MoxI is a solid Klein bottle, the genus one nonorientable handlebody. Here we see it with some typical curves on
look a rough view
when any someone who is asked to comment about: how many surface’s shapes are there? he or she could tell you this: if only closed surfaces are considered, there’re two types:
- non orientable
Orientable are the sphere (the 2-sphere), the torus, double torus, etc… But non orientable are the projective plane: an abstract simple construction given by sewing a mobiusband with a 2-disk, , by its similar border. The Kleinbottle is the surface got from sewing two mobiusbands along with their boundaries.
Let’s abstract with the symbols to the orientable surface of -genus and by the non orientable surface of -genus.
With these we have an algebra of sets that looks like that
where we have abstracted: , for a punctured surface , a closed surface with a open disk removed. And for sewing in the border
Even to them who know those ideas come into surprise when they are told that there’s no only one three dimensional shape: is the only three dimensional manifolds in math?
Even if you think and say that is another together with others little more easy to imagine.
Would you like to see a sample of the great variety of them? urgently follow this link here please!
This link gonna give you a rough idea of several types of 3-spaces looking a little more close at them,… some are now celebrated in 3d-topological technology…
Mö x I
The picture on the right is representation and a description of its parts of a 3d chunk, it is the trivial -bundle over the möbius-strip, . It is useful to determine which 3d spaces are non orientable.
A 3d-space is non orientable if it has a simple closed curve whose 3d regular neighborhood is homeomorphic to the model of the picture.
If we denote by the core of the möbius strip, you can deduce for the curve (the black pearled one) that its tangent bundle can’t be embedded in the tangent bundle of or in any other tangent bundle of an orientable one.
Remember is the Klein bottle. Let’s the image talks by itself.
Ah, il nome di questo grande pezzo è la bottiglia solida di Klein.
In math there is a geometric-topologic construction which is a source of fun and surprises. It is surface. It is a two dimensional object which can be embed in the three dimensional euclidean space, . It is a model to illustrate the concept of twisted bundle. In fact an -bundle, an interval bundle over the circle. The möbius band or möbius strip is the principal element to construct all the non orientable surfaces: the projective plane; the Klein bottle; ; ,… etc.
What happens if we remove the core central simple closed curve? look at the 2nd image.
I had spended many eons to popularize the symbol
for her. Isn’t this is beautiful?
The möbius band is employed to manufature many exotic non orientable three dimensional spaces. By the record, talking about 3d spaces: isn’t the only one.
For example is a solid Klein bottle and an innocent quiz is: can you see what is the bounding surface of this -bundle?
For a bundle of additional properties check this link
The images were rendered in Mathematica v.5
La siguiente figura te explica como ver una construcción tridimensional que involucra a la banda…
Mö x I
Aquí estamos usando para la superficie de género dos no orientable,,