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 
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 
for prime 
and square-free positive integer 
. They then show that every pure geodetic angle is uniquely
expressible as a rational multiple of 
plus an integral linear combination of the angles 
, meaning the angles 
supplemented by 
form a basis for the space of mixed geodetic angles. They
then show that if 
for integers 
,
, 
with with square-free positive 
and with relatively prime 
and 
, and if the prime factorization of 
is 
(including multiplicity), then
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(1)
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for some rational 
.
Notable values of 
include
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(2)
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(5)
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(11)
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(12)
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(13)
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(14)
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(15)
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(16)
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(17)
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(18)
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(Conway et al. 1999; OEIS A195696, A188595, and A105199),
where 
is the dihedral angle of the regular
dodecahedron, 
of the regular icosahedron, and 
of the regular tetrahedron.
Using these results, Conway et al. (1999) give Dehn invariants in terms of the basis of angles 
for unit Platonic and non-snub Archimedean
solids.
Precomputed Dehn invariants for many polyhedra are implemented in the Wolfram Language as PolyhedronData[poly, "DehnInvariant"].
