I need a larger Klein Bottle for very cool science project.
One medium sized Klein Bottle and one super small, probably the smallest Klein Bottle in the world. Both Klein Bottles purchased on eBay years ago but no longer available. I have also seen some Klein Jars / Klein Bottles on sale on amazon but often unavailable, not sure why.
We recently purchased a “Klein Bottle” Opener that Emily thinks may actually be a Klein “Bottle Opener”. I don’t know we’ll have to try it and see.
Also search online for ACME Klein Bottles at kleinbottle.com for many Klein products including apparently the worlds largest Klein Bottle. Shopping for cool science Klein products for purchase online, look for Klein Mug, Klein Bottle Decanter, Klein Bottle Hat, Klein Bottle Opener, Klein Bottle Pedal and some others I won’t mention.
Some information about Klein Bottles, Mobius Strips, Mobius Loop, etc.
kleinbottle.com In 1882, Felix Klein imagined sewing two Möbius Loops together to create a single sided bottle with no boundary. Its inside is its outside. It contains itself. Take a rectangle and join one pair of opposite sides — you’ll now have a cylinder. Now join the other pair of sides with a half-twist. That last step isn’t possible in our universe, sad to say. A true Klein Bottle requires 4-dimensions because the surface has to pass through itself without a hole. It’s closed and non-orientable, so a symbol on its surface can be slid around on it and reappear backwards at the same place.You can’t do this trick on a sphere, doughnut, or pet ferret — they’re orientable.
Klein Bottle Wikipedia:
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. Whereas 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 Klein bottle was first described in 1882 by the German mathematician Felix Klein. It may have been originally named the Kleinsche Fläche (“Klein surface”) and then misinterpreted as Kleinsche Flasche (“Klein bottle”), which ultimately may have led to the adoption of this term in the German language as well Wolfram MathWorld: The Klein bottle is a closed nonorientable surface of Euler characteristic 0 (Dodson and Parker 1997, p. 125) that has no inside or outside, originally described by Felix Klein (Hilbert and Cohn-Vossen 1999, p. 308). It can be constructed by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist, but can be physically realized only in four dimensions, since it must pass through itself without the presence of a hole. Its topology is equivalent to a pair of cross-caps with coinciding boundaries (Francis and Weeks 1999). It can be represented by connecting the side of a square in the orientations illustrated in the right figure above Water filled Klein Bottle, Filling a klein bottle with water.
The Klein bottle is a closed nonorientable surface of Euler characteristic 0 (Dodson and Parker 1997, p. 125) that has no inside or outside, originally described by Felix Klein (Hilbert and Cohn-Vossen 1999, p. 308). It can be constructed by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist, but can be physically realized only in four dimensions, since it must pass through itself without the presence of a hole. Its topology is equivalent to a pair of cross-caps with coinciding boundaries (Francis and Weeks 1999). It can be represented by connecting the side of a square in the orientations illustrated in the right figure above
Also check out youtube and websites for cool info and videos about klein bottlesNumberphile, Klein Bottles Numberphile, 1,000 Klein Bottles, Clifford Stoll, Cliff Stoll, Cliff Stoll’s Klein Bottle, 4th Klein Bottle, Klein Bottles with Cliff, Acme Klein Bottle, Cutting a Klein Bottle in Half, Cut Klein Bottles, Bottle Fluid Simulation, Klein Bottle Rubik’s Cube, How to Make a Klein Bottle, What are Klein Bottles,Mathologer,Mythologer Klein Bottle.
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
2×2 Tutorial Finished!
How To Solve 2×2 Rubik’s Cube: So Easy A 3 Year Old Can Do It:
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