Image:KleinBottle-01.png

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1 Summary
2 See also
3 Licensing
4 Parameterization
5 Mathematica source

Summary

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

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.

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}]

File links

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
Link (knot theory)
Knot group
Homotopy sphere
Nagata–Smirnov metrization theorem
Hyperbolic link
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
Split link
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
Algebraic link
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)
Unlink
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

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