Klein Bottle Viewer
See a Klein bottle: a closed surface with only one side and no edge at all. It cannot sit in ordinary 3D space without passing through itself, so every model you can hold is an immersion, a version that crosses itself. This viewer draws the two best-known ones from their published formulas: the figure-8 immersion (a ring whose figure-8 cross-section makes a half-turn on the way round) and the classic bottle shape, whose neck goes in through the side. Turn it, tilt it, press Spin, or cut half of it away to look inside. On the figure-8 form you can show the circle where it passes through itself. The blue loop shows the one-sided test: the tool carries the surface normal once round that loop, and it comes back pointing the other way. It prints on US Letter or A4, filling the page or at a width and height you type. Download a PDF, a 300 DPI PNG, an SVG or a PNG of the view. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). The SVG holds the same shaded faces and lines as the PDF.
Surface
View
Size
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. Every point and face comes from the published Klein bottle formulas, worked out by our own code. You may use what you make here, including commercially, with no credit needed.
How to view and print a Klein bottle
- Pick a preset, or choose the figure-8 or the bottle shape.
- For the figure-8, set the ring radius r.
- Tick the cut-away, the self-intersection circle or the one-sided loop.
- Set the turn and the tilt, or press Spin.
- Choose the colours and fill the page or a width and height.
- Download the PDF and print at 100% (Actual size), or the PNG, SVG or view PNG.
What you can use it for
- Topology lessons: one-sided surfaces, no boundary, immersions
- Posters and prints of a Klein bottle
- Slides and worksheets on non-orientable surfaces
- Comparing the figure-8 and the bottle shape side by side
- Plotter and laser art from the SVG
- Reference drawings for glass, ceramic or 3D-printed models
The 14 presets
| Preset | Shape | Ring radius r | Cut away | Lines drawn | Shaded faces |
|---|---|---|---|---|---|
| Figure-8 Klein bottle (r = 3) | Figure-8 | 3 | no | – | 4,608 |
| Figure-8 with its self-intersection circle | Figure-8 | 3 | no | self-intersection circle | 4,608 |
| Figure-8, cut away to show the figure 8 | Figure-8 | 3 | yes | self-intersection circle | 2,304 |
| Figure-8 with the one-sided loop | Figure-8 | 3 | no | one-sided loop | 4,608 |
| Figure-8, wide ring (r = 4.5) | Figure-8 | 4.5 | no | – | 4,608 |
| Figure-8, tight ring (r = 2.3) | Figure-8 | 2.3 | no | self-intersection circle | 4,608 |
| Figure-8 from above | Figure-8 | 3 | no | self-intersection circle | 4,608 |
| Bottle shape | Bottle shape | – | no | – | 3,456 |
| Bottle shape, cut in half to show the inside | Bottle shape | – | yes | – | 1,728 |
| Bottle shape with the one-sided loop | Bottle shape | – | no | one-sided loop | 3,456 |
| Bottle shape from the side | Bottle shape | – | no | – | 3,456 |
| Figure-8 on a dark page | Figure-8 | 3 | no | self-intersection circle + one-sided loop | 4,608 |
| Bottle shape in black ink | Bottle shape | – | no | – | 3,456 |
| Real size: figure-8 in a 120 mm square | Figure-8 | 3 | no | self-intersection circle | 4,608 |
Worked out by the tool on US Letter.
One side: the normal walk
| Shape | Closed loop followed | Normal at the end · normal at the start | Sides |
|---|---|---|---|
| Figure-8, r = 2.3 | t from 0 to 2π while v goes from π/3 to 5π/3 | -1.000000 | one |
| Figure-8, r = 3 | t from 0 to 2π while v goes from π/3 to 5π/3 | -1.000000 | one |
| Figure-8, r = 4.5 | t from 0 to 2π while v goes from π/3 to 5π/3 | -1.000000 | one |
| Bottle shape | u from 0 to π at v = 3π/2 | -1.000000 | one |
Worked out by the tool: the unit normal is carried along the loop in 720 steps without ever flipping between steps. −1 means it comes back pointing the other way.
Where the figure-8 passes through itself
| Ring radius r | Circle radius | Circle length 2πr | Plane |
|---|---|---|---|
| 2.3 | 2.3 | 14.451 | z = 0 (the xy plane) |
| 3 | 3 | 18.850 | z = 0 (the xy plane) |
| 4.5 | 4.5 | 28.274 | z = 0 (the xy plane) |
| 6 | 6 | 37.699 | z = 0 (the xy plane) |
Both formulas, the r > 2 rule and the self-intersection circle are from Wikipedia’s Klein bottle page (read 13 November 2026), which credits Alling and Greenleaf (1969).
What prints
| Download | What is in it |
|---|---|
| The shaded surface as filled faces, with the self-intersection circle and the one-sided loop if ticked, the view in the footer and a check bar | |
| PNG 300 DPI | The same sheet as a picture |
| SVG | The same faces and lines |
| View PNG | The view at the turn and tilt you set, 1600 × 1600 pixels |
For paper models of solids, use Shape Nets; to cut the outline, SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| A Klein bottle in 3D, cut away, printed | Klein Bottle Viewer (this page) |
| A Möbius strip to cut and glue | Möbius Strip Maker |
| Drawings that cannot exist as solids | Impossible Shapes |
| Nets of cubes, prisms and pyramids | Shape Nets |
Other tools: the Fractal Generator draws fractals, and the Celtic Knot Generator knotwork.
More free tools
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Frequently asked questions
What is a Klein bottle?
A closed surface with only one side: there is no inside and no outside, and no edge where the surface stops. A traveller on it can come back to the start flipped upside down. It is named after Felix Klein, who first described it in 1882.
Why does it pass through itself?
In ordinary three-dimensional space a Klein bottle cannot be built without crossing itself, so every 3D model is an immersion: a version that passes through itself. In four dimensions it can be made without any crossing.
How does the tool show it has one side?
It follows a closed loop on the surface (the blue line) and carries the surface normal, the arrow standing straight out of the surface, along it in small steps. When it gets back to the start the normal points the opposite way: the readout shows start · end = −1. On a two-sided surface such as a sphere it would come back the same way.
What is the figure-8 immersion?
A ring with a figure-8 cross-section that turns half a turn as it goes round once. Its formula is x = (r + cos(t/2) sin v − sin(t/2) sin 2v) cos t, y = (r + cos(t/2) sin v − sin(t/2) sin 2v) sin t, z = sin(t/2) sin v + cos(t/2) sin 2v, with r above 2. It passes through itself along a circle of radius r.
Where does the figure-8 cross itself?
Along a single circle of radius r in the flat middle plane, where sin v is zero. We checked this by searching 360,000 points of the surface for places where two different parts meet: they all lie on that circle. Tick the red circle to see it.
Why is no crossing line drawn on the bottle shape?
Its neck does pass through the side, but the curve where it does so is not given by a simple formula, and the tool does not draw anything it has not worked out. Use the cut-away to see the neck going through.
How is a Klein bottle related to a Möbius strip?
Cutting the classic bottle shape in half along its plane of symmetry gives two Möbius strips that are mirror images of each other. The cut-away on the bottle shape uses that plane.
What is its Euler characteristic?
0, the same as a torus: the square it is glued from has one corner point, two edges and one face, and 1 − 2 + 1 = 0.
Is this the same as the impossible shapes tool?
No. Impossible shapes are drawings that cannot exist as solid objects. A Klein bottle is a real surface: it just needs four dimensions to avoid crossing itself.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every point, face, corner and edge is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.