Geometric topology | Manifolds | Topological spaces | Surfaces

Klein bottle

In topology, a branch of mathematics, the Klein bottle (/ˈklaɪn/) is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary. The concept of a Klein bottle was first described in 1882 by the German mathematician Felix Klein. (Wikipedia).

Klein bottle
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Round Klein Bottle (Large)

This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/3kIo

From playlist 3D printing

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Round Klein Bottle (Small)

This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/2p3Z

From playlist 3D printing

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A mirror paradox, Klein bottles and Rubik's cubes

The Mathologer puts the latest $2000 addition to his Klein bottle collection to work. A couple of first-ever fun mathematical stunts in this video. This video finishes with a puzzle for you to think about. We posted a video with the solution on 1 August 2015: https://youtu.be/ZMC61C5tigA

From playlist Recent videos

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How to make a Klein Bottle (in three dimensions) - Numberphile

Cliff Stoll on how he makes Klein Bottles from glass. More links & stuff in full description below ↓↓↓ More videos on Klein Bottles: http://bit.ly/KleinBottles ACME Klein Bottles: http://bit.ly/ACME_Klein Support us on Patreon: http://www.patreon.com/numberphile NUMBERPHILE Website: htt

From playlist Klein Bottles on Numberphile

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The Coca-Cola Klein Bottle - Numberphile

Cliff Stoll - the King of Klein Bottles - shows us more designs, including the Coke Klein Bottle... Full Klein Bottle Playlist: http://bit.ly/KleinBottles More links & stuff in full description below ↓↓↓ More videos with Cliff Stoll: http://bit.ly/Cliff_Videos Cliff's Klein Bottle shop:

From playlist Klein Bottles on Numberphile

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Your Klein Bottle is in the Post - Numberphile

Klein bottle enthusiast Cliff Stoll shows us his unique system for sending bottles to customers. More links & stuff in full description below ↓↓↓ Cliff's online store: https://www.kleinbottle.com More videos on our Klein Bottle playlist: http://bit.ly/KleinBottles Cliff Stoll playlist:

From playlist Klein Bottles on Numberphile

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Klein Bottles - Numberphile

Cliff Stoll is passionate about Klein Bottles. More links & stuff in full description below ↓↓↓ Don't miss the video about how he uses a robot to store 1,000 bottles UNDER his house... https://youtu.be/-k3mVnRlQLU More videos on Klein Bottles: http://bit.ly/KleinBottles ACME Klein Bottl

From playlist Klein Bottles on Numberphile

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The 17-Klein Bottle - Numberphile

17 Klein Bottles become 1 - featuring Cliff Stoll and the glasswork of Lucas Clarke. More links & stuff in full description below ↓↓↓ More Cliff videos: http://bit.ly/Cliff_Videos More Klein Bottle videos: http://bit.ly/KleinBottles Buy glassware from Cliff (if you dare): https://www.kl

From playlist Klein Bottles on Numberphile

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Hunt for the Elusive 4th Klein Bottle - Numberphile

Carlo Séquin on his search for the elusive "fourth type of Klein bottle". More videos on Klein Bottles: http://bit.ly/KleinBottles More links & stuff in full description below ↓↓↓ Carlo's paper: http://bit.ly/Carlo_Klein Also featuring Carlo: Mobius House https://youtu.be/iwo7JReFTeg Tor

From playlist Carlo Séquin on Numberphile

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Cutting a Klein Bottle in Half - Numberphile

Klein Bottle enthusiast Cliff Stoll uses a diamond saw on a glass Klein Bottle. More links & stuff in full description below ↓↓↓ More videos about his bottles: http://bit.ly/KleinBottles Cliff's site (selling bottles): http://bit.ly/ACME_Klein Support us on Patreon: http://www.patreon.c

From playlist Klein Bottles on Numberphile

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Klein Bottles with Cliff (extra footage)

Some extras with Cliff Stoll including his unique sales pitch - find the main videos at: http://bit.ly/KleinBottles Bottles under the house: https://youtu.be/-k3mVnRlQLU NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberph

From playlist Klein Bottles on Numberphile

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Super Bottle - Numberphile

Professor Carlo Séquin explains super bottles - and a super duper bottle. More Klein Bottle videos: http://bit.ly/KleinBottles More Carlo videos: http://bit.ly/carlo_videos More links & stuff in full description below ↓↓↓ Carlo Séquin is based at the University of California, Berkeley. S

From playlist Carlo Séquin on Numberphile

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Can you solve THE Klein Bottle Rubik's cube?

In this video I tell you about Klein bottle Rubik’s cubes, Torus Rubik's Cubes and Klein Quadric Rubik's cubes as an introduction to a whole new universe of twisty puzzles. Get your own Klein bottle Rubik’s cube, as well as more than 800 other topological twisty puzzles by downloading th

From playlist Recent videos

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Applied topology 4: An introduction to the torus and Klein bottle

Applied topology 4: An introduction to the torus and Klein bottle Abstract: We explain how the torus (the surface of the donut) is a 2-dimensional manifold that fits in 3D, whereas the Klein bottle is a 2-dimensional manifold that doesn't fit in 3D but does fit in 4D. This video accompan

From playlist Applied Topology - Henry Adams - 2021

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How to make a Klein Bottle from an old pair of jeans - Numberphile

More Klein bottle videos: http://bit.ly/KleinBottles More links & stuff in full description below ↓↓↓ Visit Cliff Stoll's online store: http://www.kleinbottle.com/ More Cliff videos: http://bit.ly/Cliff_Videos Numberphile is supported by the Mathematical Sciences Research Institute (MSR

From playlist Klein Bottles on Numberphile

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