# Image:KleinBottle-01.png

No higher resolution available.

KleinBottle-01.png (240 × 300 pixel, file size: 64 KB, MIME type: image/png) This is a file from the Wikimedia Commons. The description on its description page there is shown below. Commons is attempting to create a freely licensed media file repository. You can help.

## Summary

Standard immersion of a Klein bottle into R3. Made with Mathematica.

Image:Klein bottle.svg

## Licensing I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide. In case this is not legally possible:I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. Afrikaans | Alemannisch | Aragonés | العربية‎ | Български | Català | Česky | Cymraeg | Dansk | Deutsch | Ελληνικά | English | Español | Esperanto | Euskara | فارسی‎ | Français | Galego | 한국어 | हिन्दी | Hrvatski | Ido | Bahasa Indonesia | Íslenska | Italiano | עברית‎ | Kurdî / كوردي | Latina | Lietuvių | Magyar | Bahasa Melayu | Nederlands | Norsk (bokmål) | Norsk (nynorsk) | 日本語 | Polski | Português | Ripoarish | Română | Русский | Shqip | Slovenčina | Slovenščina | Српски | Svenska | ไทย | Türkçe | Українська | Tiếng Việt | Walon | 简体中文 | 繁體中文 | 粵語 | +/-

## Parameterization

This immersion of the Klein bottle into R3 is given by the following parameterization. Here the parameters u and v run from 0 to 2π and r is a fixed positive constant.

For $0 \leq u < \pi$: $x = 6 \cos u(1 + \sin u) + 4r\left(1 - \frac{\cos u}{2}\right) \cos u \cos v$ $y = 16 \sin u + 4r\left(1 - \frac{\cos u}{2}\right) \sin u \cos v$ $z = 4r\left(1 - \frac{\cos u}{2}\right) \sin v$

For $\pi \leq u < 2\pi$: $x = 6 \cos u(1 + \sin u) - 4r\left(1 - \frac{\cos u}{2}\right) \cos v$ $y = 16 \sin u\,$ $z = 4r\left(1 - \frac{\cos u}{2}\right) \sin v$

## Mathematica source

KleinBottle[r_:1] =
Function[{u, v},
UnitStep[Sin[u]]
{
6 Cos[u](1 + Sin[u]) + 4r(1 - Cos[u]/2) Cos[u]Cos[v],
16 Sin[u] + 4r(1 - Cos[u]/2) Sin[u]Cos[v],
4r(1 - Cos[u]/2) Sin[v]
}
+ (1 - UnitStep[Sin[u]])
{
6 Cos[u](1 + Sin[u]) - 4r(1 - Cos[u]/2) Cos[v],
16 Sin[u],
4r(1 - Cos[u]/2) Sin[v]
}
]

ParametricPlot3D[Evaluate[KleinBottle[][u, v]], {u, 0, 2Pi}, {v, 0, 2Pi},
PlotPoints -> {50, 19}, Boxed -> False, Axes -> False,
ViewPoint -> {0.454, -2.439, -2.301}]


The following pages on the English Wikipedia link to this file (pages on other projects are not listed):
• Klein bottle
• Partition of unity
• Talk:Klein bottle
• Euler characteristic
• Hsiang-Lawson's conjecture
• Meagre set
• Fibre bundle
• Zero set
• Knot polynomial
• Sigma-compact space
• Spinor bundle
• Affine representation
• Submersion (mathematics)
• Unicoherent
• Continuum (mathematics)
• F-sigma set
• Toroid
• Weak operator topology
• Spectral space
• Banach-Alaoglu theorem
• Plus construction
• Nerve of an open covering
• Sober space
• Fork (topology)
• Conifold
• Von Neumann conjecture
• Noncommutative topology
• Fundamental class
• Closed manifold
• Cohomotopy group
• Lyusternik–Schnirelmann category
• Lebesgue's number lemma
• Bundle (mathematics)
• Cocycle class
• Upper topology
• HOMFLY polynomial
• Critical value
• Whitney immersion theorem
• Reidemeister move
• Knot group
• Homotopy sphere
• Nagata–Smirnov metrization theorem
• Relative homology
• Computational topology
• Solid torus
• Poincaré–Hopf theorem
• Property P conjecture
• Homotopical algebra
• Local property
• Plateau (mathematics)
• Portal:Mathematics/Featured picture archive
• Uniform property
• Gauge space
• Hyperbolic manifold
• Uniformly connected space
• Topological module
• Mereotopology
• Signature of a knot
• Cofibration
• Perverse sheaf
• Local diffeomorphism
• Uniformizable space
• Graph manifold
• Semiregular space
• Manifold
• Talk:Manifold
• Deformation retract
• Evolutionary graph theory
• Moore space (algebraic topology)
• Obstruction theory
• Hauptvermutung
• Kirby-Siebenmann class
• Donaldson's theorem
• Pseudo-Anosov map
• Higher dimension
• Hilbert manifold
• Vietoris–Begle mapping theorem
• Compactly-supported homology
• Topological algebra
• Pair of pants
• Ancestral graph
• Stable homotopy theory
• Equivariant cohomology
• Finite type invariant
• Torsion (topology)
• CR manifold
• Analytic manifold
• Decomposable continuum
• Indecomposable continuum
• Pseudo-arc
• Composant
• Circle-valued Morse theory
• 2-bridge knot
• Stable homotopy category
• Baire function
• Tube lemma
• Set-theoretic topology
• Carathéodory-Jacobi-Lie theorem
• Mutation (knot theory)
• Manifold/old2
• Template:Topology-stub
• Homotopy extension property
• Cartan model
• Slice knot
• Degree sum formula
• Amphichiral knot
• Dehn's lemma
• House with two rooms
• Geometric topology (object)
• Smith conjecture
• Centered tree
• Tameness conjecture
• Virtually Haken conjecture
• Solid Klein bottle
• P²-irreducible
• Fox n-coloring
• Metacompact space
• Orthocompact space
• Supercompact space
• Pseudonormal space
• Collectionwise Hausdorff space
• Volterra space
• A-paracompact space
• Perfect space
• Door space
• Realcompact space
• Locally Hausdorff space
• Shrinking space
• Hemicompact space
• Bitopological space
• Slice genus
• Closed extension topology
• Resolvable space
• Submaximal space
• Dowker notation
• Extremely disconnected space
• Paranormal space
• Quasitopological space
• Nodec space
• Vanishing cycle
• Digital topology
• E8 manifold
• Hybrid topology
• Virtual knot
• Gromov's compactness theorem (topology)
• Locally connected space
• User:Naufana/Images
• Immersion (mathematics)
• Aspherical space
• Borel conjecture
• Isotopy invariant
• Homology manifold
• Ambient isotopy
• Regular isotopy
• Legendrian knot
• Bennequin number
• Dispersion point
• 5-manifold
• Countably generated
• Costa's minimal surface
• Moore space (topology)
• Semimetric space
• Dendroid (topology)
• Dendrite (mathematics)
• Leray-Hirsch theorem
• Weakly contractible
• Fixed point property
• Misner space
• Satellite knot
• Cable knot
• Odd number theorem
• Milnor conjecture (topology)
• Topologically stratified space
• Whitney conditions
• Kauffman polynomial
• Ribbon knot
• (-2, 3, 7) pretzel knot
• User:Dr Finkbottle
• Gordon-Luecke theorem
• Cerf theory
• Portal:Mathematics/Featured picture/2006 11