Space city with tunnel(Wormhole)

In space a city is build on a flat square.At first sight it is a normal city which contains building which are symmetrically built: a building left also occures on the right and viceversa.

But this is not a normal city. There is a very dangerous place or square in the city.In front of the builing with a roof in the form of a star( top of the print) there is a hole. Fortunatelly one has build a fence around the hole so that one can not fall in it. But let us see what happens if someone fallls into the hole. When one falls into the hole, after a while one notice that there is no bottom, it is a tunnel. The diameter of the tunnel decreases first, then increases back to normal. Then you see the end of the tunnel. When you leave the tunnel you see yhat you are back at your startingplace with the same square, the same fence, the same building with that strange roof. so you have entered a tunnel, walk through it but nevertheless arrived at your startingpoint. Now you can walk home because you recognize all the buildings, there are at exactly the same position and look exactly the same as before your tunnel-adventure. What you don't know is that you are arrived in another city, which is built on the backside of the spacesquare. until now it is not clear which kind of gravity-field is valid in the tunnel. In the beautiful wood-cut "Band van Möbius II ( Rode mieren )" by M.C.Escher, red ants are moving around on a band which has only one side and one border.

Schematic drawing of a Möbiusband

If we take one of these ants and let him move around in the upper-city, after a while the ant arrives at the tunnel. The change of the surfface of the sttreet and the tunnel is so gradual thaaat the ant does not notice it, so in terms of topology we can say that the ant is still on the same surface. The same is true at the end of the tunnel. which means that as a Möbius-band the spacecity has only one surface. If we further assume that in the center of the print the buildings and the tunnel fits perfectly (without any space between them, then the space-city has only one border.(the border of the spacesquare).

The difference with a Möbiusband is that the tunnel runs through the surface of the city. For aa long time there are other mathematical models which do the same like tthe Bottle of Klein.

Schematic drawing of a cross-section of a Bottle of Klein

A Bottle of Klein can be made out of glass as a 3-dimensionaal bottle.

As we now draw a cross-section of the print as suppose that the tunnel is completely elastic( this does not change the topological properties of the model), one becomes folllowing drawing:

Now we can place a cupola (dome) or a large soapbubble over the "under-city" and again make a cross-section the result is the cross-sektion of a Bottle of Klein.

This print of a space-city with tunnel has some correspondence with "worm-holes" which, according the some cosmological theories, could occure in real universe. If we represent the universe as a 2-dimensional elastic sheet and the material (stars etc.) as spheres on this sheet, the spheres causes introversions (I am not shore if this word is correct) or pits. If the material is heavier, the introversions will be deeper. These introversions have influences on the ways the way the light goes if it passes near such material.

For very compact material the introversion will be so deep thet the light will be attracted to it and will fall into the introversion and will never come out again. Then the object is called a black hole. If these is a connection between two such introversions , you get a tunnel, a "worm-hole". If you travel through such "worm-hole", you would be able to travel from one time-space into another. Then time-travelling would be possible. If we accept this model in place it in a "real" world we have science-fiction, and the print can be seen as an illustration of it. Then the "under-city" is a space in another time. But shall we ever be able to leave the tunnel and his "fatal attraction"??? (Perhaps this is a good titel for the print.)

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