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The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and that story is a must in every introduction to graph theory.

My question is whether it is also known what led Euler to investigate on polyhedra; was it a 'recreational' question, or was it a mathematical research question of his contemporaries?

A further question would be whether Euler continued his investigations on polyhedra after having discovered his famous characterisation of polyhedra of genus 0.

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What I didn't mention in my question in order not to give possibly misleading suggestions, is that I was wondering whether the question about the completeness of the list of platonic solids may have played a role; there is for example a polyhedron depicted on Duerer's "Melencolia I" which is an 'almost' platonic solid; suspecting that it is related to attempts to construct a regular pentahedron is a matter of personal opinion. – Manfred Weis Apr 19 '14 at 6:43
up vote 16 down vote accepted

May I suggest this article by Joseph Malkevitch, an AMS Feature Column. Note particularly his point below (screen snapshot--not searchable) that polyhedra were simply not viewed, at the time, in terms of vertices, edges, and faces:

  Malkevitch excerpt
This point is also made in Imre Lakotos's Proofs and Refutations (Cambridge Univ. Press, 1976).
Addendum. Joe Malkevitch sent me this relevant quote in a letter from Euler to Goldbach: "Recently it occurred to me to determine the general properties of solids bounded by plane faces, because there is no doubt that general theorems should be found for them, just as for plane rectilinear figures, whose properties are: (1): that in every plane figure the number of sides is equal to the number of angles, and (2) the sum of the angles is equal to twice as many right angles as there are sides, less four." (Biggs, Lloyd, and Wilson, Graph Theory 1736-1936, Oxford Univ Press, pp.76-77.)

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Is there a caption for the picture? Based on the manner of dress, my guess is that it is a picture of Cauchy, but I don't recall Cauchy looking like that. – The Masked Avenger Apr 18 '14 at 19:03
@TheMaskedAvenger: There is no caption, but, Yes, that is Cauchy. See this bio (link), which includes the same picture. – Joseph O'Rourke Apr 18 '14 at 20:14

For a reasonably complete discussion, see Richeson's "Euler's Gem" (available on Amazon, very well written)

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