proofs

Hilbert’s third problem: decomposing polyhedra 61
Two polytopes P, P ′ ⊆ R d are congruent if there is some length-preserving
affine map that takes P to P ′ . Such a map may reverse the orientation
of space, as does the reflection of P in a hyperplane, which takes P to
a mirror image of P . They are combinatorially equivalent if there is a
bijection from the faces of P to the faces of P ′ that preserves dimension
and inclusions between the faces. This notion of combinatorial equivalence
is much weaker than congruence: for example, our figure shows a unit cube
and a “skew” cube that are combinatorially equivalent (and thus we would
call any one of them “a cube”), but they are certainly not congruent.
A polytope (or a more general subset of R d ) is called centrally symmetric
if there is some point x 0 ∈ R d such that
x 0 + x ∈ P ⇐⇒ x 0 − x ∈ P.
In this situation we call x 0 the center of P .
Combinatorially equivalent polytopes
References
[1] V. G. BOLTIANSKII: Hilbert’s Third Problem, V. H. Winston & Sons (Halsted
Press, John Wiley & Sons), Washington DC 1978.
[2] D. BENKO: A new approach to Hilbert’s third problem, Amer. Math. Monthly,
114 (2007), 665-676.
[3] M. DEHN: Ueber raumgleiche Polyeder, Nachrichten von der Königl.
Gesellschaft der Wissenschaften, Mathematisch-physikalische Klasse (1900),
345-354.
[4] M. DEHN: Ueber den Rauminhalt, Mathematische Annalen 55 (1902),
465-478.
[5] C. F. GAUSS: “Congruenz und Symmetrie”: Briefwechsel mit Gerling,
pp. 240-249 in: Werke, Band VIII, Königl. Gesellschaft der Wissenschaften
zu Göttingen; B. G. Teubner, Leipzig 1900.
[6] D. HILBERT: Mathematical Problems, Lecture delivered at the International
Congress of Mathematicians at Paris in 1900, Bulletin Amer. Math. Soc. 8
(1902), 437-479.
[7] B. KAGAN: Über die Transformation der Polyeder, Mathematische Annalen
57 (1903), 421-424.
[8] G. M. ZIEGLER: Lectures on Polytopes, Graduate Texts in Mathematics 152,
Springer, New York 1995/1998.