The 600-Cell is a convex regular 4-polytope (it is also called C_{600}, hexacosichoron, hexacosihedroid). Its boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices.

The 600-cell is regarded as the 4-dimensional analog of the icosahedron, since it has five tetrahedra meeting at every edge. → wikipedia

You can build a "3D-shadow" (vertex-first parallel projection → eusebeia) of the 600-Cell with Zometool.

To construct the model, you need:

75

120 (B1)

120 (Y1)

72 (R1), 72 (R0)

Total parts count: 359

The images were generated with ZomePad.

Click on the image to enlarge the view.

Eusebeia describes the structural relation between the 4D 600-Cell and its 3D-shadow by vertex-first parallel projection in detail. Here the structure of this 3D-shadow with regard to the individual cells and complete model made with Zometool will be described. The model will be build in 7 layers. The B1, Y1, R1 and R0 struts will be used in this example. You can scale down (B0, Y0, R0 and R00 struts), or scale up to a super size model using B2, Y2, R2 and R1 struts.

Layer | Cell | No. of cells |
---|---|---|

Layer 1 |
Vertex | - |

Layer 2 + 14* |
Tetrahedron Type 1 | 20 + 20 |

Layer 3 + 13* |
Tetrahedron Type 2 | 20 + 20 |

Layer 4 + 12* |
Tetrahedron Type 3 | 30 + 30 |

Layer 5 + 11* |
Tetrahedron Type 4 | 60 + 60 |

Layer 6 + 10* |
Tetrahedron Type 5 | 60 + 60 |

Layer 7 + 9* |
Tetrahedron Type 6 Tetrahedron Type 7 |
60 + 60 20 + 20 |

Layer 8† |
Tetrahedron Type 8 | 60 |

Total 600 |

* not build as both layers coincide

† This layer coincides with Layer 7. This layer is also build as a separate model

270 Cells (Layer 1-7) have to be build to obtain the 3D-shadow of the 600-Cell with Zometool. The nodes and struts of Layer 8 coincide with the nodes and struts of Layer 7. This layer, which is the outer hull of the model, will be build separately. The 270 cells of layer 9-14 coincide with layers 2-7. This coincidence is an artifact of the 4D to 3D projection.

You will see that the form of the tertrahedron will change going from the centre to the outer layers. The tetrahedron will be squished and skewed. This is also an artifact from the 4D to 3D projection.

Because of the vertex-first projection we will start with one node as the first layer. :-)

You will need:

1

The second layer consists of 20 Tetrahedra Type 1 which are added to the node (vertex).

To add the 20 Type 1 Tetrahedra, you need:

12

30 (B1)

12 (R1)

The third layer consists of 20 Tetrahedra Type 2 which fit onto the 20 triangels of the "regular isocahedron".

To add the 20 Tetrahedra Type 2 you need:

20

60 (Y1)

The fourth layer consists of 30 Tetrahedra Type 3.

To add the 30 Tetrahedra Type 3 you need:

30 (B1)

The fifth layer consists of 60 Tetrahedra Type 4.

To add the 60 Tetrahedra Type 4 you need:

12

60 (R1), 12 (R0)

The sixth layer consists of 60 Tetrahedra Type 5.

To add the 60 Tetrahedra Type 5 you need:

30

60 (Y1)

60 (R0)

The seventh layer consists of 60 Tetrahedra Type 6 and 20 Tetrahedra Type 7.

To add both 60 Tetrahedra Type 6 and 20 Tetrahedra Type 7, you need:

60 (B1)

The Zometool model is actual complete in Layer 7.

The eigth layer, consisting of 60 Tetrahedra Type 8, coincides with the existing nodes and struts of layer 7.

Because of the projection, Tetrahedron Type 8 appears to be flat, but in reality (4D) it is a perfect regular Tetrahedron.

The image shows the arrangement of the 60 Type 8 Tetrahedra, which forms the outer hull of the Zometool model.

To build this outer hull you need:

42

60 (B1)

60 (Y1)

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