List of D4 polytopes

In 4-dimensional geometry, there are 7 uniform 4-polytopes with reflections of D4 symmetry, all are shared with higher symmetry constructions in the B4 or F4 symmetry families. there is also one half symmetry alternation, the snub 24-cell.

Visualizations

Each can be visualized as symmetric orthographic projections in Coxeter planes of the D4 Coxeter group, and other subgroups. The B4 coxeter planes are also displayed, while D4 polytopes only have half the symmetry. They can also be shown in perspective projections of Schlegel diagrams, centered on different cells.

D4 polytopes related to B4
index Name
Coxeter diagram
=
= 		Coxeter plane projections Schlegel diagrams Net
B4
[8]		 D4, B3
[6]		 D3, B2
[4]		 Cube
centered		 Tetrahedron
centered
12 demitesseract
(Same as 16-cell)
= = h{4,3,3}
= = {3,3,4}
{3,31,1}
17 cantic tesseract
(Same as truncated 16-cell)
= = h2{4,3,3}
= = t{3,3,4}
t{3,31,1}
11 runcic tesseract
birectified 16-cell
(Same as rectified tesseract)
= = h3{4,3,3}
= = r{4,3,3}
2r{3,31,1}
16 runcicantic tesseract
bitruncated 16-cell
(Same as bitruncated tesseract)
= = h2,3{4,3,3}
= = 2t{4,3,3}
2t{3,31,1}
D4 polytopes related to F4 and B4
index Name
Coxeter diagram
= = 		Coxeter plane projections Schlegel diagrams Parallel
3D		 Net
F4
[12]		 B4
[8]		 D4, B3
[6]		 D3, B2
[2]		 Cube
centered		 Tetrahedron
centered		 D4
[6]
22 rectified 16-cell
(Same as 24-cell)
=
=
{31,1,1} = r{3,3,4} = {3,4,3}
23 cantellated 16-cell
(Same as rectified 24-cell)
=
=
r{31,1,1} = rr{3,3,4} = r{3,4,3}
24 cantitruncated 16-cell
(Same as truncated 24-cell)
=
=
t{31,1,1} = tr{3,31,1} = tr{3,3,4} = t{3,4,3}
31 (Same as snub 24-cell)
=
=
s{31,1,1} = sr{3,31,1} = sr{3,3,4} = s{3,4,3}

Coordinates

The base point can generate the coordinates of the polytope by taking all coordinate permutations and sign combinations. The edges' length will be √2. Some polytopes have two possible generator points. Points are prefixed by Even to imply only an even count of sign permutations should be included.

# Name(s) Base point Johnson Coxeter diagrams
D4 B4 F4
[12] hγ4 Even (1,1,1,1) demitesseract
[11] h3γ4 Even (1,1,1,3) runcic tesseract
[17] h2γ4 Even (1,1,3,3) cantic tesseract
[16] h2,3γ4 Even (1,3,3,3) runcicantic tesseract
[12] t3γ4 = β4 (0,0,0,2) 16-cell
[22] t2γ4 = t1β4 (0,0,2,2) rectified 16-cell
[17] t2,3γ4 = t0,1β4 (0,0,2,4) truncated 16-cell
[11] t1γ4 = t2β4 (0,2,2,2) cantellated 16-cell
[23] t1,3γ4 = t0,2β4 (0,2,2,4) cantellated 16-cell
[24] t1,2,3γ = t0,1,2β4 (0,2,4,6) cantitruncated 16-cell
[31] s{31,1,1} (0,1,φ,φ+1)/√2 Snub 24-cell

References

    External links

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