The 4D Kleinbottle Model

It would be possible to represent a spatially ‘flat surface’ in 4D as a Kleinbottle, which could be seen as combined Mobius strips.

In this case, the Earth itself is a spatially without boundary and so is like a sphere, easiest to visualize a non-boundary geometry as non-euclidean in 3D, but is a 4D euclidean object.

Consider a flat disc, you walk across it and fall off the edge, this means the surface has a boundary. Now, consider a sphere, you walk across it and come back to the same spot, this means the surface has no boundary. A kleinbottle may represent a flat surface with no boundary. This is the case in four spatial dimensions, but we will only visualize it in three (2D representations of 3D from a screen, which is for representing 4D).

Imagine a 2D world, where you can only perceive in two dimensions. Now visualize this:

A Mobius strip, which has one boundary.

Now imagine a 2D observer looking at it, or basically, imagine the Mobius strip in 2D, it could appear something like an 8, where lines intersect. However, that’s only in your limited 2D view, in 3D, they don’t intersect; they cross underneath and meet with only one boundary, which can only be represented in 3D, but it still represents a 2D manifold.

And even more general, a kleinbottle intersecting itself:

But remember the case of the Mobius strip appearing to intersect itself but really crossing underneath as one boundary geometry, it’s an illusion of our 3D perception. In fact, a surface with no boundary may be interpreted as a spheroid or convex in 3D, since that is a basic no boundary surface, but it's best just to say it would be visualized as a surface with no boundary.

Now, consider a Kleinbottle that represents a Mobius strip with no boundary.

Any traversal will in fact meet itself since no boundary of any sort exists, but it is a 2D manifold, able to be represented in 4D Euclidean space.

It becomes apparent that the Kleinbottle is attached Mobius strips.

*There is no information available as to whether any compact manifold which is equipped with an intrinsically consistent riemannian metric may be embedded in euclidean space so that this metric is induced on it by the metric of the euclidean space. The simplest manifold which seemed to be a possible counterexample is a flat Klein bottle. An example of a flat Klein bottle embedded in euclidean 4- space is given here. The manifold of this example intersects itself. The example is offered to remove the strongest contender from the list of possible counterexamples. The embedding equations of the Klein bottle are x\ — cos v cos u1 #2 = cos v sin u, u u %z = 2 sin v cos — ; x4 = 2 sin v sin — • 2 2 The following identities are clearly satisfied : Xi(u, v + 2TT) = Xi(u, v), Xi(u + 2x, 2ir — v) = Xi(u, v). These are just the identifications of points which by definition of the Klein bottle convert the u, y-plane into a Klein bottle. The components of the metric tensor are £=1 , F = 0, G= l + 3 cos2 v, and these can be transformed into the more usual form 2=1 , 7 = 0, 5= 1 by an obvious transformation involving an elliptic function.*http://www.ams.org/journals/bull/1941-47-06/S0002-9904-1941-07501-4/S0002-9904-1941-07501-4.pdf*A true Klein Bottle lives in 4-dimensions. But every tiny patch of the Klein Bottle is 2-dimensional. In this sense, a Klein Bottle is a 2-dimensional manifold which can only exist in 4-dimensions!*

Alas, our universe has only 3 spatial dimensions, so even Acme's dedicated engineers can't make a true Klein Bottle.

A photograph of a stapler is a 2-dimensional immersion of a 3-dimensional stapler. The true stapler has been flattened into the flatland of the photo. In the same way, our glass Klein Bottles are 3-D immersions of the 4-D Klein Bottle. Acme's Klein Bottle is a 3-dimensional photograph of a "true" Klein Bottle.

A Klein Bottle cannot be embedded in 3 dimensions, but you can immerse it in 3-D. (An immersion may have self-intersections; Embeddings have no self-intersections. Neither an embedding nor an immersion has folds or cusps.)

We represent a Klein Bottle in glass by stretching the neck of a bottle through its side and joining its end to a hole in the base. Except at the side-connection (the nexus), this properly shows the shape of a 4-D Klein Bottle. And except at the nexus, any small patch follows the laws of 2-dimensional Euclidean geometry.

Contrast this with a corked bottle -- say, a wine bottle. It has two sides: inside and outside. You can't get from one to the other without drilling a hole or popping the top. Once uncorked, it has a lip which separates the inside from the outside. If you make the glass arbitrarily thin, that lip won't go away. It'll become more prominent. The lip divides one side of the bottle from the other. So an uncorked bottle is topologically the same as a disc ... it has two sides, separated by a boundary -- an edge.

But a Klein Bottle does not have an edge. It's boundary-free, and an ant can walk along the entire surface without ever crossing an edge. This is true of both theoretical Klein Bottles and our glass ones. And so, a Klein Bottle is one-sided.

A Klein Bottle has one hole. This, in turn, causes it to have one handle. The genus number of an object is the number of holes (well, it's more subtle than that, but I'm not allowed to tell you why). Other genus-1 objects include innertubes, bagels, wedding rings, and teacups. A wine bottle has no holes and so is genus 0. (The genus of a human being is difficult to define because it depends on what you consider a hole -- I'd estimate most people have a genus of 0 to 4, slightly higher when yawning. Pierce your ear and you'll increase your genus by one.)

As an alternative to buying an Acme Klein Bottle, you can save money by just memorizing this set of parametric equations, since it defines the surface of every Klein Bottle.:

x = cos(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))

y = sin(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))

z = -1*sin(u/2)*(sqrt_2+cos(v))+cos(u/2)*sin(v)*cos(v)

or in polynomial form:There would be no 'edge' or cross section, it would be a one sided 4D Euclidean surface.