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An invariant of polyhedra in three-dimensional space that decides whether two polyhedra of the same volume are  "scissors congruent"  (see [[Equal content and equal shape, figures of|Equal content and equal shape, figures of]]; [[Hilbert problems|Hilbert problems]]; [[Polyhedron|Polyhedron]]).
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{{TEX|done}}{{MSC|52B10|52B45}}
  
Quite generally, a scissors-congruence invariant assigns to a polytope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d1200901.png" /> in space an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d1200902.png" /> in a [[Group|group]] such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d1200903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d1200904.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d1200905.png" /> is degenerate, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d1200906.png" /> if there is a motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d1200907.png" /> of the space such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d1200908.png" />.
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An invariant of polyhedra in three-dimensional space that decides whether two polyhedra of the same volume are  "scissors congruent" (see [[Equal content and equal shape, figures of]]; [[Hilbert problems]]; [[Polyhedron]]).
  
For the Dehn invariant, the group chosen is the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d1200909.png" />. To a polytope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d12009010.png" /> with edges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d12009011.png" /> one associates the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d12009012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d12009013.png" /> is the length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d12009014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d12009015.png" /> is the [[Dihedral angle|dihedral angle]] of the planes meeting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120090/d12009016.png" />. The Sydler theorem states that two polytopes in three-dimensional space are scissors equivalent if and only if they have equal volume and the same Dehn invariant, thus solving Hilbert's third problem in a very precise manner (cf. also [[Hilbert problems|Hilbert problems]]).
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Quite generally, a [[scissors-congruence]] invariant assigns to a polytope $P$ in space an element $D(P)$ in a [[group]] such that $D(P\cap P') + D(P \cup P') = D(P) + D(P')$, $D(P) = 0$ if $P$ is degenerate, and $D(P) = D(P')$ if there is a motion $g$ of the space such that $P' = gP$.
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For the Dehn invariant, the group chosen is the tensor product $\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{R} / 2\pi \mathbb{Z}$. To a polytope $P$ with edges $L_i$ one associates the element $D(P) = \sum_i |L_i| \otimes \delta_i$, where $|L_i|$ is the length of $L_i$ and $\delta_i$ is the [[dihedral angle]] of the planes meeting at $L_i$. The Sydler theorem states that two polytopes in three-dimensional space are scissors equivalent if and only if they have equal volume and the same Dehn invariant, thus solving Hilbert's third problem in a very precise manner (cf. also [[Hilbert problems]]).
  
 
For higher dimensions there is a generalization, called the Hadwiger invariant or Dehn–Hadwiger invariant. This has given results in dimension four, [[#References|[a4]]], and for the case when the group consists of translations only, [[#References|[a2]]].
 
For higher dimensions there is a generalization, called the Hadwiger invariant or Dehn–Hadwiger invariant. This has given results in dimension four, [[#References|[a4]]], and for the case when the group consists of translations only, [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Cartier,  "Decomposition des polyèdres: le point sur le troisième problème de Hilbert"  ''Sem. Bourbaki'' , '''1984/5'''  (1986)  pp. 261–288</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.H. Sah,  "Hilbert's third problem: scissors congruence" , Pitman  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.G. Boltianskii,  "Hilbert's third problem" , Wiley  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Jessen,  "Zur Algebra der Polytope"  ''Göttinger Nachrichte Math. Phys.''  (1972)  pp. 47–53</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Cartier,  "Decomposition des polyèdres: le point sur le troisième problème de Hilbert"  ''Sem. Bourbaki'' , '''1984/5'''  (1986)  pp. 261–288</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.H. Sah,  "Hilbert's third problem: scissors congruence" , Pitman  (1979)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  V.G. Boltianskii,  "Hilbert's third problem" , Wiley  (1978)</TD></TR>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Jessen,  "Zur Algebra der Polytope"  ''Göttinger Nachrichte Math. Phys.''  (1972)  pp. 47–53</TD></TR>
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</table>

Latest revision as of 21:33, 22 December 2014

2010 Mathematics Subject Classification: Primary: 52B10 Secondary: 52B45 [MSN][ZBL]

An invariant of polyhedra in three-dimensional space that decides whether two polyhedra of the same volume are "scissors congruent" (see Equal content and equal shape, figures of; Hilbert problems; Polyhedron).

Quite generally, a scissors-congruence invariant assigns to a polytope $P$ in space an element $D(P)$ in a group such that $D(P\cap P') + D(P \cup P') = D(P) + D(P')$, $D(P) = 0$ if $P$ is degenerate, and $D(P) = D(P')$ if there is a motion $g$ of the space such that $P' = gP$.

For the Dehn invariant, the group chosen is the tensor product $\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{R} / 2\pi \mathbb{Z}$. To a polytope $P$ with edges $L_i$ one associates the element $D(P) = \sum_i |L_i| \otimes \delta_i$, where $|L_i|$ is the length of $L_i$ and $\delta_i$ is the dihedral angle of the planes meeting at $L_i$. The Sydler theorem states that two polytopes in three-dimensional space are scissors equivalent if and only if they have equal volume and the same Dehn invariant, thus solving Hilbert's third problem in a very precise manner (cf. also Hilbert problems).

For higher dimensions there is a generalization, called the Hadwiger invariant or Dehn–Hadwiger invariant. This has given results in dimension four, [a4], and for the case when the group consists of translations only, [a2].

References

[a1] P. Cartier, "Decomposition des polyèdres: le point sur le troisième problème de Hilbert" Sem. Bourbaki , 1984/5 (1986) pp. 261–288
[a2] C.H. Sah, "Hilbert's third problem: scissors congruence" , Pitman (1979)
[a3] V.G. Boltianskii, "Hilbert's third problem" , Wiley (1978)
[a4] B. Jessen, "Zur Algebra der Polytope" Göttinger Nachrichte Math. Phys. (1972) pp. 47–53
How to Cite This Entry:
Dehn invariant. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dehn_invariant&oldid=35803
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article