# D4.16 (EntityTopic, 13)

### From Hi.gher. Space

Most of the content on this page is taken directly from the forum thread where it was discovered.

**D4.16** is a CRF polytope discovered by Klitzing and constructed by User:Quickfur in March 2016.

Its symbol is: xfox-2-oxfo-3-ooox&#xt

It has 3 mibdies and 2 tetrahedra on one side, 1 trigonal prism, 2 octahedra, 3 square pyramids, and 6 tetrahedra on the other side, for a total of 17 cells. It also has 17 vertices, the same as the cell count.

## Coordinates

The coordinates are:

# x2o3o <±1, 0, 0, -phi^2/√3> # f2x3o <±phi, ±1, -1/√3, 0> <±phi, 0, 2/√3, 0> # o2f3o <0, ±phi, -phi/√3, phi/√3> <0, 0, 2*phi/√3, phi/√3> # x2o3x <±1, ±1, 1/√3, phi^2/√3> <±1, 0, -2/√3, phi^2/√3>

## Images

Here's a look at the 3 mibdies surrounding an edge:

The 2 tetrahedra that fill in the gaps between the mibdies should be obvious.

Now here's the far side, which has a most interesting configuration of cells:

I highlighted the trigonal prism which lies antipodal to the edge shared by the 3 mibdies. We see a most interesting pattern of tetrahedron-square pyramid-tetrahedron interfacing each mibdi to the trigonal prism.

Now let's take a look at the side view, which is quite interesting:

Here, I turned off visibility clipping so that you can see the entire structure of the polychoron. The two octahedral cells are highlighted in green. They happen to project concentrically from this particular 4D viewpoint. The square pyramids aren't very obvious, but you can find them if you first look for the triangular prism (foreshortened to be quite squished) at the bottom of the projection.

Here's another side view, showing how two mibdies are folded together with a square pyramid sandwiched between them:

I left the octahedra colored, even though they are on the far side of this projection, so that it's easier to orient yourself w.r.t. the polychoron in this 4D viewpoint.

## Software models

## Incidence matrix

Dual: D4.16 dual

# | TXID | Va | Vb | Vc | Vd | Ea | Eb | Ec | Ed | Ee | Ef | Eg | Eh | 5a | 3a | 3b | 3c | 3d | 3e | 3f | 3g | 3h | 4a | C1a | C2a | C3a | C4a | C5a | C2b | Type | Name |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | Va | = point | ; in digon | ||||||||||||||||||||||||||||

1 | Vb | = point | ; outermost | ||||||||||||||||||||||||||||

2 | Vc | = point | ; tips | ||||||||||||||||||||||||||||

3 | Vd | = point | ; in trip | ||||||||||||||||||||||||||||

4 | Ea | 2 | 0 | 0 | 0 | = digon | ; digon | ||||||||||||||||||||||||

5 | Eb | 1 | 1 | 0 | 0 | = digon | ; digon to outermost | ||||||||||||||||||||||||

6 | Ec | 0 | 1 | 1 | 0 | = digon | ; outermost to tips | ||||||||||||||||||||||||

7 | Ed | 0 | 1 | 0 | 1 | = digon | ; outermost to trip | ||||||||||||||||||||||||

8 | Ee | 0 | 0 | 1 | 1 | = digon | ; tips to trip | ||||||||||||||||||||||||

9 | Ef | 0 | 2 | 0 | 0 | = digon | ; between outermost | ||||||||||||||||||||||||

10 | Eg | 0 | 0 | 0 | 2 | = digon | ; ends of trip | ||||||||||||||||||||||||

11 | Eh | 0 | 0 | 0 | 2 | = digon | ; sweep of trip | ||||||||||||||||||||||||

12 | 5a | 2 | 2 | 1 | 0 | 1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | = pentagon | ; | ||||||||||||||||

13 | 3a | 1 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 1 | 0 | 0 | = triangle | ; digon to outermost | ||||||||||||||||

14 | 3b | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | = triangle | ; equatorial caps | ||||||||||||||||

15 | 3c | 0 | 2 | 0 | 1 | 0 | 0 | 0 | 2 | 0 | 1 | 0 | 0 | = triangle | ; equatorial cap-petals | ||||||||||||||||

16 | 3d | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | = triangle | ; equatorial sweep-petals | ||||||||||||||||

17 | 3e | 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 1 | = triangle | ; equatorial sweeps | ||||||||||||||||

18 | 3f | 0 | 1 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 1 | 0 | = triangle | ; cap guard | ||||||||||||||||

19 | 3g | 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 1 | 0 | = triangle | ; sweep guard | ||||||||||||||||

20 | 3h | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | = triangle | ; ends of trip | ||||||||||||||||

21 | 4a | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | = square | ; sweep of trip | ||||||||||||||||

22 | C1a | 2 | 4 | 2 | 2 | 1 | 4 | 4 | 4 | 4 | 2 | 0 | 1 | 2 | 2 | 0 | 2 | 4 | 2 | 0 | 0 | 0 | 0 | = metabidiminished icosahedron | ; | ||||||

23 | C2a | 1 | 3 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | = tetrahedron | ; cap tets | ||||||

24 | C3a | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 | = triangular prism | ; | ||||||

25 | C4a | 0 | 3 | 0 | 3 | 0 | 0 | 0 | 6 | 0 | 3 | 3 | 0 | 0 | 0 | 1 | 3 | 0 | 0 | 3 | 0 | 1 | 0 | = octahedron | ; | ||||||

26 | C5a | 0 | 0 | 1 | 4 | 0 | 0 | 0 | 0 | 4 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 1 | = square pyramid | ; | ||||||

27 | C2b | 0 | 1 | 1 | 2 | 0 | 0 | 1 | 2 | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 0 | = tetrahedron | ; sweep tets | ||||||

28 | H4.1a | 2 | 6 | 3 | 6 | 1 | 6 | 6 | 12 | 12 | 6 | 6 | 3 | 3 | 6 | 2 | 6 | 12 | 6 | 6 | 6 | 2 | 3 | 3 | 2 | 1 | 2 | 3 | 6 | = D4.16 |
; |

## Usage as facets

*This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.*