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Dehn Invariant


The Dehn invariant is a constant defined using the angles and edge lengths of a three-dimensional polyhedron. It is significant because it remains constant under polyhedron dissection and reassembly.

Dehn (1902) showed that two interdissectable polyhedra must have equal Dehn invariants, settling the third of Hilbert's problems. Later, Sydler (1965) showed that two polyhedra can be dissected into each other iff they have the same volume and the same Dehn invariant.

Having Dehn invariant zero is necessary (but not sufficient) for a polyhedron to be space-filling. In general, as a result of the above, a polyhedron is either itself space-filling or else can be cut up and reassembled into a space-filling polyhedron iff its Dehn invariant is zero.

Zonohedra have Dehn invariant 0.

Conway et al. (1999) call an angle theta a "pure geodetic angle"' if any one (and therefore each) of its six squared trigonometric functions is rational (or infinite), use "mixed geodetic angle" to mean a linear combination of pure geodetic angles with rational coefficients, and define certain angles <p>_d for prime p and square-free positive integer d. They then show that every pure geodetic angle is uniquely expressible as a rational multiple of pi plus an integral linear combination of the angles <p>_d, meaning the angles <p>_d supplemented by pi form a basis for the space of mixed geodetic angles. They then show that if tantheta=bsqrt(d)/a for integers a, b, d with with square-free positive d and with relatively prime a and b, and if the prime factorization of a^2+db^2 is p_1p_2...p_n (including multiplicity), then

 theta=tpi+/-<p_1>_d+/-<p_2>_d+/-...+/-<p_n>_d
(1)

for some rational t.

Notable values of <p>_d include

<3>_2=sin^(-1)(sqrt(2/3))
(2)
=1/2pi-1/2alpha_t
(3)
=54 degrees44^'8.2^('')
(4)
<3>_5=1/2sin^(-1)(sqrt(5/9))
(5)
=1/4pi-1/2alpha_i
(6)
=24 degrees5^'41.4^('')
(7)
<5>_1=sin^(-1)(sqrt(4/5))
(8)
=pi-alpha_d
(9)
=63 degrees26^'5.8^('')
(10)

(Conway et al. 1999), where alpha_d is the dihedral angle of the regular dodecahedron, alpha_i of the regular icosahedron, and alpha_t of the regular tetrahedron.

Using these results, Conway et al. (1999) give Dehn invariants in terms of the basis of angles <p>_d for unit Platonic and non-snub Archimedean solids.

Precomputed Dehn invariants for many polyhedra are implemented in the Wolfram Language as PolyhedronData[poly, "DehnInvariant"].


See also

Dihedral Angle, Dissection, Ehrhart Polynomial, Hilbert's Problems, Polyhedron Dissection

Explore with Wolfram|Alpha

References

Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22, 321-332, 1999.Dehn, M. "Über raumgleiche Polyeder." Nachr. Königl. Ges. der Wiss. zu Göttingen f. d. Jahr 1900, 345-354, 1900.Dehn, M. "Über den Rauminhalt." Math. Ann. 55, 465-478, 1902.Kagan, B. "Über die Transformation der Polyeder." Math. Ann. 57, 421-424, 1903.Sydler, J.-P. "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidean à trois dimensions." Comment. Math. Helv. 40, 43-80, 1965.

Referenced on Wolfram|Alpha

Dehn Invariant

Cite this as:

Weisstein, Eric W. "Dehn Invariant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DehnInvariant.html

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