# D4 polytope

In 4-dimensional geometry, there are 7 uniform 4-polytopes with reflections of D$_4$ symmetry, all are shared with higher symmetry constructions in the B$_4$ or F$_4$ 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 D$_4$ Coxeter group, and other subgroups. The B$_4$ coxeter planes are also displayed, while D$_4$ polytopes only have half the symmetry. They can also be shown in perspective projections of Schlegel diagrams, centered on different cells.
D$_4$ polytopes related to B$_4$ :
| index | Name Coxeter diagram = = | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Schlegel diagrams | Schlegel diagrams | Net |
| index | Name Coxeter diagram = = | B$_4$ [8] | D$_4$, B$_3$ [6] | D$_3$, B$_2$ [4] | Cube centered | Tetrahedron centered | Net |
| 1 | demitesseract (Same as 16-cell) = = h{4,3,3} = = {3,3,4} {3,3$^{1,1}$} | | | | | | |
| 2 | cantic tesseract (Same as truncated 16-cell) = = h$_2${4,3,3} = = t{3,3,4} t{3,3$^{1,1}$} | | | | | | |
| 3 | runcic tesseract birectified 16-cell (Same as rectified tesseract) = = h$_3${4,3,3} = = r{4,3,3} 2r{3,3$^{1,1}$} | | | | | | |
| 4 | runcicantic tesseract bitruncated 16-cell (Same as bitruncated tesseract) = = h$_{2,3}${4,3,3} = = 2t{4,3,3} 2t{3,3$^{1,1}$} | | | | | | |
D$_4$ polytopes related to F$_4$ and B$_4$ :
| index | Name Coxeter diagram = = | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Schlegel diagrams | Schlegel diagrams | Parallel 3D | Net |
| index | Name Coxeter diagram = = | F$_4$ [12] | B$_4$ [8] | D$_4$, B$_3$ [6] | D$_3$, B$_2$ [2] | Cube centered | Tetrahedron centered | D$_4$ [6] | Net |
| 5 | rectified 16-cell (Same as 24-cell) = = {3$^{1,1,1}$} = r{3,3,4} = {3,4,3} | | | | | | | | |
| 6 | cantellated 16-cell (Same as rectified 24-cell) = = r{3$^{1,1,1}$} = rr{3,3,4} = r{3,4,3} | | | | | | | | |
| 7 | cantitruncated 16-cell (Same as truncated 24-cell) = = t{3$^{1,1,1}$} = tr{3,3$^{1,1}$} = tr{3,3,4} = t{3,4,3} | | | | | | | | |
| 8 | (Same as snub 24-cell) = = s{3$^{1,1,1}$} = sr{3,3$^{1,1}$} = 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 | Coxeter diagrams | Coxeter diagrams |
| # | Name(s) | Base point | Johnson | D$_4$ | B$_4$ | F$_4$ |
| 1 | hγ$_4$ | Even (1,1,1,1) | demitesseract | | | |
| 3 | h$_3$γ$_4$ | Even (1,1,1,3) | runcic tesseract | | | |
| 2 | h$_2$γ$_4$ | Even (1,1,3,3) | cantic tesseract | | | |
| 4 | h$_{2,3}$γ$_4$ | Even (1,3,3,3) | runcicantic tesseract | | | |
| 1 | t$_3$γ$_4$ = β$_4$ | (0,0,0,2) | 16-cell | | | |
| 5 | t$_2$γ$_4$ = t$_1$β$_4$ | (0,0,2,2) | rectified 16-cell | | | |
| 2 | t$_{2,3}$γ$_4$ = t$_{0,1}$β$_4$ | (0,0,2,4) | truncated 16-cell | | | |
| 6 | t$_1$γ$_4$ = t$_2$β$_4$ | (0,2,2,2) | cantellated 16-cell | | | |
| 9 | t$_{1,3}$γ$_4$ = t$_{0,2}$β$_4$ | (0,2,2,4) | cantellated 16-cell | | | |
| 7 | t$_{1,2,3}$γ = t$_{0,1,2}$β$_4$ | (0,2,4,6) | cantitruncated 16-cell | | | |
| 8 | s{3$^{1,1,1}$} | (0,1,φ,φ+1)/√2 | Snub 24-cell | | | |