Disclaimer: i am bioinformatician and programmer, please excuse if my wording and definitions are far from elegant and occasionally imprecise.
Intro: I am interested in space tessellations of n dimensional spaces with following properties:
1 a tessellation must be made with only one type of element, with identical angles and edge lengths
e.g. cubic honeycomb is okay, pythagorean tiling is not (one type but two sub-types of different size) and https://en.wikipedia.org/wiki/Truncated_trihexagonal_tiling is also not (three types of elements)
2 for n dimensional space, each n dimensional element is built out of one type of n-1 dimensional elements,
e.g. cubic honeycomb is okay, hexagonal prismatic honeycomb is not
3 elements must connect only through n-1 dimensional elements
e.g. hexagons on a 2D plane are okay, squares are not okay for 2D plane (they connect to 8 others, 4 through edges, 4 through vertices)
Question: For 2D, hexagons are an example which satisfy these conditions. Are there any examples for N > 2?
Sub-question: Does https://en.wikipedia.org/wiki/Rhombic_dodecahedral_honeycomb satisfy these conditions? (i cannot come up with an idea on how to verify this)
Note 1: I am mainly interested in N = 3 and N = 4, but general solution would be best.
Note 2: If they exist, pointers what to look for (keywords, books, articles, etc) would be greatly appreciated
EDIT: A related question: Are there higher-dimensional tessellations touching only nearest neighbours?
Here is an example that satisfies your criteria, I think:
The (equilateral, most symmetric) diplo-simplex with $2N+2$ vertices tiles space in 2D, 3D, and 7D.
2D diplo-simplex is hexagon, vertices of tesselation correspond to non-lattice hexagonal packing.
3D diplo-simplex is cube, vertices correspond to trivial cubic lattice. Doesn't the cubic lattice in arbitrary dimension satisfy your criteria?
7D diplo-simplex tesselation's vertices correspond to E7* lattice, i.e. the sole Delaunay polytope of the E7* lattice is the 7D diplo-simplex.