Wikipedia:Reference desk/Archives/Mathematics/2016 October 14

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

October 14

The ancient Greeks and Euler's polyhedron formula

The Ancient greeks were very into the 5 regular solids and similar things in Three dimension geometry. Is it reasonable to consider that had things gone slightly differently that Euler's formula (V-E+F=2)? I don't think the exceptions on the non-convex Polyhedra would have made a difference, they wouldn't have really considered them polyhedra...Naraht (talk) 16:26, 14 October 2016 (UTC)

I can't understand your 2nd sentence ("Is it reasonable to consider that had things gone slightly differently that Euler's formula (V-E+F=2)?"). Are some words missing ? StuRat (talk) 17:27, 14 October 2016 (UTC)

• The only meaning I can give to that question is "if Euler's formula did not hold, would there be five platonic solids?". To which no reasonable answer can be given, because it depends on the consequences of some basic arithmetic being wrong. Tigraan^{Click here to contact me} 17:48, 14 October 2016 (UTC)

No, what I'm meaning is 1) if someone in Ancient Greece had gone "Hey for each of the 5 Platonic Solids, if you add together the Vertices and Faces and subtract the Edges you get 2 and these even works for non pure solids like this Pentagonal Pyramid that I chopped off the top", would it be viewed as something special and memorable that would have been communicated on with other things of that time period? 2) Would it be within the realm of possibility for someone involved in Mathematics in that time period to have had that "Aha!"Naraht (talk) 18:59, 14 October 2016 (UTC)

Much of ancient mathematics has been lost to history, so it's perfectly possible that someone did discover the formula. It doesn't seem to be the kind of thing the Greeks were very interested in though; they where more concerned with measuring things like lengths, areas and volumes than with counting things. Any definitive answer to the question would be unprovable speculation. --RDBury (talk) 11:24, 15 October 2016 (UTC)

Technically speaking, we genuinely don't know. But, if you ask me, there is an inherent `abnormality` to Euler's formula, inasmuch as adding faces to points and lines makes as little sense as adding apples to goats and tractors. Although Euler looked at the same things as the ancients, his eyes saw things in a completely different light. Perspective is everything. — 79.113.236.217 (talk) 09:53, 16 October 2016 (UTC)

I agree. As these articles The Geometry Junkyard: Twenty Proofs of Euler's Formula: V-E+F=2 and Descartes's Lost Theorem & Euler's Polyhedral Formula explain, Euler's theorem has interesting antecedents. What was arguably a form of Euler's formula, one which also foreshadowed the Gauss-Bonnet Theorem, was discovered a century earlier by Descartes. Not even Euler thought about polyhedra & his formula in an entirely "modern" way. As the third article says "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. It appears that Euler did not view them this way either!".John Z (talk) 01:17, 17 October 2016 (UTC)