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 for n ≥ 5, with = and for quarter n-cubic honeycombs = .[1]

n Name Schläfli
symbol
Coxeter diagrams Facets Vertex figure
3 Square tiling uniform coloring 4.png
quarter square tiling
q{4,4} CDel nodes 11.pngCDel iaib.pngCDel nodes 10l.png or CDel nodes 11.pngCDel iaib.pngCDel nodes 01l.png

CDel nodes 10r.pngCDel iaib.pngCDel nodes 11.png or CDel nodes 01r.pngCDel iaib.pngCDel nodes 11.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h1.png

h{4}={2} { }×{ } Regular polygon 4 annotated.svg
{ }×{ }
4 Tetrahedral-truncated tetrahedral honeycomb slab.png
quarter cubic honeycomb
q{4,3,4} CDel branch 10r.pngCDel 3ab.pngCDel branch 10l.png or CDel branch 01r.pngCDel 3ab.pngCDel branch 01l.png
CDel branch 11.pngCDel 3ab.pngCDel branch.png or CDel branch.pngCDel 3ab.pngCDel branch 11.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
Tetrahedron.png
h{4,3}
Truncated tetrahedron.png
h2{4,3}
T01 quarter cubic honeycomb verf.png
Elongated
triangular antiprism
5 quarter tesseractic honeycomb q{4,32,4} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png or CDel nodes 01rd.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png or CDel nodes 01rd.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
Schlegel wireframe 16-cell.png
h{4,32}
Schlegel half-solid rectified 8-cell.png
h3{4,32}
Rectified tesseractic honeycomb verf.png
{3,4}×{}
6 quarter 5-cubic honeycomb q{4,33,4} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
Demipenteract graph ortho.svg
h{4,33}
5-demicube t03 D5.svg
h4{4,33}
Quarter 5-cubic honeycomb verf.png
Rectified 5-cell antiprism
7 quarter 6-cubic honeycomb q{4,34,4} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
Demihexeract ortho petrie.svg
h{4,34}
6-demicube t04 D6.svg
h5{4,34}
{3,3}×{3,3}
8 quarter 7-cubic honeycomb q{4,35,4} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
Demihepteract ortho petrie.svg
h{4,35}
7-demicube t05 D7.svg
h6{4,35}
{3,3}×{3,31,1}
9 quarter 8-cubic honeycomb q{4,36,4} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
Demiocteract ortho petrie.svg
h{4,36}
8-demicube t06 D8.svg
h7{4,36}
{3,3}×{3,32,1}
{3,31,1}×{3,31,1}
 
n quarter n-cubic honeycomb q{4,3n-3,4} ... h{4,3n-2} hn-2{4,3n-2} ...

See alsoEdit

ReferencesEdit

  1. ^ Coxeter, Regular and semi-regular honeycoms, 1988, p.318-319
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
    1. pp. 122–123, 1973. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
    2. pp. 154–156: Partial truncation or alternation, represented by q prefix
    3. p. 296, Table II: Regular honeycombs, δn+1
  • Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318 [2]
  • Klitzing, Richard. "1D-8D Euclidean tesselations".
Fundamental convex regular and uniform honeycombs in dimensions 2-9
          /   /  
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21