# Quarter hypercubic honeycomb

In geometry, the quarter hypercubic honeycomb (or quarter n-cubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb. It is given a Schläfli symbol q{4,3...3,4} or Coxeter symbol qδ4 representing the regular form with three quarters of the vertices removed and containing the symmetry of Coxeter group ${\tilde {D}}_{n-1}$ for n ≥ 5, with ${\tilde {D}}_{4}$ = ${\tilde {A}}_{4}$ and for quarter n-cubic honeycombs ${\tilde {D}}_{5}$ = ${\tilde {B}}_{5}$ .

qδn Name Schläfli
symbol
Coxeter diagrams Facets Vertex figure
qδ3
quarter square tiling
q{4,4} or

or

h{4}={2} { }×{ }
{ }×{ }
qδ4
quarter cubic honeycomb
q{4,3,4} or
or

h{4,3}

h2{4,3}

Elongated
triangular antiprism
qδ5 quarter tesseractic honeycomb q{4,32,4} or
or

h{4,32}

h3{4,32}

{3,4}×{}
qδ6 quarter 5-cubic honeycomb q{4,33,4}

h{4,33}

h4{4,33}

Rectified 5-cell antiprism
qδ7 quarter 6-cubic honeycomb q{4,34,4}

h{4,34}

h5{4,34}
{3,3}×{3,3}
qδ8 quarter 7-cubic honeycomb q{4,35,4}

h{4,35}

h6{4,35}
{3,3}×{3,31,1}
qδ9 quarter 8-cubic honeycomb q{4,36,4}

h{4,36}

h7{4,36}
{3,3}×{3,32,1}
{3,31,1}×{3,31,1}

qδn quarter n-cubic honeycomb q{4,3n-3,4} ... h{4,3n-2} hn-2{4,3n-2} ...