Klein Stien

[Plane]

http://www.kleinbottle.com/drinking_mug_klein_bottle.htm


  You need a little math to know why this is funny.

•- Displacement 825 ml (28 fluid ounces) About half a pint inside, half a pint outside.
•- Actual volume 0.0 ml (0.000 fluid ounces) Zero inside, zero outside.

[Xavier_Onassis]

So it is an impractical drinking mug that costs $80.00?

What is it good for, other than as a weird conversation piece?

[Plane]

  It is actually a morph of a Klein bottle.

  A Klein bottle is a conceptual demonstration that a three dimensional object can have a connection between its inner and outer surface that causes it to have a single surface over its inner and outer parts.

   This is to a torus or a   coffee mug as a Mobius strip is to a paper sheet.


    When I was 14 or so I discovered these and made one of paper , the type I made was kinda fun for me , but I totally failed to explain it to anyone .

[Xavier_Onassis]

A Mr Spock would say, "Fascinating".

[Plane]

http://www.youtube.com/watch?v=REanBpPSViI



http://www.youtube.com/watch?v=zPqEEZa2Gis


  It can be hard to explain why a Mobius strip or a Klein  bottle is so fascinating.

    Can you see that a strip of paper that has one and only one edge is exceptional?
    That you can cut a strip of paper in half exactly and it will be a single piece of paper?
     That a strip of paper with one surface only is fascinating?

      How about in three dimensions an object that encloses no volume is the container of all the universe ?

[Plane]

http://www.youtube.com/watch?v=sRTKSzAOBr4

http://www.youtube.com/watch?v=aTZBO8Ql6rc

http://www.youtube.com/watch?v=GGlmppx-2M8

Best is last here.