  We construct a group $\Gamma_{n}^{4}$ corresponding to the motion of points
in $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We study
homomorphisms from pure braids on $n$ strands to the product of copies of
$\Gamma_{n}^{4}$. We will also study the group of pure braids in
$\mathbb{R}^{3}$, which is described by a fundamental group of the restricted
configuration space of $\mathbb{R}^{3}$, and define the group homomorphism from
the group of pure braids in $\mathbb{R}^{3}$ to $\Gamma_{n}^{4}$. In the end of
this paper we give some comments about relations between the restricted
configuration space of $\mathbb{R}^{3}$ and triangulations of the 3-dimensional
ball and Pachner moves.
