It is paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity.
[1] A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps.
They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs.
There are nine uniform honeycombs in the [4,4,4] Coxeter group family, including this regular form.
It has cube and square tiling facets, with a triangular prism vertex figure.
It contains cube and truncated square tiling facets, with a mirrored sphenoid vertex figure.