The Omnitruncated 24-Cell or great prismatotetracontoctachoron is a uniform 4-polytope derived from the 24-cell. It is composed of 1152 vertices, 2304 edges, and 1392 faces (864 squares, 384 hexagons, and 144 octagons). It has 240 cells: 48 truncated cuboctahedra, 192 hexagonal prisms.→ wikipedia.

You can build a "3D-shadow" (cell-first parallel projection → eusebeia) of the Omnitruncated 24-Cell with Zometool.

To construct the model you need:

576

288 (B0), 144 (B1)

288 (G0)

384 (Y1)

Total parts count: 1680

The images were generated with ZomePad.

Click on the image to enlarge the view.

The structural relation between the 4D Omnitruncated 24-Cell and its 3D-shadow by cell-first parallel projection is described in detail by → eusebeia.

Layer | Cell(s) | No. of cells |
---|---|---|

Layer 1 + 17* |
Regular Truncated Cuboctahedron | 1 + 1 |

Layer 2 + 16* |
Hexagonal Prism Type 1 | 8 + 8 |

Layer 3 + 15* |
Hexagonal Prism Type 2 | 12 + 12 |

Layer 4 + 14* |
Truncated Cuboctahedron Type 1 | 6 + 6 |

Layer 5 + 13* |
Hexagonal Prism Type 3 | 12 + 12 |

Layer 6 + 12* |
Hexagonal Prism Type 4 | 24 + 24 |

Layer 7 + 11* |
Hexagonal Prism Type 5 | 24 + 24 |

Layer 8 + 10* |
Truncated Cuboctahedron Type 2 | 8 + 8 |

Layer 9† |
Truncated Cuboctahedron Type 3 Truncated Cuboctahedron Type 4 Hexagonal Prism Type 6 Hexagonal Prism Type 7 |
12 6 8 24 |

Total 240 |

*not build as both layers coincide

†this layer is also build as a separate model

95 cells (Layer 1-8) have to be build to obtain the 3D-shadow of the Omnitruncated 24-Cell with Zometool. The nodes and struts of Layer 9 coincide with the nodes and struts of Layer 7 and 8. This layer, which is the outer hull of the model, will be build separately. The 95 cells of layer 10-17 coincide with layers 1-8. This coincidence is an artifact of the 4D to 3D projection.

You will see that the form of the Truncated Cuboctahedra and Hexagonal Prisms will change going from the centre to the outer layers. They will be squished and skewed. This is also an artifact from the 4D to 3D projection.

A regular Truncated Cuboctahedron is the only cell in the first layer.

To construct the Truncated Cuboctahedron you need:

48

24 (B1)

48 (G1)

The second layer consists of 8 Hexagonal Prisms Type 1 which are added to the 8 hexagonal sides of the regular Truncated Cuboctahedron.

To add the 8 Type 1 Hexagonal Prisms to the regular Truncated Cuboctahedron you need:

48

48 (G0)

48 (Y1)

The third layer consists of 12 Hexagonal Prisms Type 2 which are added to the between the Hexagonal Prisms added in Layer 2.

To add the 12 Type 2 Hexagonal Prisms you need:

48

24 (B1)

24 (G0)

48 (Y1)

The fourth layer consists of 6 Truncated Cuboctahedra Type 1 which are added between the Hexagonal Prisms added in Layer 2 and 3.

To add the 6 Truncated Cuboctahedra you need:

144

48 (B0), 48 (B1)

48 (G0)

96 (Y1)

The fifth layer consists of 12 Hexagonal Prisms Type 3 which are added between the Truncated Cuboctahedra added in Layer 4.

To add the 12 Type 3 Hexagonal Prisms you need:

48

48 (B0), 24 (B1)

24 (G0)

The sixth layer consists of 24 Hexagonal Prisms Type 4 which are added to the between the Hexagonal Prisms added in Layer 2.

To add the 24 Type 4 Hexagonal Prisms you need:

96

96 (B0), 24 (B1)

96 (Y1)

The seventh layer consists of 24 Hexagonal Prisms Type 5 which are added to the between the Hexagonal Prisms added in Layer 2.

To add the 24 Type 5 Hexagonal Prisms you need:

96

96 (B0)

48 (G0)

48 (Y1)

The eigth layer consists of 8 Truncated Cuboctahedra Type 2 which are added between the Hexagonal Prisms added in the previous layers.

To add the 8 Truncated Cuboctahedra you need:

48

48 (G0)

48 (Y1)

The Zometool model is actual complete in Layer 8.

The ninth layer, consisting of 12 Truncated Cuboctahedra Type 3, 6 Truncated Cuboctahedra Type 4, 8 Hexagonal Prisms Type 6, and 24 Hexagonal Prisms Type 7 which coincides with the existing nodes and struts.

The image shows the arrangement of the cells which forms the outer hull of the Zometool model.

To build this outer hull you need:

336; 144 (B0), 72 (B1); 144 (G1); 192 (Y1)

Total parts count: 888

If you don't know this technique, here a small tutorial (YouTube): 3D without glasses, Cross-Eye HD