This euclidean honeycomb uses that (for cells) in the sense of an infinite horohedron.

The infinite stack of this slab, then blending out those horohedra, would either result in rich or in gyrich.

Incidence matrix according to Dynkin symbol

s2s6o3x   (N → ∞)

...

s2s6x3o   (N → ∞)

...

xo3ox3xx3*a &#x   (N → ∞) → height = sqrt(2/3) = 0.816497

o.3o.3o.3*a     & | 6N |  2  2  2 |  1  2  1  3  2 | 1 1  3
------------------+----+----------+----------------+-------
x. .. ..        & |  2 | 6N  *  * |  1  1  0  1  0 | 1 1  1
.. .. x.        & |  2 |  * 6N  * |  0  1  1  0  1 | 1 0  2
oo3oo3oo3*a &#x   |  2 |  *  * 6N |  0  0  0  2  1 | 0 1  2
------------------+----+----------+----------------+-------
x.3o. ..        & |  3 |  3  0  0 | 2N  *  *  *  * | 1 1  0
x. .. x.3*a     & |  6 |  3  3  0 |  * 2N  *  *  * | 1 0  1
.. o.3x.        & |  3 |  0  3  0 |  *  * 2N  *  * | 1 0  1
xo .. ..    &#x & |  3 |  1  0  2 |  *  *  * 6N  * | 0 1  1
.. .. xx    &#x   |  4 |  0  2  2 |  *  *  *  * 3N | 0 0  2
------------------+----+----------+----------------+-------
x.3o.3x.3*a     & ♦ 3N | 3N 3N  0 |  N  N  N  0  0 | 2 *  *
xo3ox ..    &#x   ♦  6 |  6  0  6 |  2  0  0  6  0 | * N  *
xo .. xx3*a &#x & ♦  9 |  3  6  6 |  0  1  1  3  3 | * * 2N