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?