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How can you model a Klein bottle, such as this one?
Kleine bottle

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There are most likely several methods, and they probably all have both drawbacks and advantages. I will show how to use the XYZ Math surface for this.

First of all, unless you have already done this, enable the Add Mesh: Extra Objects addon. Go to user preferences (CtrlAltU), click the Addons tab, then Add Mesh. Make sure the check box for Add Mesh: Extra Objects is ticked, then click Save user settings.
Enable extra objects

Then Add > Mesh > Extra Objects > Math Function > XYZ Math Surface.
This is what you will see
XYZ Math Surface

The default object created by this is a snail shell. To get the Klein bottle, enter these equations. Just copy and paste from here.

X equation:

$$-\frac{2}{15}\cos(u)\left(3\cos(v)-30\sin(u)+90\left(\cos(u)\right)^4\sin(u)\\ -60\left(\cos(u)\right)^6\sin(u)+5\cos(u)\cos(v)\sin (u)\right)$$

-2/15*cos(u)*(3*cos(v)-30*sin(u)+90*cos(u)**4*sin(u)-60*cos(u)**6*sin(u)+5*cos(u)*cos(v)*sin(u))

Y equation:

$$-\frac{1}{15}\sin(u)\left(3\cos(v)-3\left(\cos(u)\right)^2\cos(v)-48\left(\cos(u)\right)^4\cos(v)+48\left(\cos(u)\right)^6\cos(v)\\ -60\sin(u)+5\cos(u)\cos(v)\sin(u)-5\left(\cos(u)\right)^3\cos(v)\sin(u)\\ -80\left(\cos(u)\right)^5\cos(v)\sin(u)+80\left(\cos(u)\right)^7\cos(v)\sin(u)\right)$$

-1/15*sin(u)*(3*cos(v)-3*cos(u)**2*cos(v)-48*cos(u)**4*cos(v)+48*cos(u)**6*cos(v)-60*sin(u)+5*cos(u)*cos(v)*sin(u)-5*cos(u)**3*cos(v)*sin(u)-80*cos(u)**5*cos(v)*sin(u)+80*cos(u)**7*cos(v)*sin(u))

Z equation:

$$\frac{2}{15}\left(3+5\cos(u)\sin(u)\right)\sin(v)$$

2/15*(3+5*cos(u)*sin(u))*sin(v)

U min: 0
U max: pi or (3.141592653589793)

V min: 0
V max: 2*pi or (6.28318530718)

U steps and V steps control the number of vertices in each direction, the same as the number of segments and rings on a UV sphere.

If you enable U wrap, the opening of the bottle will be closed. You decide which you prefer. If you enable V wrap, some vertices will be lost, but I haven't been able to find where. I don't know at all what Close V does. It doesn't change either the appearance or the number of vertices and faces.

As you can see, these equations are very complex, and this has the drawback of making them difficult to modify. I will try to show you how to overcome that, using proportional editing and a scaling technique I learned from the answer to my own question in this thread: Making a curved tube thinner or thicker without scaling its overall size

Go into Edit mode, and select all. Remove doubles (W R). Deselect all and make sure Limit selection to visible is disabled.

  1. Make sure proportional editing (O) is set to Connected and proportional editing falloff (ShiftO) is set to smooth.

  2. Select the vertices you want to modify.
    Selection

  3. If you want to make any part of the mesh bigger or smaller, Shrink Fatten (AltS) is the tool. Scroll the mouse wheel to change the size of the sphere of influence of the proportional editing.
    Fatten

  4. If you want to move any part of the bottle, keep proportional to its aforementioned settings, select the parts you want to move and drag them to where you want them. Like so:
    Move

  5. Repeat steps 2 through 4 until the bottle is to your liking.

You're done.

N.B. A Klein bottle is self intersecting, and therefore won't work as a fluid container with the current fluid simulation of Blender.

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  • $\begingroup$ Very nice, now you only have to show us how to ge consistent normals ;) $\endgroup$ – maddin45 Nov 17 '14 at 10:28
  • $\begingroup$ @maddin45 I have tried and tried, to no avail. :( $\endgroup$ – user7952 Nov 17 '14 at 10:32
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    $\begingroup$ It was a joke ;) There is no way to get consistent normals on a Kleine bottle (or even a Mobius strip) because at some point the inside will become the outside. I might work if you first give the surface some thickness by applying a solidify modifier, because then the surface would flip twice. I haven't tried it though. $\endgroup$ – maddin45 Nov 17 '14 at 10:41
  • $\begingroup$ I'm working on an animation where I'll use a cheat. Using an opaque bottle containing a fluid outflow and making the vertices by the opening an inflow. This way, when the bottle is standing, the fluid will disappear into the outflow, and when it's tilted the fluid will flow as if the bottle contained it. In theory at least. I'm not quite there yet. $\endgroup$ – user7952 Nov 17 '14 at 10:48
  • $\begingroup$ Just noticed that an inflow can't use vertex groups (hadn't actually come to that specific bit until now), so perhaps a fully transparent object as an inflow just at the opening and an outflow just beneath the inflow. And doesn't the solidify modifier also require consistent normals? At least I get very bad results when trying to solidify. Anyway, this is getting OT now, as this was about the actual modelling. I may have to post another question to explore this further. One that I can't answer myself. $\endgroup$ – user7952 Nov 17 '14 at 11:53
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Beside math, there can be a simpler way to model it:

  • Add Bezier Curve, do not make it cyclic, then tweak points;
  • Adjust Radius (Select one or more points, AltS)
  • Convert to mesh (AltC - Convert Mesh from Curve/Meta/Surf/Text)
  • Bridge the two open loops (W - Bridge Edge Loops)
  • Cut off the intersect faces: Mesh > Faces > Intersect, then delete them. (optional)

enter image description here enter image description here enter image description here enter image description here

P.S.:

The consistency of normals has to be broken in this case. It is very interesting, which means you cannot make thickness simply by applying one Solidify modifier. It actually isn't a 3-dimensional thing.

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  • $\begingroup$ A good one, no doubt. :) Upvote for that. $\endgroup$ – user7952 Nov 18 '14 at 12:33
  • $\begingroup$ That's a smart use of "Bridge Edge Loops". I'm sure I will borrow that technique for something someday. :-) $\endgroup$ – Mentalist Jan 10 '16 at 15:00
  • $\begingroup$ "Beside math, there are simpler ways..." I was getting all geared up to not believe you and then looked further, and did a quiet face-palm. That's really nice! To make a finite-thickness walled object you have to make two copies - one moving "out", one moving "in". The one that starts "out" will end in, and vice versa. Then delete the original and splice the two together. $\endgroup$ – uhoh Oct 7 '16 at 16:48
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Since it came up in the comments about the normals being a total mess after Remove Doubles is used because the mesh folds through itself I have posted an answer here on how to fix this.
Recently a tool named Intersect was added that can fix this issue by performing what is similar to a Boolean Intersect operation except that this tool works in Edit mode and can handle self-intersections.

The first image shows the messy mesh after Remove Doubles is used.

enter image description here

Now press spacebar and type Intersect and click that entry to activate the tool and change the menu entry to Self-Intersect.

enter image description here

Next use Remove Doubles again. This may require an adjustment to the Merge Distance.

Once that's done you can select everything and use Ctrl n to Recalculate Normals and it should look good.(aside from the quads meeting up with nGons.)

There are also some interior faces created at the junction which should be deleted.

Here's the result rendered with a glass shader and a bit of volume.

enter image description here

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  • $\begingroup$ And an upvote for that one too. $\endgroup$ – user7952 Nov 18 '14 at 12:34
  • $\begingroup$ Looks like this is rendered as a solid. The Klein Bottle is actually hollow. The messed-up normals are then a topological necessity, the Klein Bottle being famously non-orientable. So, the intersect tool actually doesn't the physically “right”, but mathematically wrong thing. $\endgroup$ – leftaroundabout Feb 27 '18 at 1:33

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