Klein bottle
The Klein bottle iscertain non-orientable surface, i.e.surface (a two-dimensional topological space),which thereno distinction between"inside" and"outside" ofsurface. The Klein bottle was first described1882 byGerman mathematician Felix Klein. Itclosely related toMöbius stripembeddings ofprojective plane such as Boy's surface.
Picturebottle withhole inbottom. Now extendneck. Curveneck back on itself, insertthroughside ofbottle (a true Klein bottlefour dimensions would not require this step, but itnecessary when representingin three-dimensional Euclidean space),connecttohole inbottom.
Unlikedrinking glass, this object has no "rim" wheresurface stops abruptly. Unlikeballoon,fly can go fromoutside toinside without passing throughsurface (so there isn't really an "outside""inside").
Topologically,Klein bottle can be defined assquare [0,1] × [0,1]sides identified byrelations (0,y) ~ (1,y)0 ≤ y ≤ 1 (x,0) ~ (1-x,1)0 ≤ x ≤ 1, as infollowing diagram:
----> ^ ^ | | <----LikeMöbius strip,Klein bottle istwo-dimensional differentiable manifold whichnot orientable. UnlikeMöbius strip,Klein bottle isclosed manifold, meaningiscompact manifold without boundary. WhileMöbius strip can be embeddedthree-dimensional Euclidean space R3,Klein bottle cannot. It can be embeddedR4, however.
The Klein bottle can be constructed (inmathematical sense) by joiningedgestwo Möbius strips together, as described infollowing anonymous limerick:
- A mathematician named Klein
- ThoughtMöbius band was divine.
- Said he: "If you glue
- The edgestwo,
- You'll getweird bottle like mine."
External links
- Acme Klein Bottles - Actual Klein bottles! (Or at least 3D projectionsthem).
- Andrew Lipson's Mathematical LEGO Sculptures - Lego constructionsMöbius stripKlein bottle structures.
- Klein Bottle Images by John Sullivan
