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?

While you already accepted an answer, it seems not superfluous to add another one, in particular since you are implicitly asking for a better translation/understanding of the passage you quoted.

I will give you a translation here, and some comments of mine.

Beforehand, though, I think it is fair to say hat you seem to have been misled when you wrote

website with good historical information about the contributions of Gauss to Analysis situs

I think it is fair to say that this website gives good motivation, yet one should not say it gives "good historical information".

It does not meet the standards of professional historical writing.

Needless to say,

  • I am not blaming the author of the text on the website, who might perhaps have consciously written in a way that they just thought fit for the purpose, without even aspiring to produce a piece of professional historical writing,

  • I myself do not claim that what comes below is of professional quality; I also will focus more or less on your mathematical question, and there are some flaws in the below, too.

Yet to me it seems one should tell you that the text you link to, while nice, has its flaws.

The text you cite is nice, yet it can hardly be considered a scientific source; in parts, it is misleading, historically and mathematically, sometimes it is quite loosely crafted, and the German has stylistic flaws.

Literal copy-paste-quote of a superset1 of the text from the website you cited:

[...] Damit warf er [Gauß] die gesamte Newtonsche Mechanik in hohem Bogen zum Fenster hinaus, denn die Bewegung eines Körpers wird durch seine Lage im Raum und durch die gesamten ihn beeinflussenden anderen Körper erklärt und nicht nur durch die Anziehungs- und Abstoßungskräfte einer Zentralkraft.

Darüber hinaus behandelt Gauß in seinen Arbeiten zur Theorie der krummen Flächen die Frage der kürzesten Linie, genau die gleiche Frage, die Bernoulli schon 1697 den Geometern gestellt hatte. Denn beide Fragen hängen sehr eng zusammen. Der sich bewegende Körper verfolgt erstens nach dem Gesetz der kleinsten Wirkung [comment: this seems an inaccuracy in the text from the website: usually Gauß is credited with the Prinzip des kleinsten Zwangs (principle of least constraint), which is similar to, yet clearly kept similar from the principle of least action ], den kürzesten Weg von einem Punkt zum anderen, wird aber dabei abgelenkt durch die "Rahmenbedingungen", die ihm durch seine Lage im Raum gesetzt sind.

In diesem Zusammenhang ist es erwähnenswert, daß Gauß sehr früh begann, sich mit allen möglichen Verschlingungen oder Umschlingungen eines Fadens oder Verkettungen von Kurven im Raum zu beschäftigten. Schon 1794 macht er eine Liste von 13 sauber gezeichneten Ansichten von Knoten, die er in einem Buch gefunden hat. Diese werden später wichtig bei der Untersuchung elektrischer und magnetischer Ströme. Auch in der Astronomie finden sich bei der Untersuchung von Sternennebeln oder den Atmosphären von Planeten oft eigentümliche Kurven und Verkettungen. Erst in den letzten Jahren hat man aus Satellitenaufnahmen im Saturnring unerklärliche Verschlingungen entdeckt.

Gauß war sich im klaren darüber, daß die Euklidische Geometrie nur eine Geometrie von vielen möglichen darstellt. Wenn man nämlich einen gekrümmten Raum annimmt, so fallen die zeitlich kürzesten Bahnen mit den räumlich kürzesten zusammen. Er hat sich zum Beispiel immer mit der Frage beschäftigt, welche Gesetze für solche gekrümmten Bewegungen gelten, bzw. wie die Koordinatensysteme wohl aussehen, wenn Körper alle Arten von verschlungenen Bahnen annehmen. Auch bei den Planetoiden tritt ja mehrfach das kettenartige Ineinandergreifen von verschiedenen Bahnen auf!

Als Beispiel dafür, wie sich die Gesetze bei anderen "Rahmenbedingungen" für die Möglichkeiten der Bewegung eines Körpers ändern, ist folgende Konstruktion, die Möbius nach Äußerungen von Gauß überliefert hat. Möbius hatte seit dem Herbst 1813 ein Semester lang unter Leitung von Gauß auf der Göttinger Sternwarte gearbeitet. In seinen Aufzeichnungen über die krummen Flächen bezieht er sich ausdrücklich auf eine mündliche Mitteilung von Gauß über die Eigenschaften eines Doppelringes:

[Abbildung Doppelring] "Man kann sich so einen Doppelring leicht zur Anschauung bringen, wenn man ein Blatt Papier in Form eines Kreuzes ausschneidet und hierauf die Enden FH und F'H' (siehe Abbildung) des einen Paares gegenüberliegender Arme etwa oberhalb der anfänglichen Ebene des Kreuzes und die Enden BD und B'D' des anderen Paares unterhalb dieser Ebene mit einander vereinigt. Es besitzt diese nur von einer Linie ABB' IHH' GD'D EF'FA begrenzte Fläche noch die merkwürdige Eigenschaft (nach einer mündlichen Mitteilung von Gauß; wodurch Gauß zur Betrachtung der Fläche geführt worden ist, ist mir unbekannt), daß man von irgend vier auf ihrem Perimeter auf einander folgenden Punkten P, Q, R, S den ersten mit dem dritten und den zweiten mit dem vierten durch zwei Linien PTT'R und QUU'S verbinden kann, welche in der Fläche selbst liegen und dennoch einander nicht schneiden - wie dies doch immer geschehn würde, wenn die Fläche eine Grundform der ersten Klasse (zum Beispiel ein Zylinder, C.H.) wäre."

Gauß hinterließ also nicht nur ein unermeßliches Werk von großen Entdeckungen und Erkenntnissen; vielmehr warfen seine Gedanken und Ideen noch einen Lichtstrahl einer erahnten Erkenntnis auf alle möglichen unerforschten Naturerscheinungen. [...]

Now my translation and comments:

In doing so [the author is referring to Gauss' formulation of what is nowadays called Gauss's principle of least constraint he threw the entire Newtonian mechanics out of the window, [This is misleading hyperbole. I am not a professional historian, yet I am sure that one cannot justifiably say that Gauss threw out Newtonian mechanics out of the window. I have no time to adduce references, yet I have a faint recollection that there are original documents in which Gauss expresses his appreciation of Newton. One can now quibble that the author is not referring to Newton's mathematics but to Newton's mechanics only, yet still I think that the author here is misleadingly hyperbolic.] since the movement of a body is determined by its position in space, and by all other bodies which influence it---not by the attractive or repulsive forces of a central force.

Moreover, Gauss in his work on the theory of curved surfaces treated the question of geodetics, precisely the same question that already in 1697 had been posed by Bernoulli [there is more than one notable mathematician named 'Bernoulli'; the author should have made it clear which one was meant] to the geometers. For both questions are closely related. The moving body takes (according to the law of least action) the shortest path from one point to another, yet is deflected by the 'boundary conditions' which are imposed on it by its position in space.

In this respect it is worth mentioning that Gauss began very early on to occupy himself with all possible entanglings and loopings [I do not care much about translating the words 'Verschlingungen' and 'Umschlingungen' here: this is one of the worst parts of the text you linked to: while 'Verschlingung' is indeed the usual German translation of the technical term 'link' in contemporary knot theory, 'Umschlingung' does not have any usual meaning, not even in the vernacular, and to juxtapose it with 'Verschlingug' is misleading; moreover, the 'allen moglichen Verschlingungen' is misleading, or at least, inappropriate German for a scientific text; it seems that the author has a penchant to let the German colloquialism 'alle möglichen' (roughly corresponding to the English: 'all sorts of linkings' or 'any old linking', with a dismissive connotation) slip into the text] of a thread, or linkages of curves in space. Already in 1794 he makes a list of 13 neatly drawn pictures of knots which he found in a book. These will become important later on, in investigations of electric and magnetic currents. Also in astronomy one often finds, in investigations of nebulas or the atmospheres of planets, peculiar curves or linkages. Only a few years ago one has discovered [in the original, saying 'man hat aus... etwas entdeckt' is not correct German: one 'entdeckt' something 'in' something else, not 'aus] inexplicable links in photographs made by space probes of the rings of Saturn [the term 'Satellitenaufnahmen' in the text from the website is just wrong: in German one has to say 'Raumsonde' in this context: what made these photographs of the rings of Saturn were not 'satellites' but 'space probes'.]

Gauss was aware that the Euclidean geometry represents but one of many possible geometries. For if one assumes a curved space, then the timelike geodesics coincide with the spacelike geodesics. [I am not sure what the author means here, especially in view of the vaguely general hypothesis 'if one assumes a curved space'.] He has, for example, always occupied himself what laws hold for such curved movements, or rather, how the coordinate systems might look, if bodies follow all sorts of linked orbits. [The author is very vague and confusing when writing 'alle Arten von verschlugenen Bahnen annehmen'. This almost sounds as if the author would be thinking of Feynman path integrals... The universal quantifier 'alle Arten' seems misleading to me, it might be a German colloquialism having slipped into this text] Also in the case of minor planets the linking of distinct orbits occurs.

An example for how the laws change. if different 'boundary conditions' are imposed on the possibilities of movement of a body [The German sentence I tranlsated here is grammatically incorrect: it starts with "Als Beispiel dafur, wie ..." which one cannot finish as the author tries to do with "ist folgende Konstruktion". It just does not parse, while the meaning is clear of course.] is the following construction, which Möbius related after having heard it from Gauss. Möbius had worked since the autumn of 1813 for one semester under Gauss' direction in the observatory in Göttingen. In his notes on curved surfaces he explicitly refers to an oral communication of Gauss on the propertiese of a 'double ring':

: It is easily possible to visualize such a 'double ring' if one cuts out a cross-shaped piece of paper and then identifies the edges FH and FH' (see Figure) of one pair of opposing 'arms' above (say) the initial plane and identifies the edges BD and B'D' of the other pair below that plane. The resulting surface, which has only one boundary ABB'IHH'GD'DEFFA the remarkable property (according to an oral communication of Gauss'; what led Gauss to consider this surface is not known to me) that it is possible to choose a sequence of four arbitrary distinct points P,Q,R,S on its boundary(-line), and then connect the third first in the sequence with the third in the sequence, and the second in the sequence with the fourth in the sequence by two lines PTT'R and QUU'S, the lines being contained in the surface itself, and not intersecting each other --- such an intersection would however always occur if the surface were a 'Grundform der ersten Klasse' [I do not know what Möbius means by 'Grundform der ersten Klasse'] (for example a cylinder , C.H.) [Presumably the C.H. are the initials of the author of the text on the website; C.H. should, I think, rather have written 'one-way-infinite cyclinder' or 'semi-infinite cylinder', i.e. the cartesian product $\{x\in\mathbb{R}\colon 0<x\}\times\mathbb{S}^1$, where $\mathbb{S}^1$ denotes the circle; reason: C.H.'s comment only makes sense if all the four vertices are on the same boundary component of the cyclinder, and then refers to a topological property which is hinted at in the following quick sketch I just made for you

enter image description here

and wherein I used the same letters as in Möbius' original paper.

Please note:

  • The most mathematically substantive part of Möbius' remarks is not the statement

daß man von irgend vier auf ihrem Perimeter auf einander folgenden Punkten P, Q, R, S den ersten mit dem dritten und den zweiten mit dem vierten durch zwei Linien PTT'R und QUU'S verbinden kann, welche in der Fläche selbst liegen und dennoch einander nicht schneiden

but what comes afterwards, i.e., the only nontrivial statement (to my mind) is

  • wie dies doch immer geschehn würde, wenn die Fläche eine Grundform der ersten Klasse (zum Beispiel ein Zylinder, C.H.) wäre.

(My hunch is that this is equivalent to the Jordan curve theorem, but please not that this is a hunch and I have not looked closely into this.)

I am not sure what Möbius meant exactly with 'Grundform der ersten Klasse', yet the following, taken from his collected works, comes close to a definition:

enter image description here

The highlighted passage reads

Addendum. A piece of a 2-sphere which is bounded by a closed line --- or, in other words: a 2-sphere with a hole --- is a 'Grundform erster Klasse. Likewise, a 2-sphere with two, three, etc, $n$ holes is a 'Grundform der zweiten, dritten, etc, $n$ten Klasse.

This is not modern language.

What seems most ambiguous about Möbius's passage is

  • whether he regards a '2-sphere with a hole' to be a punctured sphere (i.e. sphere with a point removed; this is homeomorphic via stereographic projection to $\mathbb{R}^2$) or rather a sphere with an open disc removed (which is itself homeomorphic to a closed disc, yet is not a manifold, only a manifold-with-boundary). These two notions are not the same; more precisely, the two interpretations one can give Möbius's formulation 'Kugelfläche mit einer Öffnung' are not homeomorphic. However, his formulation "Ein von einer geschlossenen Linie begrenztes Stück einer Kugelfläche" rather sounds like he intends the boundary to be part of the 'Stück', yet is is simply ambiguous to contemporary eyes.

E.g. in the Handbook of Geometric Topology you find a usual definition of 'punctured surface':

enter image description here

The point of my above 'C.H. should' is that 'cylinder' is vague and makes people think of the ordinary finite cylinder surace which has two boundary components and then the reader is perhaps confused what the correspondence to the one-boundary-component case of Möbius' surface is.

The point that C.H. is making that on a semi-infinite cylinder, which by the way is homeomorphic to the 'plane with an open disc removed', the intersection of the P->R arc with the Q->S arc cannot be avoided. And this is intuitively quite obvious (though this is not a proof, of course), for the 'semi-infinite cylinder with boundary' is homeomorphic to the 'plane with an open disc removed', represented by the following picture:

enter image description here

The 'open disc' is represented by the gray circle. The intersection of the blue and red line cannot be avoided if these lines are supposed to be arcs in the complement of the 'open disc'.

Now let's go on with the translation:

So Gauß did not only leave an inestimable body of work, full of great discoveries and insights; his thoughts and ideas rather threw a ray of light of an anticipated understanding on all possible uninvestigated natural phenomena. [I think it is fair to say that the German in this last sentence, could be better, to say the least. I did not make much effort to render this into decent English. First of all, the German colloquialism 'alle möglichen' (roughly: 'all sorts of'; it has a pejorative ring to it), inappropriate in a scientific text, and irritating at least to mathematicians' sensitivities in its abuse of a universal quantifier ("allen"), appears again. Second, 'Lichtstrahl einer erahnten Erkenntnis', to my mind, is (an example of the strained-use-of-words-version-of-the-literary-concept-of) a catachresis.

Now to the mathematics.

  • The object that Möbius was told about by Gauß to me essentially seems to be the punctured torus. You can easily find much on this by searching. It is not completely standardized what precisely 'punctured torus'. For the present purposes, the following excert from H.B.Griffiths: Surfaces, Cambridge University Press. 2nd edition (1981) should suffices (I suppose you do not know the usual notions of basic algebraic topology, which would be needed to give a definition of 'punctured torus', so I won't try to do so)

On the topic of why Möbius' paper-surface is homotopic to the punctured torus see also

this thread

Relevant pictures are

enter image description here enter image description here enter image description here

(source: "Ian Mallet", cf. https://www.youtube.com/watch?v=j2HxBUaoaPU)

While you might have found this yourself, it might still be useful to you to here cite the relevant passage from Möbius [I reproduce this from the Cambridge University Press 2011 edition of Gauss's works, Volume 10, Part 2]

enter image description here

Regarding the larger question which to me seems to be behind your question (and its title), it seems useful to cite from the Cambridge University Press edition of Gauss' collected works in greater detail:

enter image description here

enter image description here

I won't translate this; you can do so with a search engine if you are really interested in it. The author is in particular saying here that Gauss influenced each of Listing, Möbius and Riemann, and a letter of Gauss to Schumacher is cited in which Gauss (in free translation) says that he thought that Möbius was someone who wanted and could think for himself instead of being spoon-fed with knowledge, and should only be given some hint here and there. Moreover, it seems there are many speculations as to what Gauß told whom when, in particular in the context of Riemann's 1854 Habilitation lecture. I know little about this, yet much of this seems speculation.

To end with addressing your follow-up question in your comment

O.k I accept your answer, since i noticed that the surface has two sides so it cannot be a mobius strip. But i still dont understand what is the "remarkable property" which Gauss refers to in the last part of the passage: – user2554 Aug 22 at 9:27

and your comment

can you explain to me what the last part of the passage means? (that i cited in my last comment) - i need the answer to close the discussion and understand the property which Gauss describes. – user2554 Aug 22 at 10:28

Let me try to summarize that: in modern terms, the "remarkable property" that Möbius wrote he was told by Gauss is this:

  • whatever ordered sequence of four distinct points $P$, $Q$, $R$, $S$ you pick on the one boundary $\partial( \{x\in\mathbb{R}\colon 0<x\}\times \mathbb{S}^1 )$ of the surface $S$ Möbius is describing a construction of, there exist, for both $i\in2$, arcs $[0,1]\xrightarrow[]{A_i}S$ such that

$A_0(0)=P$

$A_0(1)=R$

$A_1(0)=Q$

$A_1(1)=S$

and

for all $(t_0,t_1)\in[0,1]\times[0,1]$ we have $A_0(t_0)$ not equal to $A_1(t_1)$.

The term 'arc' here means 'continuous and injective map $[0,1]\rightarrow$(the surface in question).

An illustration of this is

enter image description here

Let me also mention that that the historical literature seems (I am no expert on this historical matter) to be saying that no student-teacher-relationship in a conventional sense of these words existed between Möbius and Gauß. In particular, Möbius's doctoral advisor was Johann Friedrich Pfaff. And this Pfaff was also the doctoral advisor of, guess who, Gauß. So the title of your question seems not quite correct: formally-genealogically, Gauß and Möbius were academic brothers.

Please let me know briefly whether your question in your comments is sufficiently answered now.

To sum up, if I would have to answer all your question in one sentence, my answer would be:

Gauß gave his academic brother Möbius a hint that the punctured torus is not simply-connected.

1 I am doing this to give you examples of properties of this text which, I think, you should be warned against.

Update September 17, 2017. By chance I found out that there is a new book by Étienne Ghys, entitled A singular mathematical promenade, which contains a chapter relevant to your question. (The book is amazing by itself, too, something of a 'Book 2.0'.) For your convenience, I here reproduce a few parts of Ghys' book relevant to your question:

enter image description here

enter image description here

enter image description here

enter image description here

  • Peter Heinig - Wow!! what a comprehensive answer!! i have to tell you thanks a thousand times! Obviously i voted your answer, despite that i haven't yet made a comprehensive reading of it. Topology is really one of the weak points in my mathematical knowledge, and that's why i'm willing to go it's beginings - understanding (at least partially) it's founding fathers - Leibniz, Euler, Gauss , Riemann, Poincare and several others. – user2554 Aug 29 '17 at 14:40
  • @user2554: glad I could help a bit. Let me reiterate that the title of your question, by asking 'what', presupposes that Gauß taught Möbius, and it seems that little such teaching actually occured. Moreover, let me reiterate that it seems that from a modern point of view, the main substance of the 'Bemerkung' that Möbius mentioned Gauß made to him seems to be equivalent to the Jordan Curve Theorem, and yet (0) this is insufficiently treated in my answer, (1) neither Möbius nor Gauß were probably aware of the complexities involved. – Peter Heinig Aug 29 '17 at 15:09

No, this surface is actually less interesting than the Moebius strip, it is just a torus (if you do not know what this means, think of the surface of a doughnut) with a (rather large) hole in it. However, it does suggest a nontrivial (at the time of writing, but quite standard now) and useful mathematical idea, namely, constructing surfaces by identifying sides of some (not necessarily convex) polygons.

  • O.k I accept your answer, since i noticed that the surface has two sides so it cannot be a mobius strip. But i still dont understand what is the "remarkable property" which Gauss refers to in the last part of the passage: – user2554 Aug 22 '17 at 9:27
  • "the latter still possesses the remarkable property 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.). " – user2554 Aug 22 '17 at 9:27
  • can you explain to me what the last part of the passage means? (that i cited in my last comment) - i need the answer to close the discussion and understand the property which Gauss describes. – user2554 Aug 22 '17 at 10:28

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