Decomposition of balls in \(\mathbb{R}^{d}\) (Q2810731)

The paper is devoted to the problem of the decomposition of the \(d\)-dimensional ball into a finite number of disjoint congruent pieces. Remark that such a decomposition of a set differs from the concept of a tiling. In a tiling, the interiors of tiles must be disjoint, whereas a boundary point of a tile may belong to two or more distinct tiles.NEWLINENEWLINEIn the paper under review, first it is proved that the \(2s\)-dimensional ball (either open or closed) can be decomposed into \(m\) disjoint congruent pieces for every \(m \geq 4(2s+1)+2\) if \(s \geq 2\) and \(s \neq 3\). The proof uses: a subgroup of the four-dimensional special orthogonal group, a rational parametrization of special orthogonal matrices in \(\mathbb{R}^{d \times d}\), decomposition of infinite graphs defined by isometries.NEWLINENEWLINEAlso odd-dimensional cases are investigated and then combined with the even-dimensional results, thus yielding that the \(d\)-dimensional ball can be decomposed into a finite number of disjoint congruent pieces for \(d \geq 6\) and \(d = 3\), 4. It is shown that the minimal number of required pieces is less than \(20d\) if \(d > 10\).