The formula $V-E+F=2$ is so simple that I can't believe that it was really Euler (or perhaps Descartes) who first observed it (I mean the formula itself in some generality, not necessarily a valid proof). To have a concrete question: Is there any reference to this formula in ancient mathematics?
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asked Jan 14, 2014 at 8:48
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16$\begingroup$ “It is tempting to speculate about why all the able mathematicians […] who investigated polyhedra in the years before Euler did not notice the polyhedral formula. There certainly are results in Euclid's Elements and in the work of later Greek geometers that appear more complex […]. Presumably a major factor, in addition to the lack of attention paid to counting problems in general up to relatively recent times, was that people who thought about polyhedra did not see them as structures with vertices, edges, and faces.” - ams.org/samplings/feature-column/fcarc-eulers-formula $\endgroup$– WaldemarCommented Jan 14, 2014 at 9:24
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$\begingroup$ Descartes came close. How close? $\endgroup$– Pratik DeoghareCommented Jan 14, 2014 at 16:32
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$\begingroup$ @PratikDeoghare -- very close, see ams.org/samplings/feature-column/fcarc-descartes3 $\endgroup$– Carlo BeenakkerCommented Jan 15, 2014 at 0:41
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1$\begingroup$ I was under the impression that the ancients were loath to do arithmetic combining objects of different dimensions, e.g., adding lengths to areas, or areas to volumes. $F-E+V$ combines something two-dimensional with something one-dimensional with something zero-dimensional, and would have been anathema to the ancients, no? $\endgroup$– Gerry MyersonCommented Mar 9, 2020 at 2:03
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3 Answers
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there is no doubt the answer to your question is "no"; for a wonderful and scholarly recent book on the whole story, see Euler's Gem: The Polyhedron Formula and the Birth of Topology by David Richeson.
They all missed it. The ancient Greeks -- mathematical luminaries such as Phythagoras, Theaetetus, Plato, Euclid, and Archimedes, who where infatuated with polyhedra -- missed it. Johannes Kepler, the great astronomer, so in awe of the beauty of polyhedra that he based an early model of the solar system on them, missed it. In his investigation of polyhedra the mathematician and philosopher René Descartes was but a few logical steps away from discovering it, yet he too missed it. These mathematicians, and so many others, missed a relationship that is so simple that it can be explained to any schoolchild, yet is so fundamental that it is part of the fabric of modern mathematics.
The great Swiss mathematician Leonhard Euler did not miss it. On November 14, 1750, in a letter to his friend, the number theorist Christian Goldbach, Euler wrote, "It astonishes me that these general properties of stereometry have not, as far as I know, been noticed by anyone else".
Centuries later, we remain astonished.
answered Jan 14, 2014 at 10:18
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13$\begingroup$ As usual, we can't be that sure about what Greek Hellenistic mathematicians knew, because of the subsequent dramatic loss of knowledge. The answer is unfortunately doomed to remain "no, as far as we know". $\endgroup$ Commented Jan 14, 2014 at 11:40
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10$\begingroup$ @jwg -- I don't want to be glib, but what evidence of absence can one give in this case, other than absence of evidence? Richeson's monograph is a scholarly piece of work, which thoroughly and critically examines all available evidence for early discoveries of the polyhedron formula, and there is none. $\endgroup$ Commented Jan 14, 2014 at 13:24
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2$\begingroup$ @CarloBeenakker: are you saying Richeson thoroughly and critically examines the empty set? $\endgroup$ Commented Jan 14, 2014 at 13:37
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4$\begingroup$ @Carlo: exact, we can only remark absence of evidence, which is why I think we can't say "no doubt, the answer is no". Absence of evidence, of course, does not allow "why-not conjectures" about lost ancient knowledges. But there is a number of examples that show that our knowledge of what the ancients knew is not that complete. One example for all: the story of the Shroeder numbers, in this beautiful short paper by R.Stanley math.mit.edu/~rstan/papers/hip.pdf $\endgroup$ Commented Jan 14, 2014 at 13:39
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2$\begingroup$ Also, if we even look at Archimedes' greatest discoveries, we have to admit it was not at all obvious that they should have all survived till our times. For instance, the recovery of the Method dates 1840, by chance, and it was only attributed to Archimedes in 1906. $\endgroup$ Commented Jan 14, 2014 at 14:00
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Today almost nobody shares anymore old Leibnitz' optimistic idea There is no ignorabimus in mathematics (in Hilbert's words). We know that there are true facts in mathematics that will never be proved, either because our maths society (if not the whole mankind) will extinguish before, or because the shortest proof has by far more symbols than there are atoms in the universe, or just because there is no proof at all. Yet, it seems we still reject the analogous ignorabimus about questions in the history of mathematics. We tend to think our knowledge of what the ancients knew is quite detailed: If some piece of maths is missing from the ancient sources, the reason has to be, just because it was not known, otherwise, it would certainly had arrived to us in some way. This optimistic point of view is unreasonable too, as it has been proved, among others, by the work of Lucio Russo.
It's a fact that a relevant part of what we know about ancient scientific knowledge came to us in the most erratic and totally unpredictable way through fires, floods, and wars. Fundamental Archimedes's Method survived many centuries in a monastry only because its parchment happened to be useful to write a religious text, but in the meanwhile it happened not to be completely erased. It was discovered by chance in 1840 and attributed to Archimedes only in 1906, and lost again till 1998.
A popular striking example is the story of the Schroeder numbers. See this beautiful article by Richard Stanley . Before the year 1990 no or few examples of Hellenistic combinatorics were known. Then the numbers 103,049 and 310,952 were noticed in a passage of Plutarch, proving, as irrefutably as fingerprints, Hellenistic combinatorics was by far more advanced than ever thought before. Had Plutarch omitted that mathematical aside note, today conjecturing the knowledge of the Schroeder numbers in the Greek mathematics should be considered at least uncalled for, exactly as it is conjecturing the knowledge of the Euler formula. Of course, the former fact does not make the latter conjecture more likely to be true. We can only compare the two and say that Euler formula is elementary and simple enough that it could be well-known and proved.
I think a more reasonable question is: whether Euler formula was missed or not by the advanced Hellenistic mathematics, how was that it was not discovered/rediscovered/remembered till Euler? Possibly, the reason is, till that moment that formula had not such a relevant role as it had later. On the contrary, say, the properties of the parabola have always been something of interest, so that it was never forgotten despite centuries of wars (and maybe even thanks to wars: to throw projectiles).
In conclusion I somehow differ from Carlo Benakker's opinion, and think that a more cautious, though less satisfying answer to this question would be: no, as far as we know; and if the true answer is no, we will probably never know. If the true answer is yes, we may hope in a new achievement of historiography that gives us a proof, like in the case of Schroeder numbers.
answered Jan 14, 2014 at 19:28
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$\begingroup$ @PietroMajer About "the shortest proof has by far more symbols than there are atoms in the universe" : I have a question, maybe too informal to be asked in MO (I don't know): is it possible to prove that there exist provable statements, let's say in ZFC, whose minimal proof (if such a thing makes sense) can be as long as we want ? $\endgroup$ Commented Jan 15, 2014 at 9:06
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2$\begingroup$ well, I think otherwise there would be just a finite number of provable statements, right? I guess experts could say something more precise about this. $\endgroup$ Commented Jan 15, 2014 at 9:21
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2$\begingroup$ @PietroMajer It is actually easy to find a provable statement with a very long proof. I read somewhere that there are more than 10000 new theorems proved by year in the world. Take these 10000 theorems $T_1,\dots T_{10000}$. Then the proof of the statement $T_1 \hbox{ and } \dots \hbox{ and }T_{10000}$ should be very long. More interesting is the question whether the quotient of the length of the minimal proof by the length of the statement can be arbitrarily large. But I doubt very much that this question can be formalized. $\endgroup$ Commented Jan 15, 2014 at 14:55
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$\begingroup$ Even if we know that (sadly) the shortest proof for $T_{101}\mathrm{and}T_{102}\dots\mathrm{and} T_{10000}$ is maybe not that long :) $\endgroup$ Commented Jan 15, 2014 at 15:02
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7$\begingroup$ @PhilippeGaucher If you fix a formal language in which you express your theorems and proofs, then the function $n \mapsto \sup_{\text{theorems }T\text{ of length } n} \inf_{\text{proofs }P\text{ of }T} \text{length}(P)$ grows faster than any computable function. If it didn't, we would be able to solve the Halting Problem. $\endgroup$– S. Carnahan ♦Commented Jan 16, 2014 at 21:45
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Once I was lecturing to high school students about this theorem. My proof began with the words: "suppose that the net is drawn on a surface of a rubber ball...". One student asked: "Did rubber exist at the time of Euler?"
I think the moral of this story is that the very statement of the question was foreign to ancient Greek mathematics. Did they have a definition of ARBITRARY polytope? Or even an arbitrary convex polytope? I doubt it. (See this related post).
Descartes proof was earlier than that of Euler, and his statement was in terms of solid angles (which is equivalent to the Gauss-Bonnet theorem, but stated in terms of elementary geometry). I think it was Euler who recognized for the first time that we have a topological fact here. And this was a great insight. Like the problem of the same Euler about the Konigsberg bridges. It is not difficult, but before Euler these things apparently were not considered part of mathematics. That's why these questions were not raised in antiquity.
EDIT. Many things were not discovered for the simple reason that no one was looking for these things. Let me give a more recent example. Could Fermat, Descartes, Huygens, Euler, etc. discover linear programming and simplex method? There is no doubt they could. After all, this is elementary mathematics which can be easily taught in high school. But look what had really happen. Fourier had a paper on systems of linear inequalities (where he invented what is called Fourier-Motzkin elimination nowadays). And in the end of XIX century (!) the editor of Complete works of Fourier had to APOLOGIZE for including this paper: he wrote that "every work of such a grand master has to be included", where "every" evidently means "even such trivial nonsense":-)
And linear programming became a part of mainstream mathematics only during WWII, when it was really needed.
answered Jan 16, 2014 at 15:46
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3$\begingroup$ I like your rubber story, and particularly your expression of doubt about "arbitrary" polytopes. And yet the absence of evidence is not less wonderful when restricted to, say, the five Platonic solids, or their stellations, or any other of the small number of polytopes that the ancients might have considered. $\endgroup$ Commented Jan 16, 2014 at 16:02
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1$\begingroup$ The absence of evidence. Yes. It is a subject of discussion, how much of the ancient science works survived. I side with Lucio Russo who thinks that most of it is lost. $\endgroup$ Commented Jan 16, 2014 at 20:22
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3$\begingroup$ Actually Euclid does have a sort of general definition of polyhedron (Elements, XI) although he immediately focuses on a list of the most common ones: prism, pyramid, cube, etc. Archimedes made an exhaustive study of the semi-regular polyhedra now called after him (in a work that we known by secondary sources), employing certain general operations to build new polyhedra out of polyhedra. $\endgroup$ Commented Jan 16, 2014 at 22:39
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1$\begingroup$ So, at least we know that Hellenistic mathematicians did have a various enough class of examples (some of them quite complicated) in order that the problem of looking for a pattern between V,E,F could be considered. $\endgroup$ Commented Jan 16, 2014 at 22:45
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$\begingroup$ @Pietro Maier: I edited and added a reference on a funny example about Euclid's "definition" of polyhedra. $\endgroup$ Commented Mar 8, 2020 at 22:58