UDC 513.3

MATHEMATICS

V. B. Zylev

ON \(G\)-COMPOSABILITY AND \(G\)-COMPLEMENTABILITY

(Presented by Academician P. S. Novikov on 10 V 1967)

In this note only compact sets that are closures of open sets will be considered; for brevity we shall call them bodies.

Let a group \(G\) act in a topological space \(R\), possessing the following property: if \(A\) and \(B\) are bodies, then in the group \(G\) there exists a transformation \(g\) such that \(gA \subset B\).

If the intersections \(P_i \cap P_j\), \(a_iP_i \cap a_jP_j\) \((i \ne j)\), where \(a_i \in G\), are either empty or consist of points that are boundary points for the intersecting bodies, and
\[ C=\bigcup_{i=1}^{n} P_i,\qquad C'=\bigcup_{i=1}^{n} a_iP_i, \]
then \(C\) and \(C'\) will be called \(G\)-composable, and we shall write
\[ C \sim_g C', \]
or simply \(C \sim C'\).

If for bodies \(A\) and \(A'\) there can be found such bodies \(P_i\) and such transformations \(a_i \in G\) \((i=1,\ldots,n)\) that the intersections \(P_i \cap P_j\), \(a_iP_i \cap a_jP_j\) \((i \ne j)\), \(A \cap P_i\), \(A' \cap a_iP_i\) \((i=1,\ldots,n)\), are either empty or consist of boundary points, and
\[ A \cup \bigcup_{i=1}^{n} P_i = A' \cup \bigcup_{i=1}^{n} a_iP_i, \]
then \(A\) and \(A'\) will be called \(G\)-complementable.

Theorem 1. Two \(G\)-complementable sets \(A\) and \(A'\) are always \(G\)-composable.

Proof. Given
\[ A \cup P = A' \cup P', \]
where
\[ P=\bigcup_{i=1}^{n} P_i,\qquad P'=\bigcup_{i=1}^{n} a_iP_i,\qquad a_i\in G\ (i=1,\ldots,n). \]

If \(A=A'\), there is nothing to prove; hence one may assume that one of them is not a part of the other; let, for example, \(A' \not\subset A\), then \(\overline{A' \setminus A}\) is a body.

By the property of the group \(G\) formulated above, there exists a transformation \(\alpha \in G\) such that \(\alpha P \subset A\), and such an \(\alpha' \in G\) that
\[ \alpha'P' \subset \overline{A' \setminus A}. \]
Perform this transformation \(\alpha\) on \(P\), and remove those parts of \(A\) which have been covered by the body \(\alpha P\); denote them by \(P^*\), and send them by the transformation \(\alpha^{-1}\) to the place where the body \(P\) was previously located.

We have:
\[ \left(\overline{A \setminus \alpha P}\right)\cup \alpha^{-1}P^* \sim A, \]
since the body
\[ \left(\overline{A \setminus \alpha P}\right)\cup \alpha^{-1}P^* \]
is obtained from \(A\) by rearranging parts. In the same way we perform the transformation \(\alpha'\) on \(P'\), and remove those parts of \(A'\) which have become covered by the body \(\alpha'P'\); denote them by \(P'^*\), and by the transformation \((\alpha')^{-1}\) send them to the places freed by the removal of \(P'\).

We have:
\[ \left(\overline{A' \setminus \alpha'P'}\right)\cup(\alpha')^{-1}P'^* \sim A', \]
since the body
\[ \left(\overline{A' \setminus \alpha'P'}\right)\cup(\alpha')^{-1}P'^* \]
is obtained from \(A'\) by rearranging parts.

Introduce the new bodies
\[ Q=A\cup P=A'\cup P' \]
and
\[ R=\overline{Q\setminus \alpha P\setminus \alpha'P'}. \]
Since \(\alpha P \cap \alpha'P'\) is empty, the bodies
\[ R\cup \alpha P \]
and
\[ R\cup \alpha'P' \]
will be \(G\)-composable.

We have the equalities
\[ \left(\overline{A \setminus \alpha P}\right)\cup \alpha^{-1}P^* = R \cup \alpha'P',\qquad \left(\overline{A' \setminus \alpha'P'}\right)\cup \]

\[ \frac{\cup (a')^{-1}P^*}{(A'\setminus a'P')\cup (a')^{-1}P^*} =\overline{R}\cup aP, \]
but earlier we had \((\overline{A\setminus aP})\cup a^{-1}P^*\sim A\), \((A'\setminus a'P')\cup (a')^{-1}P^*\sim A'\), and, by the transitivity of \(G\)-composability, \(A\sim A'\). The theorem is proved.

Let us derive from this theorem the following:

Theorem 2. In three-dimensional Euclidean space, any two polyhedra can be decomposed into a finite number of pairwise similar polyhedra.

Consider two equal tetrahedra \(T\) and \(T'\). It is known that the tetrahedron \(T\) can be decomposed into two similar-to-\(T\) tetrahedra \(T_1\) and \(T_2\), and into two prisms \(\Pi_1\) and \(\Pi_2\). The tetrahedron \(T'\) can likewise be decomposed into three tetrahedra similar to it and into four prisms:
\(T'\sim T_1'\cup T_2'\cup T_3'\cup \Pi_1'\cup \Pi_2'\cup \Pi_3'\cup \Pi_4'\).
a) \(T\sim T_1\cup T_2\cup \Pi_1\cup \Pi_2\). But it is known that there exists a prism \(\Pi_5'\) such that
\(\Pi_5'\sim \Pi_1'\cup \Pi_2'\cup \Pi_3'\cup \Pi_4'\); then, evidently:
b) \(T'\sim T_1'\cup T_2'\cup T_3'\cup \Pi_5'\).

Applying our theorem to the equalities a) and b) (taking into account that \(T=T'\), \(T_1\) is similar to \(T_1'\), \(T_2\) is similar to \(T_2'\), and \(\Pi\) and \(\Pi_5'\) can be decomposed into pairwise similar ones), we obtain

\[ \overline{T\setminus T_1\setminus T_2\setminus \Pi_1}\sim \Pi_2, \qquad \overline{T'\setminus T_1'\setminus T_2'\setminus \Pi_5'}\sim T_3'. \]

The prism \(\Pi_2\) and the tetrahedron \(T_3'\) are similarly composed by virtue of our theorem. Thus, any tetrahedron is similarly composed with a prism. It follows from this that any two three-dimensional polyhedra can be decomposed into a finite number of pairwise similar polyhedra.

In conclusion I express my gratitude to V. A. Efremovich for his help.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
30 III 1967