Ingeometry, the quarter hypercubic honeycomb (orquarter 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 
| qδn | Name | Schläfli symbol |
Coxeter diagrams | Facets | Vertex figure | ||
|---|---|---|---|---|---|---|---|
| qδ3 |  quarter square tiling |
q{4,4} |    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} | ... | |
|
Fundamental convex regular and uniform honeycombs in dimensions 2–9
| ||||||
|---|---|---|---|---|---|---|
| Space | Family | 
|
|
|
|
 /  / 
|
| E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
| E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
