Acronym  pd{3,5,3} 
Name  partial diminishing (old: truncation) of hyperbolic order 3 icosahedral tesselation 
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Circumradius  sqrt[(13 sqrt(5))/22] = 0.509376 i 
Confer 

The vertex figure is a chiral, tetrahedrally diminished dodecahedron, faceting those 4 vertices to the depth of just one reduced edge set, i.e. down to their neighbouring vertices.
As can be seen from the incidence matrix below, by the overall symmetry the edges of the pap fall into 2 classes: the 10 edges of the base plus the 5 "inclining" lacings fall in one, while the 5 "declining" lacings make up the other.
There is also a not so often partial diminishing of the starting honeycomb (cf., here we use one sixth of its vertices), respectively to the vertex figure, so that the there resulting honeycomb spd{3,5,3} uses the additional qualifier "semi". As the vertex set in both cases is a mere subset of the original tesselation, the overall curvature still remains the same.
Incidence matrix according to Dynkin symbol
(N → ∞) 5N ♦ 12 4  12 18  4 12 +++ 2  30N *  2 2  1 3 2  * 10N  0 3  0 3 +++ 5  5 0  12N *  1 1 3  2 1  * 30N  0 2 +++ ♦ 20  30 0  12 0  N * ♦ 10  15 5  2 10  * 6N
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