  We consider the 3D incompressible Euler equations in vorticity form in the
following fundamental domain for the octahedral symmetry group: $\{
(x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness
for $C^\alpha$ vorticities not necessarily vanishing on the boundary with any
$0<\alpha<1$, and establish finite-time singularity formation within the same
class for smooth and compactly supported initial data. The solutions can be
extended to all of $\mathbb{R}^3$ via a sequence of reflections, and therefore
we obtain finite-time singularity formation for the 3D Euler equations in
$\mathbb{R}^3$ with bounded and piecewise smooth vorticities.
