A formula relating the number of polyhedron vertices,faces, and polyhedron edges of a simply connected (i.e., genus
0) polyhedron (or polygon).
It was discovered independently by Euler (1752) and Descartes, so it is also known
as the Descartes-Euler polyhedral formula. The formula also holds for some, but not
all, non-convex polyhedra.
The polyhedral formula states
(1)
where
is the number of polyhedron vertices, is the number of polyhedron
edges, and
is the number of faces. For a proof, see Courant and Robbins
(1978, pp. 239-240).
The formula was generalized to -dimensional polytopes by Schläfli
(Coxeter 1973, p. 233),
(2)
(3)
(4)
(5)
(6)
and proved by Poincaré (Poincaré 1893; Coxeter 1973, pp. 166-171; Williams 1979, pp. 24-25).
is the Euler characteristic, sometimes also known as the Euler-Poincaré characteristic. The polyhedral formula corresponds
to the special case .
There exist polytopes which do not satisfy the polyhedral formula, the most prominent of which are the great dodecahedron and small
stellated dodecahedron, which no less than Schläfli
himself refused to recognize (Schläfli 1901, p. 134) since for these solids,
Aigner, M. and Ziegler, G. M. "Three Applications of Euler's Formula." Ch. 10 in Proofs
from the Book. Berlin, Germany: Springer-Verlag, 1998.Beyer,
W. H. (Ed.). CRC
Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 128,
1987.Courant, R. and Robbins, H. What
Is Mathematics?: An Elementary Approach to Ideas and Methods. Oxford, England:
Oxford University Press, 1978.Coxeter, H. S. M. "Euler's
Formula." and "Poincaré's Proof of Euler's Formula." §1.6
and Ch. 9 in Regular
Polytopes, 3rd ed. New York: Dover, pp. 9-11 and 165-172, 1973.Coxeter,
H. S. M. The
Beauty of Geometry: Twelve Essays. New York: Dover, 1999.Euler,
L. "Elementa doctrine solidorum." Novi comm. acad. scientiarum imperialis
petropolitanae4, 109-160, 1752-1753. Reprinted in Opera, Vol. 26,
pp. 71-92.Numberphile. "How Many Panels on a Soccer Ball?"
May 13, 2014. https://www.youtube.com/watch?v=Nyo3TjKyu_c.Poincaré,
H. "Sur la généralisation d'un théorème d'Euler
relatif aux polyèdres." Comptes rendus hebdomadaires des séances
de l'Académie des Sciences117, 144-145, 1893.Schläfli,
L. "Theorie der vielfachen Kontinuität." Denkschriften der Schweizerischen
naturforschenden Gessel.38, 1-237, 1901.Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 252-253, 1999.Williams,
R. The
Geometrical Foundation of Natural Structure: A Source Book of Design. New
York: Dover, 1979.
, faces 
, and polyhedron edges 
of a simply connected (i.e., genus
0) polyhedron (or polygon).
It was discovered independently by Euler (1752) and Descartes, so it is also known
as the Descartes-Euler polyhedral formula. The formula also holds for some, but not
all, non-convex polyhedra.
is the number of polyhedron vertices, 
is the number of polyhedron
edges, and 
is the number of faces. For a proof, see Courant and Robbins
(1978, pp. 239-240).
-dimensional polytopes by Schläfli
(Coxeter 1973, p. 233),
surfaces, the formula can be generalized to the Poincaré
formula
.
and small
stellated dodecahedron 
, which no less than Schläfli
himself refused to recognize (Schläfli 1901, p. 134) since for these solids,
