Recently i found a website with good historical information about the contributions of Gauss to Analysis situs (the old term for topology). The site is in german so i made a google translate to English on it. The site contains the following remark by Möbius on a certain oral communication of Gauss:

"One can easily visualize a double ring by cutting out a sheet of paper in the form of a cross, and then the ends FH and F'H '(see figure) of the pair of opposing arms approximately above the initial plane Of the cross and the ends BD and B'D 'of the other pair below this plane, the latter still possesses the remarkable property (after an oral communication by Gauss; what led Gauss to consider this surface, is unknown to me), that from any four points P, Q, R, S following each other on their perimeters, the first one is connected to the third and the second to the fourth by two lines PTT'R, and QUU'S, which lie in the surface itself and yet do not intersect one another - as would always happen if the surface had a basic form of the first class (for example, a Cylinder, C.H.). "

The site, whose link is here: http://www.solidaritaet.com/ibykus/1997/1/gauss.htm, also includes an a certain pic.

So, is this surface a kind of Möbius strip?; i checked it and it's not the classical "twisted" mobius strip - in fact i'm not sure at all this is a kind of mobius strip (i checked it). But i certainly think this surface implies some basic topological ideas introduced by Gauss. So what are these topological ideas?

Recently I found a website with good historical information about the contributions of Gauss to Analysis situs (the old term for topology). The site is in German so I made a Google translate to English on it. The site contains the following remark by Möbius on a certain oral communication of Gauss:

One can easily visualize a double ring by cutting out a sheet of paper in the form of a cross, and then the ends FH and F'H '(see figure) of the pair of opposing arms approximately above the initial plane Of the cross and the ends BD and B'D 'of the other pair below this plane, the latter still possesses the remarkable property (after an oral communication by Gauss; what led Gauss to consider this surface, is unknown to me), that from any four points P, Q, R, S following each other on their perimeters, the first one is connected to the third and the second to the fourth by two lines PTT'R, and QUU'S, which lie in the surface itself and yet do not intersect one another - as would always happen if the surface had a basic form of the first class (for example, a Cylinder, C.H.).

The site, whose link is here: http://www.solidaritaet.com/ibykus/1997/1/gauss.htm, also includes an a certain pic.

So, is this surface a kind of Möbius strip?; I checked it and it's not the classical "twisted" Möbius strip - in fact I'm not sure at all this is a kind of Möbius strip (I checked it). But I certainly think this surface implies some basic topological ideas introduced by Gauss. So what are these topological ideas?

Recently i found a website with good historical information about the contributions of Gauss to Analysis sytus (the old term for topology). The site is in german so i made a google translate to english on it. The site contains the following remark by Mobius on a certain oral communication of Gauss:

"One can easily visualize a double ring by cutting out a sheet of paper in the form of a cross, and then the ends FH and F'H '(see figure) of the pair of opposing arms approximately above the initial plane Of the cross and the ends BD and B'D 'of the other pair below this plane, the latter still possesses the remarkable property (after an oral communication by Gauss; what led Gauss to consider this surface, is unknown to me), that from any four points P, Q, R, S following each other on their perimeters, the first one is connected to the third and the second to the fourth by two lines PTT'R, and QUU'S, which lie in the surface itself and yet do not intersect one another - as would always happen if the surface had a basic form of the first class (for example, a Cylinder, C.H.). "

The site, whose link is here: http://www.solidaritaet.com/ibykus/1997/1/gauss.htm, also includes an a certain pic.

So, is this surface a kind of mobius strip?; i checked it and it's not the classical "twisted" mobius strip - in fact i'm not sure at all this is a kind of mobius strip (i checked it). But i certainly think this surface implies some basic topological ideas introduced by Gauss. So what are these topological ideas?

Recently i found a website with good historical information about the contributions of Gauss to Analysis situs (the old term for topology). The site is in german so i made a google translate to English on it. The site contains the following remark by Möbius on a certain oral communication of Gauss:

"One can easily visualize a double ring by cutting out a sheet of paper in the form of a cross, and then the ends FH and F'H '(see figure) of the pair of opposing arms approximately above the initial plane Of the cross and the ends BD and B'D 'of the other pair below this plane, the latter still possesses the remarkable property (after an oral communication by Gauss; what led Gauss to consider this surface, is unknown to me), that from any four points P, Q, R, S following each other on their perimeters, the first one is connected to the third and the second to the fourth by two lines PTT'R, and QUU'S, which lie in the surface itself and yet do not intersect one another - as would always happen if the surface had a basic form of the first class (for example, a Cylinder, C.H.). "

The site, whose link is here: http://www.solidaritaet.com/ibykus/1997/1/gauss.htm, also includes an a certain pic.

So, is this surface a kind of Möbius strip?; i checked it and it's not the classical "twisted" mobius strip - in fact i'm not sure at all this is a kind of mobius strip (i checked it). But i certainly think this surface implies some basic topological ideas introduced by Gauss. So what are these topological ideas?