Mitsugi Ohno true Klein bottle

This 3D model posted on printables.com, is a copy of the Mitsugi Ohno true Klein bottle.

Here is the story about the “true Klein bottle”.

In 1892, mathematician Felix Klein first described the Klein bottle as an object that has a single surface. It is constructed with a thought experiment in which you imagine a bottle, and then loop the bottle’s neck around to pass through the bottle’s side and connect the top of the looped bottle neck to a hole in the bottom of the bottle. From a mathematical perspective, the neck of the bottle can't contact the surface of the bottle when it passes through the side of the bottle.

Mitsugi Ohno, of Kansas State University, was the first person to have created a true Klein bottle; that is a Klein bottle with a hole in the side.

Here is text copied from the Mitsugi Ohno Wikipedia page:

                In 1961, Kansas State University Professor Cardwell asked Mitsugi Ohno to construct a true glass Klein bottle, a one-sided figure formally described as “an enclosure continuous with its outer surface constructed by twisting a tube through an opening in the side of the tube and joining it to the other end".

Klein bottles had previously been made by other glass blowers, but their versions were sealed and did not have an open hole where the tube went through the side of the bottle . After several days of failing to make such a Klein bottle with an opening, he declared it impossible. However, the solution to the problem was revealed to Ohno in a dream and he rushed to his office to make it. It was the most difficult glass piece that he made in his long career, and the accomplishment of which he was most proud. His first successful version of a Klein bottle is on permanent display at Kansas State University's Student Union.

The company ACME Klein Bottles sells many sizes and shapes of very nice glass Klein bottles. They have Klein bottle expertise.

Each glass Klein bottle comes with ample documentation. Here's a quote from ACME Klein Bottles’ product documentation: "We represent it 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, this shows the shape of a true Klein Bottle." From their own admission, ACME Klein bottles don’t sell "true Klein Bottles"; all of their glass Klein bottles are sealed without an open hole in the side.

While at Kansas State University, Mitsugi Ohno made 25 true glass Klein bottles. As of this posting, as far as I can tell, the only other person that has made a true glass Klein bottle is Lucas Clarke.

The true Klein bottle is a remarkable 3D representation of a 4D object; however, I have met several people that are not convinced that the hole in the side is the best representation, therefore the following essay argues this point.

Essay on the true Klein bottle:

We are on a quest to determine the best 3D representation of the 4D object. What is determined to be the best is not entirely subjective. What follows are arguments for the true Klein bottle like the one made by Mitsugi Ohno. There are primarily 2 contenders for the best representation: 

1. The true Klein bottle that has a hole in the side.
2. A Klein bottle without a hole.

As envisioned by Felix Klein, the surface of the bottle’s neck can’t touch the surface of the bottle’s body. The Klein bottle can only exist in 4 spatial dimensions. Since we live in 3 spatial dimensions, it is hard for us to imagine how this would look.

Maybe it’s easier to imagine if we start the thought experiment in the 2D world. This illustration is an example of a 2D object that intersects itself. 

Let’s say a professor in this 2D world says that this object does not intersect with itself and that one of the paths curves over the other in the 3D to avoid the intersection. The following illustration shows what the object would look like to us in the 3D world.

However, the people in the 2D world can’t see the portion of the object that exists in 3D, so for them the object looks like the one shown here. 

Of course, it could be that its the other edge that goes into 3D space. In that case the 2D people would see the following figure.

We can extend this imagery for our 4D Klein bottle representation in our 3D world. Here is an illustration of our 3D Klein bottle where the 2 surfaces intersect.

Of course, the illustration doesn’t look correct because Felix Klein stated that the 2 surfaces can’t intersect. This is how the no-hole Klein bottle looks, but it clearly doesn’t look right because the 2 surfaces touch each other.

The following illustration shows our 3D Klein bottle where the neck enters 4D space and so becomes invisible to us.

The neck, while it exists in 4D, passes through the surface of the bottle without touching it and rejoins the neck on the inside of the bottle in 3D space.   This view is technically correct, because the two surfaces don’t touch, but it’s confusing to look at; you can’t see what’s going on inside.

The next illustration shows our 3D Klein bottle where the surface of the bottle enters 4D space, so it is invisible to us.  

We can see the neck because it remains in 3D space. It’s clear to us that the neck continues to the inside of the bottle and does not intersect the bottle’s surface.   If we made such a bottle in our 3D world, an observer might pretend that we actually made a 4D object by saying, “The surface is continuous, but you can’t see it because it exists in the 4th spatial dimension.” That statement is interesting, but, of course, it’s invalid because we can’t make objects in the 4th spatial dimension. We are simply attempting to make the best 3D representation of a theoretical 4D object.

The logical case for this being the best way to represent the 4D Klein bottle in our 3D world seems sound, yet there are still people that object to this hole, so let me present 2 more arguments for using the hole.

Argument 1:

The no-hole people like this design:

But if you remove the neck from the no-hole Klein bottle, you see the structure shown in the following illustration.  

The no-hole people’s Klein bottle has a hole in the surface, so presumably, the no-hole people are not objecting to the existence of a hole, but rather to the size of the hole. If the model is made in glass, it’s possible to see that there is a hole without removing the neck. Their argument for no-hole seems to be less solid than it originally appeared to be.

Argument 2:

Another test for evaluating the best 3D representation of the 4D Klein bottle is a thought experiment in which you travel along the surface of the Klein bottle to prove that it contains a single surface. If you imagine an ant crawling along the surface, or your finger tracing the surface, or a paint brush painting the surface, you can complete the circuit and get back to the starting point. The following figure shows a cross section of the 3D Klein bottle with a hole in the side.

Your ant, finger, or paintbrush can easily travel along the surface of the Klein bottle and get back to where it started. In doing so, travel proceeds along the neck of the bottle to the location where the ant, finger, or paintbrush passes through the hole. Without the hole, the ant, finger, or paintbrush are blocked and cannot travel through the side of the bottle.

Another design that is rarely discussed is where there are no holes at all, as shown in the following illustration.

With this 3D representation the travel paths are blocked both inside and outside the neck.

Sometimes the no-hole people argue that the whole is forbidden. They correctly point out that the 4D Klein bottle has no edges. Just like a sphere, if you travel along the surface of the 4D Klein bottle, you will never be blocked by a wall or stopped at an edge. You can travel in any direction continuously. The 3D representation with a hole doesn’t allow for this property:

• If you try continuous travel over the surface of the 3D Klein bottle that has a hole around the neck, travel eventually stops at the edge of the hole.

However, the 3D representation without a hole also fails this test:

• If you try continuous travel over the surface of the 3D Klein bottle that has no hole, travel is blocked at the intersection of the neck and the surface of the bottle.

I can think of only 2 advantages of the no-hole Klein bottle design:

1. If you are making the 3D Klein bottle out of glass, it’s much easier to make with no-hole.
2. You can fill a no-hole 3D Klein bottle with liquid. If you try adding liquid to a Klein bottle that has a hole around the bottle neck, the liquid pours out of the hole when you rotate the bottle.

Those 2 advantages are practical and have nothing to do with the topological arguments for the best representation.

Therefore I conclude that the best 3D representation of the 4D Klein bottle is with a hole in the side. And the best shape to use in this 3D model is that made by Mitsugi Ohno in 1961 with a hole in the side.

Printing suggestions:

This 3D model, like the original Mitsugi Ohno Klein bottles, contains his signature. If you have a 2 color printer, try painting his signature in blue. Alternatively, you can pause the print after his signature is printed and color it blue with a felt tip marker.

The provided PrusaSlicer *.3mf file is optimized for a better print by using: 

1. Paint-on seams. Paint-on seams are positioned under neck and on the back of the top part of the neck to make them less visible. Paint-on seams are also used on the inside of the body.
2. Paint-on supports. Paint-on supports are for the underside of the arched neck and the top of the hole in the side. Supports are not absolutely required for the top of the hole in the side, but are useful to stabilize the rest of the support.
3. Higher temperature. Higher temperature increases the transparency of the plastic. I used Voxel clear PETG at 270°F.
4. Turned off fan. Turning off the cooling fan increases transparency.
5. Thicker layers. Increasing the Layer Height from 0.2 to 0.3 increased the transparency.
6. Slower print speed. Slowing down the print speed by 25% by setting “Max Volumetric Speed” from the default of 12 mm³/s to 3 mm³/s dramatically increased the transparency.
7. 80% size. Print at 80% size so the *.stl model fits in your build volume and creates 2 layer wall thickness.

Do not build supports on the inside of the Klein bottle because they are impossible to remove. 

Note: I found that clear PETG is clearer than clear PLA.