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Polyhedral Formula


A formula relating the number of polyhedron vertices V, faces F, and polyhedron edges E 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

 V+F-E=2,
(1)

where V=N_0 is the number of polyhedron vertices, E=N_1 is the number of polyhedron edges, and F=N_2 is the number of faces. For a proof, see Courant and Robbins (1978, pp. 239-240).

The formula was generalized to n-dimensional polytopes by Schläfli (Coxeter 1973, p. 233),

Pi_1:N_0=2
(2)
Pi_2:N_0-N_1=0
(3)
Pi_3:N_0-N_1+N_2=2
(4)
Pi_4:N_0-N_1+N_2-N_3=0
(5)
Pi_n:N_0-N_1+N_2-...+(-1)^(n-1)N_(n-1)=1-(-1)^n.
(6)

and proved by Poincaré (Poincaré 1893; Coxeter 1973, pp. 166-171; Williams 1979, pp. 24-25).

For genus g surfaces, the formula can be generalized to the Poincaré formula

 chi=V-E+F=chi(g),
(7)

where

 chi(g)=2-2g,
(8)

is the Euler characteristic, sometimes also known as the Euler-Poincaré characteristic. The polyhedral formula corresponds to the special case g=0.

There exist polytopes which do not satisfy the polyhedral formula, the most prominent of which are the great dodecahedron {5,5/2} and small stellated dodecahedron {5/2,5}, which no less than Schläfli himself refused to recognize (Schläfli 1901, p. 134) since for these solids,

 N_0-N_1+N_2=12-30+12=-6
(9)

(Coxeter 1973, p. 172).


See also

Dehn Invariant, Euler Characteristic, Descartes Total Angular Defect, Genus, Poincaré Formula, Polyhedral Graph, Polytope

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References

Aigner, M. and Ziegler, G. M. "Three Applications of Euler's Formula." Ch. 10 in Proofs from the Book. Berlin: 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 petropolitanae 4, 109-160, 1752-1753. Reprinted in Opera, Vol. 26, pp. 71-92.Poincaré, H. "Sur la généralisation d'un théorème d'Euler relatif aux polyèdres." Comptes rendus hebdomadaires des séances de l'Académie des Sciences 117, 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.

Cite this as:

Weisstein, Eric W. "Polyhedral Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolyhedralFormula.html

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