  For a class of polyhedrons denoted $\mathbb K_n(r,\varepsilon)$, we construct
a bijective continuous area preserving map from $\mathbb K_n(r,\varepsilon)$ to
the sphere $\mathbb S^{2}(r)$, together with its inverse. Then we investigate
for which polyhedrons $\mathbb K_n(r',\varepsilon)$ the area preserving map can
be used for constructing a bijective continuous volume preserving map from
$\bar{\mathbb K}_n(r',\varepsilon)$ to the ball $\bar{\mathbb S^{2}}(r)$. These
maps can be further used in constructing uniform and refinable grids on the
sphere and on the ball, starting from uniform and refinable grids of the
polyhedrons $\mathbb K_n(r,\varepsilon)$ and $\bar{\mathbb
K}_{n}(r',\varepsilon)$, respectively. In particular, we show that HEALPix
grids can be obtained by mappings polyhedrons $\mathbb K_n(r,\varepsilon)$ onto
the sphere.
