
Twenty Proofs of Euler's Formula: V-E+F=2 - memexy
https://www.ics.uci.edu/~eppstein/junkyard/euler/
======
OscarCunningham
This is also the example used in Imre Lakatos fascinating essay 'Proofs and
Refutations', about the nature of proof.

[https://math.berkeley.edu/~kpmann/Lakatos.pdf](https://math.berkeley.edu/~kpmann/Lakatos.pdf)

~~~
memexy
Yup, it's a really good exposition of the process most math goes through
before it is fully "rigorous". In one of Alan Kay's talks he mentions how
Euler basically got all his proofs wrong but his intuitions were almost all
correct and some people got PhDs by making his arguments more rigorous [0].

Gian-Carlo Rota has a similar story about most of his proofs being wrong but
generally having the right intuitions and being on the right track. The paper
is titled "Ten Lessons I Wish I Had Been Taught" and it's really good. He
gives advice applicable to any domain where sharing knowledge is the key
marker of progress: [http://www.ams.org/notices/199701/comm-
rota.pdf](http://www.ams.org/notices/199701/comm-rota.pdf).

\--

0:
[https://www.youtube.com/watch?v=oKg1hTOQXoY&feature=emb_titl...](https://www.youtube.com/watch?v=oKg1hTOQXoY&feature=emb_title)

[~ 8min 15sec] There was a mathematician by the name of Euler. Whose
speculations about what might be true formed 20 large books. That most of them
were true. Most of them were right. Almost all of his proofs were wrong. And
many PhDs in mathematics in the last, and this, century have been formed by
mathematicians going to Euler's books, finding one of his proves [and] showing
it was a bad proof. And then guessing that he, his insight was probably
correct and finding a much more convincing proof.

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eeereerews
I like Noah's Ark a lot.

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kilovoltaire
Wow yeah agreed. Direct link for convenience:

[https://www.ics.uci.edu/~eppstein/junkyard/euler/noah.html](https://www.ics.uci.edu/~eppstein/junkyard/euler/noah.html)

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antonii
The proof is elegant, but the whole flooding process on arbitrary polyhedron
may be a bit difficult to follow (especially if you are trapped by details
like saddles above some peaks, lakes with different altimeters, and how
gravity works over the polyhedron).

The following simpler formulation of the flooding process may be easier to
follow.

    
    
      initial state: num_islands=1; num_lakes=V
      final state:   num_islands=F; num_lakes=1
      how state changes: Each time a saddle sinks, either two lakes are joined (num_lakes--) or an island is split (num_islands++)
    

The whole process takes (F-1)+(V-1) events to finish, which is equal to the
number of saddles E.

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zhamisen
Thank you, you have helped to make it more clear for me!

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phkahler
Interesting that this also works for a degenerate polyhedron with 3 vertices,
3 edges, and 2 faces. In other words a 2 sided triangle.

~~~
contravariant
It works even if the vertices edges and faces aren't straight so what you just
described is also equivalent to two hemispheres.

Now maybe you've started wondering where the catch is, since it clearly
doesn't work if you just use 1 face for everything. The rule is that the
pieces need to be 'simple' which basically means that they're like a disc or
simplex, they're allowed to be deformed but you can't cut holes or do anything
that changes the topology.

