The Dehn invariant of a three-dimensional polyhedron 
is the tensor
product expression
![D(P)=sum_(e)l(e) tensor [theta_e] in R tensor _Q(R/piQ),](/images/equations/DehnInvariant/NumberedEquation1.svg) |
(1)
|
where the sum is over the edges, 
is the edge length, and ![[theta_e]](/images/equations/DehnInvariant/Inline3.svg)
is the class of the interior dihedral
angle, measured in radians, modulo rational
multiples of 
(Chentouf and Sun 2023, p. 2). It remains unchanged under polyhedron
dissection and reassembly.
Dehn (1900) 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.
Every rational tetrahedron has Dehn invariant zero because all six of its dihedral angles vanish
modulo rational multiples of 
. Rational tetrahedra therefore form a special subclass of
the Dehn-invariant-zero tetrahedra (Chentouf and Sun 2023).
Conway et al. (1999, p. 322) 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 eligible pairs of primes 
and squarefree
positive integers 
. 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 squarefree positive 
and with relatively
prime 
and 
,
and if the prime factorization of 
is 
(including multiplicity), then
 |
(2)
|
for some rational 
.
Notable values of 
include
 |  |  |
(3)
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(4)
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(5)
|
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(6)
|
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(7)
|
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(8)
|
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(9)
|
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(10)
|
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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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(19)
|
(Conway et al. 1999, p. 323; 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, p. 331) give Dehn invariants in terms of the basis of angles 
for unit Platonic
and non-snub Archimedean solids.
Precomputed representations of Dehn invariants for many polyhedra are implemented in the Wolfram Language as PolyhedronData[poly, "DehnInvariant"]. These values use scalar representations; the invariant itself is the tensor expression above.
