MATHEMATICS

V. B. ZYLYOV

ON THE EQUICOMPOSABILITY OF TWO EQUICOMPLEMENTABLE POLYHEDRA

(Presented by Academician P. S. Novikov, January 13, 1964)

If two polyhedra \(A\) and \(A'\) of equal volumes can be complemented, respectively, by a finite number of congruent polyhedra \(P_1\) and \(P_1'\), \(P_2\) and \(P_2'\), ..., \(P_k\) and \(P_k'\) (all the polyhedra introduced above have no common interior points pairwise) in such a way that both polyhedra so complemented become congruent, then \(A\) and \(A'\) are called equicomplementable. If \(A\) and \(A'\) can be cut into a finite number of respectively congruent polyhedra, then \(A\) and \(A'\) are called equicomposable.

Theorem. In any homogeneous \(n\)-dimensional Riemannian space, any two equicomplementable polyhedra are equicomposable.

An analogous theorem for the special case when the \(n\)-dimensional space is Euclidean, the group \(G\) contains the full group of parallel translations, and the bodies are polyhedra, was proved for \(n=3\) by Sydler \((^{1,2})\) and for arbitrary \(n\) by Hadwiger \((^{3})\). Moreover, Sydler and Hadwiger used very subtle special considerations essentially connected with the fact that the bodies are polyhedra and the space is Euclidean: namely, they used the method of “steps,” which had already been applied by Euclid to derive the volume of a tetrahedron, introduced the notion of the cylindricity index of a polyhedron, and applied induction on this index.

The proof in the present paper is based on the most general set-theoretic considerations and, although it is presented in the text only for polyhedra, is valid for bodies under very general assumptions concerning them.* We pass to the proof of the theorem.

Let \(F_A\) (Fig. 1a) be the polyhedron to which the white polyhedron \(A\) is complemented by the shaded polyhedra \(P_1, P_2, \ldots, P_k\) (in Fig. 1 we have taken, for simplicity, \(k=2\)), and let \(F_{A'}\) be the polyhedron to which the white \(A'\) is complemented by the shaded polyhedra \(P_1', P_2', \ldots, P_k'\), where \(F_A\) is congruent to \(F_{A'}\), and each \(P_i\) is congruent to the corresponding \(P_i'\).

Our aim is to prove the equicomposability of the white polyhedra \(A\) and \(A'\), equicomplemented by the polyhedra \(P_1, P_2, \ldots, P_k\) and, respectively, \(P_1', P_2', \ldots, P_k'\) to congruent polyhedra \(F_{A'}\) and \(F_A\). Let \(\alpha\) be a certain fixed motion carrying \(F_{A'}\) into \(F_A\). Then the polyhedron \(A\) is
\[ F_A - P_1 - P_2 - \cdots - P_k, \]
and the polyhedron \(A'\) is congruent to the polyhedron
\[ \bar A = F_A - Q_1 - Q_2 - \cdots - Q_k, \]
where \(Q_1=\alpha P_1'\), \(Q_2=\alpha P_2'\), ..., \(Q_k=\alpha P_k'\). The \(Q_1\), shown by a dotted line, may consist both of parts of the polyhedron \(A\) and of parts of some of the polyhedra \(P_i\).

Clearly, we may assume that \(2v(P_i)<v(A)\) (\(v\) is the volume of a polyhedron; \(i=1,2,\ldots,k\)), for otherwise one may subdivide \(P_i'\) by planes into smaller parts, and \(P_i\) into the same parts. Now subdivide (if necessary) that region of the polyhedron \(Q_1\) which consists of parts of the shaded polyhedra \(P_1,\ldots,P_k\) into smaller shaded polyhedra \(S_1,\ldots,S_p\) (in Fig. 1a \(p=2\))

* For example, in the plane, by bodies one may understand bounded domains with piecewise-analytic boundaries.

so that polyhedra equal to them could be placed in \(A-Q_1\) (where \(A-Q_1\) is that part of \(A\) which is not covered by \(Q_1\)), and so that \(S_1,\ldots,S_p\) would have no common interior points. This can be done by virtue of the inequality \(2v(P_1)<v(A)\).

Let us place these hatched polyhedra \(S_1,\ldots,S_p\) in \(A-Q_1\), and cut out the white parts of \(A-Q_1\) lying under them and put them into the regions of the polyhedron \(Q_1\) that have been freed by them. After this, within the boundaries of \(Q_1\) there is a white polyhedron assembled from parts of \(A\) (Fig. 1b).

Fig. 1

Let us now take out this entire composite white polyhedron, which fills the position of the polyhedron \(Q_1\), move to the place of \(Q_1\) all the parts of the hatched polyhedron \(P_1\) that are in \(F_A\), assembling them back into a polyhedron congruent to \(P_1\) (these parts in Fig. 1b are \(r_1\) and \(S_1\)), and cut the removed composite white polyhedron, also congruent to \(P_1\), in the same way as the hatched polyhedron \(P_1\), congruent to it and placed in the position of \(Q_1\), has been cut (in Fig. 1a \(P_1\) is divided into two hatched polyhedra \(r_1\) and \(S_1\)); and the white polyhedra obtained from this cutting we distribute over the places where the parts of \(P_1\) lay before their transfer to the place of \(Q_1\) (Fig. 1c).

Next we repeat exactly the same process for \(Q_2\). The fact that pieces from \(P_2,\ldots,P_k\) may already have been cut out and transferred into \(A\) in the first process does not prevent us from assembling, in the same way, all parts of the polyhedron \(P_2\) within the boundaries of the polyhedron \(Q_2\).

In this second step it is essential that \(Q_2\) will not overlap \(P_1\), which has been moved to the place of \(Q_1\), because \(P_2'\) did not overlap \(P_1'\), while \(Q_1\) and \(Q_2\) are obtained from \(P_1'\) and \(P_2'\) by one and the same motion \(a\). We proceed analogously in all subsequent steps, considering the polyhedra \(Q_3,\ldots,Q_n\). In this process the original white polyhedron \(A\) successively passes into new white polyhedra equidecomposable with one another and, finally, into the polyhedron \(\overline{A}\), which completes the proof.

Received
2 I 1964

REFERENCES

1. J. Sydler, Comm. Math. Helv., 16 (1943–1944).
2. V. G. Boltyanskii, Equicomplementary and Equidecomposable Figures, Moscow, 1956.
3. H. Hadwiger, Math. Zs., 55, 3 (1952).