bottle. It has an opening seal"
"to mechanics, classified second degree line complexes using Weierstrass's theory of elementary divisors. Klein's first important mathematical discoveries were made during 1870. In collaboration with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They later investigated W-curves, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to have a major role in his later work. Klein also learned about groups from Camille Jordan. Klein devised the ""Klein bottle"" named after him, a one-sided closed surface which cannot be embedded in three-dimensional"
"out a little disk from each surface and identifying (""gluing"") their boundary circles. Gluing two projective planes creates the Klein bottle. The article on the fundamental polygon describes the higher non-orientable surfaces. Real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, 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 R passing through the origin. The plane is also often described"
"by where formula_2 is the floor function. Replacing the genus by the Euler characteristic, we obtain a formula that covers both the orientable and non-orientable cases, This relation holds, as Ringel and Youngs showed, for all surfaces except for the Klein bottle. Philip Franklin (1930) proved that the Klein bottle requires at most 6 colors, rather than 7 as predicted by the formula. The Franklin graph can be drawn on the Klein bottle in a way that forms six mutually-adjacent regions, showing that this bound is tight. The upper bound, proved in Heawood's original short paper, is based on a"
"proved it in , and the triple cross surface is accordingly called '. Geometrically, connect-sum with a torus () adds a handle with both ends attached to the same side of the surface, while connect-sum with a Klein bottle () adds a handle with the two ends attached to opposite sides of an orientable surface; in the presence of a projective plane (), the surface is not orientable (there is no notion of side), so there is no difference between attaching a torus and attaching a Klein bottle, which explains the relation. Compact surfaces, possibly with boundary, are simply closed"
"This phenomenon is called torsion. Similarly, in the projective plane, following the unshrinkable cycle ""b"" round twice remarkably creates a trivial cycle which ""can"" be shrunk to a point; that is, ""b"" + ""b"" = 0. Because ""b"" must be followed around twice to achieve a zero cycle, the surface is said to have a torsion coefficient of 2. However, following a ""b""-cycle around twice in the Klein bottle gives simply ""b"" + ""b"" = 2""b"", since this cycle lives in a torsion-free homology class. This corresponds to the fact that in the fundamental polygon of the Klein bottle, only"
"For ""b""=0 this is Euclidean, and is the 3-torus (the surface bundle associated to the identity map of the 2-torus). Two non-orientable Euclidean Klein bottle bundles over the circle. The first homology is Z+Z+Z/2Z if ""b""=0, and Z+Z if ""b""=1. The first is the Klein bottle times ""S"" and other is the surface bundle associated to a Dehn twist of the Klein bottle. They are homeomorphic to the torus bundles {""b""; (""n"", 2);}. Homeomorphic to the non-orientable Euclidean Klein bottle bundle {1; (""n"", 2);}, with first homology Z + Z/4Z. These are the non-orientable Euclidean surface bundles associated with orientation"
"The cubic outer shell of the Large House is formed by a glass facade, which gives view into the interior and the cylindric casing of the auditorium and its stairways, and the two monumental sculptures by the French artist Yves Klein. They consist of one 7×20 m (×) monochrome sponge sculpture in a distinctive blue (""Gelsenkirchen Blue"") because Klein's International Klein Blue was unsuitable for the large walls: the acetone-based mixture evaporated, the paint lost adhesion and luminescence, and it was highly flammable. A second, slightly smaller work, is in blue and white. These intensely blue reliefs, which are visible"
"Bologna bottle A Bologna bottle, also known as a Bologna phial or philosophical vial, is a glass bottle which has great external strength, often used in physics demonstrations and magic tricks. The exterior is generally strong enough that one could pound a nail into a block of wood using the bottle as a hammer; however, even a small scratch on the interior would cause it to crumble. It is created by heating a glass bottle and then rapidly cooling the outside whilst slowly cooling the inside. This causes external compression and internal tension such that even a scratch on the"
"The uncut form, generally represented as the Boy surface, is visually complex, so a hemispherical embedding is shown in the diagram, in which antipodal points around the rim such as ""A"" and ""A′"" are identified as the same point. Again, ""a"" and ""b"" are non-shrinkable while ""c"" is. But this time, both ""a"" and ""b"" reverse left and right. Cycles can be joined or added together, as ""a"" and ""b"" on the torus were when it was cut open and flattened down. In the Klein bottle diagram, ""a"" goes round one way and −""a"" goes round the opposite way. If"
"interval of self-intersection) where the image cannot be that of an immersion. The bounding edge of a cross-cap is a simple closed loop. Like certain versions of the Möbius strip, it may take the form of a symmetrical circle. A cross-cap that has been closed up by gluing a disc to its boundary is a model of the real projective plane P (again with an interval of self-intersection, and two points where this model is not an immersion of P). Two cross-caps glued together at their boundaries form a model of the Klein bottle, this time with two intervals of"
"the torus and Klein bottle require a 4-gon (square). The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators. This is a consequence of the Seifert–van Kampen theorem. Gluing edges of polygons is a special kind of quotient space process. The quotient concept can be applied in greater generality to produce new or alternative constructions of surfaces. For example, the real projective plane can be obtained as the quotient of the sphere by identifying all"
"to discover how these objects are made. Another well known puzzle is one consisting of a cube made of two pieces interlocked in four places by seemingly inseparable links. The solutions to these are to be found in different places. There are all kinds of objects which fit this description – ""impossible bottles"" which contain objects that are far too large, Japanese hole coins with wooden arrows and rings through them, wooden spheres in a wooden frame with far too small openings and many more. The apple and arrow in the picture are made of one piece of wood each."
"is a subset of a toroidally closed spherinder (solid spheritorus). The parametrization of the 3-dimensional immersion of the bottle itself is much more complicated. Here is a version found by Robert Israel: for 0 ≤ ""u"" < π and 0 ≤ ""v"" < 2π. Regular 3D embeddings of the Klein bottle fall into three homotopy classes (four if you're allowed to paint them). The three are represented by The traditional Klein bottle embedding is achiral. The figure-8 embedding is chiral (the pinched torus embedding above is not regular as it has pinch points so it's not relevant in this section)."
"metal, glass, earthenware, or wood. Bottles are often recycled according to the SPI recycling code for the material. Bottle A bottle is a narrow-necked container made of an impermeable material (clay, glass, plastic, aluminium, etc.) in various shapes and sizes to store and transport liquids (water, milk, beer, wine, ink, cooking oil, medicine, soft drinks, shampoo, and chemicals, etc.) and whose mouth at the bottling line can be sealed with an internal stopper, an external bottle cap, a closure, or a conductive ""inner seal"" using induction sealing. Some of the earliest bottles appeared in China, Phoenicia, Crete, and Rome. First"
"× ""F"" would be a cylinder, but the Möbius strip has an overall ""twist"". Note that this twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space). A similar nontrivial bundle is the Klein bottle which can be viewed as a ""twisted"" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-torus, ""S"" × ""S"". A covering space is a fiber bundle such that the bundle projection is a local homeomorphism. It follows that the fiber is a discrete space. A special"