Editing Klein bottle (section)

==Construction==
The following square is a [[fundamental polygon]] of the Klein bottle. The idea is to 'glue' together the corresponding red and blue 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 a self-intersecting Klein bottle.<ref name=":0" />

:[[Image:Klein Bottle Folding 1.svg]]
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, one would pass one end through the side of the cylinder. This creates a curve of self-intersection; this is thus an [[Immersion (mathematics)|immersion]] of the Klein bottle in the [[three-dimensional space]].

<gallery |="" align="center">
Image:Klein Bottle Folding 1.svg
Image:Klein Bottle Folding 2.svg
Image:Klein Bottle Folding 3.svg
Image:Klein Bottle Folding 4.svg
Image:Klein Bottle Folding 5.svg
Image:Klein Bottle Folding 6.svg
</gallery>

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 [[orientability|non-orientable]], as reflected in the one-sidedness of the immersion.

[[File:Science Museum London 1110529 nevit.jpg|thumb|right|150px|Immersed Klein bottles in the [[Science Museum (London)|Science Museum in London]]]]
[[Image:Acme klein bottle.jpg|thumb|150px|right|A hand-blown Klein bottle]]
The common physical model of a Klein bottle is a similar construction. The [[Science Museum (London)|Science Museum in London]] has a collection of hand-blown glass Klein bottles on display, exhibiting many variations on this topological theme. The bottles were made for the museum by Alan Bennett in 1995.<ref>{{cite web|archive-url=https://web.archive.org/web/20061128155852/http://www.sciencemuseum.org.uk/on-line/surfaces/new.asp|archive-date=2006-11-28 |url=http://www.sciencemuseum.org.uk/on-line/surfaces/new.asp|title=Strange Surfaces: New Ideas |publisher=Science Museum London }}</ref>

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 the intersection along the fourth dimension, out of the original three-dimensional space. A useful analogy is to consider a self-intersecting curve on the plane; self-intersections can be eliminated by lifting one strand off the plane.{{sfn|Alling|Greenleaf|1969}}

[[File:Klein bottle time evolution in xyzt-space.gif|thumb|[[Time evolution]] of a Klein figure in ''xyzt''-space]]
Suppose for clarification that we adopt time as that fourth dimension.  Consider how the figure could be constructed in ''xyzt''-space.  The accompanying illustration ("Time evolution...") shows one useful evolution of the figure.  At {{nowrap|1=''t'' = 0}} the wall sprouts from a bud somewhere near the "intersection" point.  After the figure has grown for a while, the earliest section of the wall begins to recede, disappearing like the [[Cheshire Cat]] but leaving its ever-expanding smile behind.  By the time the growth front gets to where the bud had been, there is 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.{{sfn|Alling|Greenleaf|1969}}

More formally, the Klein bottle is the [[Quotient space (topology)|quotient space]] described as the [[Square (geometry)|square]] [0,1] × [0,1] with sides identified by the relations {{nowrap|(0, ''y'') ~ (1, ''y'')}} for {{nowrap|0 ≤ ''y'' ≤ 1}} and {{nowrap|(''x'', 0) ~ (1 − ''x'', 1)}} for {{nowrap|0 ≤ ''x'' ≤ 1}}.