@Misc{POLYFORM.MIZ,
  author = {Alama, Jesse},
  title = {Euler's Polyhedron Formula},
  day = {9},
  month = {October},
  year = {2007},
  address1 = {Department of Philosophy\\Stanford University},
  acknowledgement = {The author would like to thank his advisor,
                  Grigori Mints, for suggesting the project of giving
                  a formal proof of Euler's polyhedron formula, as
                  well as Peter Koepke and Bas Spitters for inviting
                  the author to present the work contained in this
                  article at seminars in Bonn, Germany and Nijmegen,
                  the Netherlands.},
  summary = {Euler's polyhedron theorem states for a polyhedron
                  $p$, that $$ V - E + F = 2, $$ where $V$, $E$, and $F$ are,
                  respectively, the number of vertices, edges, and
                  faces of p.  The formula was first stated in print
                  by Euler in 1758 \cite{euler1758a}.  The proof given
                  here is based on Poincaré's linear algebraic proof,
                  stated in \cite{poincare1893} (with a corrected
                  proof in \cite{poincare1899}), as adapted by Imre
                  Lakatos in the latter's \textsl{Proofs and
                  Refutations} \cite{proofs-and-refutations}.\par
                  As is well known, Euler's formula is not true for all
                  polyhedra.  The condition on polyhedra considered
                  here is simple connectedness, which says that
                  the cycles (chains whose boundary is zero) are exactly
                  the bounding chains (chains that are the boundary of
                  a chain of one higher dimension).\par
                  The present proof actually goes beyond the
                  three-dimensional version of the polyhedal formula
                  given by Lakatos; it is dimension-free, in the sense
                  that it gives a formula in which the dimension of the
                  polyhedron is a parameter.  The classical Euler
                  relation $V - E + F = 2$ is corresponds to the case where
                  the dimension of the polyhedron is $3$. \par
                  The main theorem, expressed in the language of the
                  present article, is
                  $$\mathtt{Sum alternating-characteristic-sequence(p) = 0},$$
                  where $p$ is a polyhedron.  The alternating
                  characteristic sequence of a polyhedron is the
                  sequence $$ -N(-1), +N(0), -N(1),
                  \dots, (-1)^(\dim(p))*N(\dim(p)),$$ where $N(k)$ is
                  the number of polytopes of $p$ of dimension $k$.  The
                  special case of $\dim(p) = 3$ yields Euler's classical
                  relation.  ($N(-1)$ and $N(3)$ will turn out to be equal,
                  by definition, to $1$.) \par
                  Two other special cases are proved in the present article:
                  the first says that a simply connected one-dimensional
                  "polyhedron" consists of just two vertices (and thus
                  consists of just a single edge); the second special
                  case asserts that a two-dimensional simply-connected
                  polyhedron (a polygon) has as many vertices as
                  edges. \par
                  A treatment of the more general version of Euler's
                  relation can be found in \cite{grunbaum2003} 
                  and \cite{brondsted1983}.  The former contains a proof 
                  of Steinitz's theorem, which shows that the abstract 
                  polyhedra treated in Poincar\'e's proof, which might not
                  appear to be about polyhedra in the usual sense of the
                  word, are in fact embeddable in $\mathbf{R}^3$ under 
                  certain conditions.  It would be valuable to formalize 
                  a proof of Steinitz's theorem and relate it to the 
                  development contained here.}
  section1 =      {Set-theoretical Preliminaries},
  section2 =      {Arithmetical Preliminaries},
  section3 =      {Preliminaries on Finite Sequences},
  section4 =      {Polyhedra and Incidence Matrices},
  section5 =      {The Chain Spaces and thei Subspaces.  Boundary of a $k$-chain},
  section6 =      {Simply Connected and Eulerian Polyhedra},
  section7 =      {The Extremal Chain Space},
  section8 =      {A Generalized Euler Relation and its $1-$, $2-$, and $3-$-dimensional Special Cases}}

@article{euler1758a,
	author = {Euler, Leonhard},
	title = {Elementa Doctrinae Solidorum},
	journal = {Novi Commentarii Academiae Scientarum Petropolitanae},
	year = {1758},
	volume = {4},
	pages = {109-140}
}

@book{proofs-and-refutations,
	author = {Imre Lakatos},
	title = {Proofs and Refutations: The Logic of Mathematical Discovery},
	publisher = {Cambridge University Press},
	year = {1976},
	note = {Edited by John Worrall and Elie Zahar}}

@book{grunbaum2003,
	author = {Gr\"unbaum, Branko},
	title = {Convex Polytopes},
	publisher = {Springer},
	year = {2003},
	edition = {2nd},
	series = {Graduate Texts in Mathematics},
	number = {221}}

@book{brondsted1983,
	author = {Br\o ndsted, Arne},
	title = {An Introduction to Convex Polytopes},
	publisher = {Springer},
	year = {1983},
	series = {Graduate Texts in Mathematics}}

@article{poincare1893,
	author = {Poincar\'e, Henri},
	title = {Sur la G\'en\'eralisation d'un Th\'er\`eme d'Euler relatif aux Poly\`edres},
	journal = {Comptes Rendus de S\'eances de l'Academie des Sciences},
	year = {1893},
	volume = {117},
	pages = {144}}

@article{poincare1899,
	author = {Henri Poincar\'e},
	title = {Compl\'ement \`a l'Analysis Situs},
	journal = {Rendiconti del Circolo Matematico di Palermo},
	year = {1899},
	volume = {13},
	pages = {285-343}}