Clifford Stoll is a man with 1000 klein bottles and only a crawl space to store them under. His solution is to build a robot to help him manage the inventory.
I also thought those wheels looked familiar.
He sells Klein bottles which have been sliced in half with a saw. The saw setup uses a Basic Stamp to control one of the motors. I think it's controlling the motor which pushes the bottle through the saw.
Oh my God, I want one of those Klien bottles. Preferably not sawn in half.
As a kid our neighbour was a maths teacher, she had made a Klien bottle out of cardboard. I found it infinitely fascinating but being only seven did not really understand the explanation of it.
So Loopy, are those two Mobius strips separate or are they intertwined?
What happens when you cut a Mobius strip down the middle? Or into thirds?
Is it possible to turn a car inner tube (Torus with a hole) inside out through it's plug hole?
These and other fascinating questions will not be answered here.
Oh yeah, I loved the plywood robot chassis.
Think about it, Heater -- they are separate with opposite twists.
And of course, if you cut a Mobius Stripe down the middle, you just get something twice as long with twice the twists. Cut in thirds, you get more length and more twists accordingly.
Is that profound? I am still trying to get a photo of the dog owner with four mutts on his motorscooter. The old guy is no longer around, but a new lady appeared with somewhat the same set up -- though one large pooch seems to lounge in a bicycle basket at the front with all four paws in the air.
Actually, homes in the San Francisco Bay Area commonly are required to have concrete in their crawl spaces for 'rat proofing' which prevents rodents from tunneling into the home. So lots of people might have good reason to acquire a crawl space 'fork lift' robot.
Much as I like Klein Bottles, I have yet to buy one as it seems nearly impossible to wash.
Loopy,
You should be less quick with the "think about" it openings. Generally when I say a thing I have thought about it already
In this case:
"if you cut a Mobius Stripe down the middle, you just get something twice as long with twice the twists"
Yep, quite so.
"Cut in thirds, you get more length and more twists accordingly."
No, you don't. Get some paper and try it. What you get is two separate linked loops, one twice as long as the other. The short loop is the middle of the original mobius strip. The long loop is the two outside thirds
"Is that profound"
Yep. It's a result that confounds most people. Until they do it.
Not sure how you are cutting into thirds. But your words are cutting indeed.
There is something in the shift of the topography from a Klein Bottle to a Mobius Strip that I feel is profound and I am trying to discover actually what it is. It is obvious that volume disappears, but not so obvious what stays.
Further cutting isn't quite as dramatic, but what you are saying indicates more strangeness. What occurs with 4, 5, 6 cuts and so on?
Yep. It's a result that confounds most people. Until they do it.
I was just cutting some Mobius strips.
Cutting one in half didn't surprise me much but when I went to cut one in thirds I thought I must have done something wrong. It took cutting a second strip in thirds to believe they end up in two pieces linked together.
I was about to point out Loopy's error but found you had already pointed this out.
I found it both profound and confounding.
Loopy,
A Mobius strip is a long thin "tape" joined end to end with a twist. You can cut it along it's length, one third of it's width in from one edge. That cut will go round once and come to the other side of the tape. Continue cutting around and you eventually meet where the cut started. Boom, the thing falls into two linked loops. One twice as long as the other.
Oddly enough I remember one day in primary school busy cutting Mobius strips this way and that. The cut at one third width I remembered the result of. The others I don't. And it hurts my brain to think about it now.
I'm going to have a restless night now trying to imagine cutting a Klien bottle along it's length in the same way.
@Duane,
Well done you did it. It is surprising isn't it?
Now here is a thing. If you try to cut it at one quarter the width what happens? Well you go around the length once and find yourself back at the start position but on the other side of the tape. Keep going and you go around again meeting with where you started the cut.
At this point you still only have two loops. A short "fat" one from out of the middle of the tape. And a long thing on from the outside. Same as the cut in thirds case!
So now you have to split that "fat" middle strip in two to get the quarter width strip.
Now what do we have?
I haven't done this one but my guess is we'd end up with one long loop with a full turn and two small mobius stips. The three loops will be linked together with the long link in the center.
Edit: And yes, I was very surprised. I had to do it a second time to convince myself I had done it correctly.
@Loopy, yes that's his website. There are more videos of him on YouTube talking about klein bottles, they're a real passion of his.
@All, I've cut a möbius strip in half and thirds. I was stunned by the outcome of cutting one in thirds. If you haven't done it, do it!
@Erco, Clifford Stoll has done an amazing number of things. Back in 1986 when he was a systems administrator at the Lawrence Berkeley National Laboratory, he set up a honeypot that led to the capture of hacker Markus Hess. Besides doing that he wrote a rather engaging book about it. Now he sets up Acme Klein bottles and as an aside builds a really cool robot. I'm not worthy.
Actually, homes in the San Francisco Bay Area commonly are required to have concrete in their crawl spaces for 'rat proofing' which prevents rodents from tunneling into the home. So lots of people might have good reason to acquire a crawl space 'fork lift' robot.
I was wondering why the crawl space had a floor!!!
Hi gang, Yes, that's me. And yes, those are Parallax wheels driving my forklift around. I built it about 4 or 5 years ago, and posted to this forum for help -- I'd used the odometery system along with a basic-stamp controller (later an Arduino) to semi-automate the system. Turned out to be much easier to run the lil' guy by radio-control. This, in turn, required five video cameras, each with its own transmitter (and all the transmitters and their harmonics had to be far from 2.4 GHz, to avoid stomping on the R/C & WiFi connections to the forklift. Answering a few questions: Yes, the concrete slab under our Oakland house is the result of an earthquake retrofit. The "rat-slab" cost an additional $4K (on top of the contractor's $25K for stemwall & foundation reinforcement). There's about 10 cubic yards of concrete down there, and oh, how the guys cursed when they had to smooth all that mud! I now keep the floor clean with a very beat-up Roomba ... when the Roomba runs out of battery, the forklift goes to the rescue, and brings it home to charge. The forklift robot is pretty simple - tilt motor is from a car window lift; up-down is a 12V Craftsman right-angle drill that bites into a foot long 1/4-20 threaded rod. The plywood lift section has a pair of 1/4-20 wing nuts that follow the lead screw. And the two Parallax drive wheels are controlled by a Parallax HB25, from a Futaba 2.4GHz spread spectrum radio control system. It has only 4 channels, which isn't enough, so I lashed together a 300MHz push-button controller to handle simple on/off commands. A 12V 8A lead acid wheelchair battery keeps things going; you'll see it (and two dead lead-acid batteries) providing counterweight at the tail of the forklift.
Surprisingly, a real problem is making the right pallets under the boxes. I have not come up with a good solution: I now glue 1x2 wood under each box, and the forklift can handle it. But removing the wood is a pain - good glue can't be removed without shredding the box, and weak glue lets the wood pallet fall off when the forklift is pushing boxes around. I've tried using rigid insulation, cardboard, plastic squares. Oh - the forks on the front of the forklift are simple corner braces from the hardware store. Someday I may weld a hinge into them, to allow them to fold up and save space. Originally, I had a line-following system: 16 led's in a row, pointing downwards. The basic stamp scanned through these, lighting them up, one at a time, left to right. A single photocell picked up reflected light. The brightest point told the stamp which Led was closest to the white line. This worked OK, but the neutral gray concrete as well as limited memory of the stamp made for difficulties; I finally tossed in the towel and went to simple R/C. For my simple purposes, that's all I need. One problem with those two Motor/Wheels is that the tires go flat after a month or two. In addition, the screws that hold the wheel sides (I think they're 1" long, 4/40 screws?) tend to loosen and fall out. (A bit of loctite has helped here). It's very difficult to put air into the tires, however, due to the direction of the valve stems. This, in turn, made for non-repeatable odometery (especially when you're carrying loads on the forklift)
And yes, I use an antique basic stamp to control my Klein bottle slicer. I have a diamond band saw, and I feed the glass bottle into the bandsaw using a home-brew linear motion table. The linear motion is generated by a long 8-32 screw, which is driven by a 12 volt drill. A Basic Stamp reads the rotation of the drill, and controls a linear power supply to drive the drill motor at about one revolution every couple seconds. It's really a kludge, but hey- I only slice a few dozen Klein bottles every few years.
Best wishes all around,-Cliff Stoll (a hazy Friday evening in Oakland)
Thanks for taking time to answer some of our question Cliff Stoll. It's great to have you here on Parallax's forum.
If you ever want to make your Roomba remote controlled I'd be happy to help.
I used a PlayStation 2 controller with my Roomba but it would be easy to modify the program to use a RC receiver. If your Roomba is one of the many with a serial port, the modification would be really easy. (I'm sure I could be persuaded to send you the programmed hardware in exchange for credit towards a Kline bottle.)
I have a set of those motors and wheels you used with your forklift robot. I'm not sure if your wheels have the earlier encoders or the latest ones. The latest version of the motor/wheel kit has a 36 tooth encoder which gives a robot pretty good precision when using odometry (assuming the tires are firm). There is a (rather boring) video on my Let's Make Robot's page showing the odometry in action.
I sure enjoyed the various Klein bottle videos you participated in on Numberphile. Thanks for stopping by. (You're of course always welcome here.)
Well, I went to YouTube and viewed a video of a fellow cutting a Mobius Strip into halves and thirds as I am too lazy to get out the materials and make a mess. (We live in armchair luxury these days)
It becomes obvious that the twist in the Mobius Strip determines the different outcomes for cutting it by 2, 3, 4, 5 and so on. Even number of cuts does progress as one might think and that may generate one rule, and Odd number of cuts generates a separate circumstance that leaves a center Mobius Strip of original character. There simply is an edge situation that lengthens the Mobius Strip and a center situation that creates a remnant that is the original length and original number of twist.
Please explore and maybe we can put forth a set of appropriate rules that predict what happens consistently as you try more and more cuts. I am looking for that insight, not just being thrilled by the surprise of what you think isn't what you actually get.
The topography is what beckons my interest. We live in a world with a mindset of inside and outside and more conventional visualizations of geometry, surface, and edge. But Klien Bottles and Mobius Strips doe exist and may have a place in nature.
Who knows what this might imply? It could model the magnetics of the core of the Earth or the Sun. I grew up when Plasma as a 4th state of matter was not in the text books. Just maybe the geometry of plasma conforms to this oddball stuff.
Loopy, you don't have paper, tape and scissors within arms reach of where you're sitting? If not you're not really lazy. I'm lazy enough to keep this stuff nearby so it was easy to try out the multiple cuts.
Since we live in a world with only three spacial dimensions a Klein bottle has to pass through its own surface. I thought Cliff Stoll's cloth Klein bottle with a zipper was a great way to show how the bottle is made from two Mobius strips.
The video should start at 7:15 when the two strips are described.
Edit: The whole video is worth watching but the part discribing how a Klein bottle is made from two Mobius strips starts at 7:15. Apparently linking to the cued up video doesn't work.
OMG, that was almost a shaggy dog story. But it is worth viewing as you will see a lot of wonderful things and get through everything that might be a bit dry in text form.
The video has this gentleman that has models of every concievable Klein bottle that ever was. But after saying there are only four kinds of Klein bottles, he holds back telling you what the elusive fourth Klein bottle is until he has a chance to take you down every blind alley there ever was.
And he even drags out the famous Klein stein.
In some ways, I don't want to agree with him about the fourth Klein bottle, but he seems to be the definitive source.
We do seem to have constructions from two mirror image Mobius strips that form valid Klein bottles and then something else from two similar Mobius strips. That is yet another opportunity to drag out the paper, scissors and tape.
We do seem to have constructions from two mirror image Mobius strips that form valid Klein bottles and then something else from two similar Mobius strips. That is yet another opportunity to drag out the paper, scissors and tape.
The half turns of two similar strips keep them from merging together. You get two side by side half turns.
In other words, do you either get a valid Klein bottle or you get a wider Mobius strip? Or do those half-turn refuse to go side-by-side? (Okaay, I really do need to get out the paper, scisors, and tape.)
I once had a huge collection of bottles, but I found a 12th step program that got rid of them ;-)
In other words, do you either get a valid Klein bottle or you get a wider Mobius strip? Or do those half-turn refuse to go side-by-side? (Okaay, I really do need to get out the paper, scisors, and tape.)
Yes, you need to get out power, scissors and tape and yes, those half turns refuse to go side by side.
HI,
I did indeed get out the tape, paper, and scissors. And I did confirm what happens when both twists are in the same direction and one tries to combine two same length Mobius strips. The twists prevent the union.
I just prefer to attempt to visualize before I build models. I can recall the visualization for thought while I am out and about. Some people build the models, follow all the procedures, but never gain any real insight. I am very much into wanting that visualization clarity.
++++++++++
The real question that I ponder is what good use may one put a Klein bottle or Mobius Strip to.
It seems that the Mobius Strip might create a fan belt that has twice the surface to extend wear, but the twist might create a different kind of wear that could shorten useful service. Much would depend on advancements in material science to optimize for the Mobius Strip.
I can seem to think of a real world need for a Klein bottle, but it does seem to want to be a fish or bug trap. I guess one would have to put a trap door in one to make it easy to empty. This conclusion is simply based on it seeming to want things to go in more easily than to go out, but that observation may apply more to liquids influenced by gravity than a fish or a bug that might just follow the wall to escape.
Another application for a Mobius Strip might be to transfer paint or ink. The drive roller would provide paint or ink to the outside surface while the inside surface is a friction drive, and the other roller would take up paint or ink on the inside surface. But I have my doubts there is any real need for all this complexity.
And the core of my frustrations is that designing devices with a twist in them is just something that I generally don't think about. And my gut reaction is that adding a twist is unnecessary. But maybe one day, I will have a break through to see the advantage of a twist. So far I just can't see any.
I can seem to think of a real world need for a Klein bottle, but it does seem to want to be a fish or bug trap.
. . . that observation may apply more to liquids influenced by gravity than a fish or a bug that might just follow the wall to escape.
I doubt Klein bottles would make good fish or bug traps but some would be worse than others. I think the shape we most commonly see wouldn't make a good trap.
In this bottle, I think the fish would find its way out easier than finding its way in.
I think this wine Klein bottle could be use as a trap.
At least the fish would have to look a bit before finding a way out.
I really like the look of Cliff Stoll's wine Klein bottles. The one pictured above looks really good but I think I like his short handled Klein wine bottle even more.
I think the short handled version would make a poorer fish trap than the long handled one.
Cliff Stoll doesn't appear to carry a good selection of fish traps but if one is willing to use a container with volume as the trap then The Acme Mystery Pyramid Teapot may be the best option.
Yes, I also think it's cheating to have more than one surface on a fish trap.
BTW, while taping together two similar Mobius strips may not work, taping together mirrored strips is easier said than done. I finally managed to get my two strips taped together but the end result looked hideous.
Mercy! Just got lost here for about a hour... What fun!Just an hour? You got off easy.
Here is a photo of a couple attempts I made to tape the edges of two Mobius strips together.
The object on the left is two Mobius strips with their edges beginning to be taped together. It obviously didn't produce a Klein bottle. The paper isn't flexible enough to allow me to continue taping the edges.
I tried to make a pair of Mobius strips shaped like the two strips which can be zipped together as shown in the Numberphile video. I had a bit more success with these other Mobius strips. The compound curve of the "handle" is hard to make with paper. The paper ends up collapsing on itself (apparently this also a tricky part to make in a glass Klein bottle).
There are examples of origami Klein bottles on the internet but I wanted to make one with paper scissors and tape. I want to start with two strips of paper which are each made into Mobius strips with a half twist as the ends are taped together. I then want to tape these two mirrored strips together to form a Klein bottle. I'll probably give this another try sometime in the future.
A pattern to make the necessary strips would need to have a lot of "darts" (I think that's the right term) in order for the paper to be formed into the complex curves of the Klein bottle.
I'm intrigued by Cliff Stoll's knitted Klein bottle.
My wife is an avid crocheter and I've helped her come up with patterns for various geometric shapes. I've been trying to think of how one could crochet a Klein bottle. I don't know if there are crochet stitches which produce the same stitch pattern on both sides of the resulting material. Of course one can't go around making two surfaced Klein bottles.
I may stick with the paper variety for now.
Well, I recently had a bit of breakthrough in the utility of a Klein bottle.
If one is allowed to just consider it a contain and put doors and windows in it, a cabin with a rather interesting convention ventilation system comes about.
Any sort of a trap would partially involve gravity or the addition of other appendages to the interior surface of the neck to work well. And one would likely need a door to remove whatever is trapped.
I am enjoying exploring the issue of real-world functionality of the shape. The geometry just may have been ignored up until now due to its complexity.
+++++++++
BTW - Linguistically and logically one isn't permitted to use the plural of surface when discussing a Klein bottle. Just another odd tid-bit.
@Duane: That Klein hat is hysterical.
@CliffS: Thanks for checking in! We all enjoyed your energy, creativity and passion, all which came through loud & clear in your video. Welcome to the forums, and come back to visit!
@CliffS: Thanks for checking in! We all enjoyed your energy, creativity and passion, all which came through loud & clear in your video. Welcome to the forums, and come back to visit!
Comments
He sells Klein bottles which have been sliced in half with a saw. The saw setup uses a Basic Stamp to control one of the motors. I think it's controlling the motor which pushes the bottle through the saw.
That might surprise some, but that's what happens.
As a kid our neighbour was a maths teacher, she had made a Klien bottle out of cardboard. I found it infinitely fascinating but being only seven did not really understand the explanation of it.
So Loopy, are those two Mobius strips separate or are they intertwined?
What happens when you cut a Mobius strip down the middle? Or into thirds?
Is it possible to turn a car inner tube (Torus with a hole) inside out through it's plug hole?
These and other fascinating questions will not be answered here.
Oh yeah, I loved the plywood robot chassis.
And of course, if you cut a Mobius Stripe down the middle, you just get something twice as long with twice the twists. Cut in thirds, you get more length and more twists accordingly.
Is that profound? I am still trying to get a photo of the dog owner with four mutts on his motorscooter. The old guy is no longer around, but a new lady appeared with somewhat the same set up -- though one large pooch seems to lounge in a bicycle basket at the front with all four paws in the air.
http://www.kleinbottle.com/klein_bottle_hats.htm
Actually, homes in the San Francisco Bay Area commonly are required to have concrete in their crawl spaces for 'rat proofing' which prevents rodents from tunneling into the home. So lots of people might have good reason to acquire a crawl space 'fork lift' robot.
Much as I like Klein Bottles, I have yet to buy one as it seems nearly impossible to wash.
You should be less quick with the "think about" it openings. Generally when I say a thing I have thought about it already
In this case:
"if you cut a Mobius Stripe down the middle, you just get something twice as long with twice the twists"
Yep, quite so.
"Cut in thirds, you get more length and more twists accordingly."
No, you don't. Get some paper and try it. What you get is two separate linked loops, one twice as long as the other. The short loop is the middle of the original mobius strip. The long loop is the two outside thirds
"Is that profound"
Yep. It's a result that confounds most people. Until they do it.
There is something in the shift of the topography from a Klein Bottle to a Mobius Strip that I feel is profound and I am trying to discover actually what it is. It is obvious that volume disappears, but not so obvious what stays.
Further cutting isn't quite as dramatic, but what you are saying indicates more strangeness. What occurs with 4, 5, 6 cuts and so on?
I was just cutting some Mobius strips.
Cutting one in half didn't surprise me much but when I went to cut one in thirds I thought I must have done something wrong. It took cutting a second strip in thirds to believe they end up in two pieces linked together.
I was about to point out Loopy's error but found you had already pointed this out.
I found it both profound and confounding.
A Mobius strip is a long thin "tape" joined end to end with a twist. You can cut it along it's length, one third of it's width in from one edge. That cut will go round once and come to the other side of the tape. Continue cutting around and you eventually meet where the cut started. Boom, the thing falls into two linked loops. One twice as long as the other.
Oddly enough I remember one day in primary school busy cutting Mobius strips this way and that. The cut at one third width I remembered the result of. The others I don't. And it hurts my brain to think about it now.
I'm going to have a restless night now trying to imagine cutting a Klien bottle along it's length in the same way.
@Duane,
Well done you did it. It is surprising isn't it?
Now here is a thing. If you try to cut it at one quarter the width what happens? Well you go around the length once and find yourself back at the start position but on the other side of the tape. Keep going and you go around again meeting with where you started the cut.
At this point you still only have two loops. A short "fat" one from out of the middle of the tape. And a long thing on from the outside. Same as the cut in thirds case!
So now you have to split that "fat" middle strip in two to get the quarter width strip.
Now what do we have?
I haven't done this one but my guess is we'd end up with one long loop with a full turn and two small mobius stips. The three loops will be linked together with the long link in the center.
Edit: And yes, I was very surprised. I had to do it a second time to convince myself I had done it correctly.
@All, I've cut a möbius strip in half and thirds. I was stunned by the outcome of cutting one in thirds. If you haven't done it, do it!
@Erco, Clifford Stoll has done an amazing number of things. Back in 1986 when he was a systems administrator at the Lawrence Berkeley National Laboratory, he set up a honeypot that led to the capture of hacker Markus Hess. Besides doing that he wrote a rather engaging book about it. Now he sets up Acme Klein bottles and as an aside builds a really cool robot. I'm not worthy.
Actually, homes in the San Francisco Bay Area commonly are required to have concrete in their crawl spaces for 'rat proofing' which prevents rodents from tunneling into the home. So lots of people might have good reason to acquire a crawl space 'fork lift' robot.
I was wondering why the crawl space had a floor!!!
Surprisingly, a real problem is making the right pallets under the boxes. I have not come up with a good solution: I now glue 1x2 wood under each box, and the forklift can handle it. But removing the wood is a pain - good glue can't be removed without shredding the box, and weak glue lets the wood pallet fall off when the forklift is pushing boxes around. I've tried using rigid insulation, cardboard, plastic squares. Oh - the forks on the front of the forklift are simple corner braces from the hardware store. Someday I may weld a hinge into them, to allow them to fold up and save space. Originally, I had a line-following system: 16 led's in a row, pointing downwards. The basic stamp scanned through these, lighting them up, one at a time, left to right. A single photocell picked up reflected light. The brightest point told the stamp which Led was closest to the white line. This worked OK, but the neutral gray concrete as well as limited memory of the stamp made for difficulties; I finally tossed in the towel and went to simple R/C. For my simple purposes, that's all I need. One problem with those two Motor/Wheels is that the tires go flat after a month or two. In addition, the screws that hold the wheel sides (I think they're 1" long, 4/40 screws?) tend to loosen and fall out. (A bit of loctite has helped here). It's very difficult to put air into the tires, however, due to the direction of the valve stems. This, in turn, made for non-repeatable odometery (especially when you're carrying loads on the forklift)
And yes, I use an antique basic stamp to control my Klein bottle slicer. I have a diamond band saw, and I feed the glass bottle into the bandsaw using a home-brew linear motion table. The linear motion is generated by a long 8-32 screw, which is driven by a 12 volt drill. A Basic Stamp reads the rotation of the drill, and controls a linear power supply to drive the drill motor at about one revolution every couple seconds. It's really a kludge, but hey- I only slice a few dozen Klein bottles every few years.
Best wishes all around,-Cliff Stoll (a hazy Friday evening in Oakland)
If you ever want to make your Roomba remote controlled I'd be happy to help.
I used a PlayStation 2 controller with my Roomba but it would be easy to modify the program to use a RC receiver. If your Roomba is one of the many with a serial port, the modification would be really easy. (I'm sure I could be persuaded to send you the programmed hardware in exchange for credit towards a Kline bottle.)
I have a set of those motors and wheels you used with your forklift robot. I'm not sure if your wheels have the earlier encoders or the latest ones. The latest version of the motor/wheel kit has a 36 tooth encoder which gives a robot pretty good precision when using odometry (assuming the tires are firm). There is a (rather boring) video on my Let's Make Robot's page showing the odometry in action.
I sure enjoyed the various Klein bottle videos you participated in on Numberphile. Thanks for stopping by. (You're of course always welcome here.)
It becomes obvious that the twist in the Mobius Strip determines the different outcomes for cutting it by 2, 3, 4, 5 and so on. Even number of cuts does progress as one might think and that may generate one rule, and Odd number of cuts generates a separate circumstance that leaves a center Mobius Strip of original character. There simply is an edge situation that lengthens the Mobius Strip and a center situation that creates a remnant that is the original length and original number of twist.
Please explore and maybe we can put forth a set of appropriate rules that predict what happens consistently as you try more and more cuts. I am looking for that insight, not just being thrilled by the surprise of what you think isn't what you actually get.
The topography is what beckons my interest. We live in a world with a mindset of inside and outside and more conventional visualizations of geometry, surface, and edge. But Klien Bottles and Mobius Strips doe exist and may have a place in nature.
Who knows what this might imply? It could model the magnetics of the core of the Earth or the Sun. I grew up when Plasma as a 4th state of matter was not in the text books. Just maybe the geometry of plasma conforms to this oddball stuff.
Since we live in a world with only three spacial dimensions a Klein bottle has to pass through its own surface. I thought Cliff Stoll's cloth Klein bottle with a zipper was a great way to show how the bottle is made from two Mobius strips.
The video should start at 7:15 when the two strips are described.
Edit: The whole video is worth watching but the part discribing how a Klein bottle is made from two Mobius strips starts at 7:15. Apparently linking to the cued up video doesn't work.
The video has this gentleman that has models of every concievable Klein bottle that ever was. But after saying there are only four kinds of Klein bottles, he holds back telling you what the elusive fourth Klein bottle is until he has a chance to take you down every blind alley there ever was.
And he even drags out the famous Klein stein.
In some ways, I don't want to agree with him about the fourth Klein bottle, but he seems to be the definitive source.
We do seem to have constructions from two mirror image Mobius strips that form valid Klein bottles and then something else from two similar Mobius strips. That is yet another opportunity to drag out the paper, scissors and tape.
We do seem to have constructions from two mirror image Mobius strips that form valid Klein bottles and then something else from two similar Mobius strips. That is yet another opportunity to drag out the paper, scissors and tape.
The half turns of two similar strips keep them from merging together. You get two side by side half turns.
I once had a huge collection of bottles, but I found a 12th step program that got rid of them ;-)
Yes, you need to get out power, scissors and tape and yes, those half turns refuse to go side by side.
I did indeed get out the tape, paper, and scissors. And I did confirm what happens when both twists are in the same direction and one tries to combine two same length Mobius strips. The twists prevent the union.
I just prefer to attempt to visualize before I build models. I can recall the visualization for thought while I am out and about. Some people build the models, follow all the procedures, but never gain any real insight. I am very much into wanting that visualization clarity.
++++++++++
The real question that I ponder is what good use may one put a Klein bottle or Mobius Strip to.
It seems that the Mobius Strip might create a fan belt that has twice the surface to extend wear, but the twist might create a different kind of wear that could shorten useful service. Much would depend on advancements in material science to optimize for the Mobius Strip.
I can seem to think of a real world need for a Klein bottle, but it does seem to want to be a fish or bug trap. I guess one would have to put a trap door in one to make it easy to empty. This conclusion is simply based on it seeming to want things to go in more easily than to go out, but that observation may apply more to liquids influenced by gravity than a fish or a bug that might just follow the wall to escape.
Another application for a Mobius Strip might be to transfer paint or ink. The drive roller would provide paint or ink to the outside surface while the inside surface is a friction drive, and the other roller would take up paint or ink on the inside surface. But I have my doubts there is any real need for all this complexity.
And the core of my frustrations is that designing devices with a twist in them is just something that I generally don't think about. And my gut reaction is that adding a twist is unnecessary. But maybe one day, I will have a break through to see the advantage of a twist. So far I just can't see any.
. . . that observation may apply more to liquids influenced by gravity than a fish or a bug that might just follow the wall to escape.
I doubt Klein bottles would make good fish or bug traps but some would be worse than others. I think the shape we most commonly see wouldn't make a good trap.
In this bottle, I think the fish would find its way out easier than finding its way in.
I think this wine Klein bottle could be use as a trap.
At least the fish would have to look a bit before finding a way out.
I really like the look of Cliff Stoll's wine Klein bottles. The one pictured above looks really good but I think I like his short handled Klein wine bottle even more.
I think the short handled version would make a poorer fish trap than the long handled one.
Cliff Stoll doesn't appear to carry a good selection of fish traps but if one is willing to use a container with volume as the trap then The Acme Mystery Pyramid Teapot may be the best option.
Yes, I also think it's cheating to have more than one surface on a fish trap.
BTW, while taping together two similar Mobius strips may not work, taping together mirrored strips is easier said than done. I finally managed to get my two strips taped together but the end result looked hideous.
Here is a photo of a couple attempts I made to tape the edges of two Mobius strips together.
The object on the left is two Mobius strips with their edges beginning to be taped together. It obviously didn't produce a Klein bottle. The paper isn't flexible enough to allow me to continue taping the edges.
I tried to make a pair of Mobius strips shaped like the two strips which can be zipped together as shown in the Numberphile video. I had a bit more success with these other Mobius strips. The compound curve of the "handle" is hard to make with paper. The paper ends up collapsing on itself (apparently this also a tricky part to make in a glass Klein bottle).
There are examples of origami Klein bottles on the internet but I wanted to make one with paper scissors and tape. I want to start with two strips of paper which are each made into Mobius strips with a half twist as the ends are taped together. I then want to tape these two mirrored strips together to form a Klein bottle. I'll probably give this another try sometime in the future.
A pattern to make the necessary strips would need to have a lot of "darts" (I think that's the right term) in order for the paper to be formed into the complex curves of the Klein bottle.
I'm intrigued by Cliff Stoll's knitted Klein bottle.
My wife is an avid crocheter and I've helped her come up with patterns for various geometric shapes. I've been trying to think of how one could crochet a Klein bottle. I don't know if there are crochet stitches which produce the same stitch pattern on both sides of the resulting material. Of course one can't go around making two surfaced Klein bottles.
I may stick with the paper variety for now.
If one is allowed to just consider it a contain and put doors and windows in it, a cabin with a rather interesting convention ventilation system comes about.
Any sort of a trap would partially involve gravity or the addition of other appendages to the interior surface of the neck to work well. And one would likely need a door to remove whatever is trapped.
I am enjoying exploring the issue of real-world functionality of the shape. The geometry just may have been ignored up until now due to its complexity.
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BTW - Linguistically and logically one isn't permitted to use the plural of surface when discussing a Klein bottle. Just another odd tid-bit.
@CliffS: Thanks for checking in! We all enjoyed your energy, creativity and passion, all which came through loud & clear in your video. Welcome to the forums, and come back to visit!
Amen!