Proof 12: Shelling
interprets a polyhedron or polytope as a complex of polyhedral
faces of varying dimensions. He lets fi denote the number
of faces of dimension i, so f0=V,
f1=E, and f2=F,
but he also adds faces of dimension d (the whole polytope)
and -1 (the empty face); f-1=fd=1.
defines a shelling of a complex of polyhedral faces
to be a linear ordering on the maximal-dimension faces (facets)
such that, if the facets are of dimension at least one,
the ordering satisfies the following properties:
Every convex polytope has a shelling found by traveling in a straight line
from some point near one face of the polyhedron, and ordering the faces
by the sequence of points at which the line crosses the plane of a face
and the face becomes visible. (The line must be imagined to pass
"to infinity and beyond" through to the other side of the polyhedron.)
The intersection of a facet Fj with previous facets
can be found
geometrically as the portion of the boundary of the facet
visible from the intersection point of the viewing line with the plane
of the facet; this shows that the lower-dimensional shelling required by
property 2 is of the same geometric type.
Ziegler then proves that any polytope
has Euler characteristic
by an induction on dimension and shelling length:
- The boundary of the first facet F0 has a shelling.
- The intersection of each facet Fj (j > 0) with the
union of previous
facets is nonempty and forms the prefix of a shelling of Fj.
The base case, in which f-1=f0=1, is clear.
Now if P is a d-polytope with shelling order
F1,F2,... then we have
more precisely that
X(F1 u F2 u ... u Fj)
is 0 for j < fd-1
and 1 for j = fd-1
which follows by induction on j,
since the facets we add in are (d-1)-polytopes, the Euler
characteristic is additive, and the intersection of Fj with
previous facets is a shellable part of, but (for j <
not the whole boundary of
Fj (Ziegler shows that this last observation is true
in general, but it is evident geometrically for the
shelling defined above).
Removing the two "extra" faces f-1 and
f3 from this sum
gives us the usual Euler formula.
Proofs of Euler's Formula.
From the Geometry Junkyard,
and recreational geometry pointers.
from a common source file.
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