Zometool - 4-Polytopes - 24-Cell
Introduction
The 24-Cell is a convex regular 4-polytope (it is also called C24, icositetrachoron, octaplex, icosatetrahedroid,octacube, or polyoctahedron).
The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube → wikipedia.
Zometool Model - Cell-First Projection
You can build a "3D-shadow" (cell-first parallel projection → eusebeia) of the 24-Cell with Zometool.
To construct the model you need:
18 
24 (B1)
36 (G1)
Total parts count: 78
The images were generated with ZomePad.
Click on the image to enlarge the view.
Construction Overview
The structural relation between the 4D 24-Cell and its 3D-shadow by cell-first parallel projection is described in detail by → eusebeia.
| Layer | Cell | No. of cells |
|---|---|---|
| Layer 1 + 5* | Regular Octahedron | 1 + 1 |
| Layer 2 + 4* | Octahedron Type 1 | 8 + 8 |
| Layer 3† | Octahedron Type 2 | 6 |
| Total 24 |
*not build as both layers coincide
†this layer is also build as a separate model
Only 9 cells (Layer 1-2) have to be build to obtain the 3D-shadow of the 24-Cell with Zometool. The nodes and struts of Layer 3 coincide with the nodes and struts of Layer 1 and 2 This layer, which is the outer hull of the model, will be build separately. The 9 cells of layer 4-5 coincide with the first 2 layers. This coincidence is an artifact of the 4D to 3D projection.
You will see that the form of the octahedra will change going from the centre to the outer layers. The octahedra will be squished and skewed. This is also an artifact from the 4D to 3D projection.
Construction - Layer 1
A regular Octahedron is the only cell in the first layer.
To construct the Octahedron you need: 8 12 (G1)
Construction - Layer 2 - Complete Zometool Model
The second layer consists of 8 squished Octahedra Type 1 which are added to the 8 triangular sides of the regular Octahedron.
To add the 8 Type 1 Octahedra to the regular Octahedron you need: 12 24 (B1) 24 (G1)
Construction - Layer 3 - Outer Hull
The Zometool model is actual complete in Layer 2.
The third layer, consisting of 6 Octahedra Type 2, coincides with the existing nodes and struts of layer 1 and 2.
Because of the projection, Octahedron Type 2 appears to be flat, but in reality (4D) it is a perfect regular Octahedron.
The image shows the arrangement of the 6 Type 2 Octahedra, which forms the outer hull of the Zometool model.
To build this outer hull you need: 18 24 (B1) 24 (G1)
Total parts count: 66
Zometool Model - Vertex-First Projection
You can build a "3D-shadow" (vertex-first parallel projection → eusebeia) of the 24-Cell with Zometool.
To construct the model you need: 15 12 (B1)
32 (Y1)
Total parts count: 78
The images were generated with ZomePad. Click on the image to enlarge the view.
Construction Overview
The structural relation between the 4D 24-Cell and its 3D-shadow by vertex-first parallel projection is described in detail by → eusebeia.
| Layer | Cell | No. of cells |
|---|---|---|
| Layer 1 + 3* | Octahedron Type 3 | 6 + 6 |
| Layer 2† | Octahedron Type 4 | 12 |
| Total 24 |
*not build; †this layer is also build as a separate model
Only 6 cells (Layer 1-2) have to be build to obtain the 3D-shadow of the 24-Cell (vertex-first projection) with Zometool. The nodes and struts of Layer 2 coincide with the nodes and struts of Layer 1. Layer 2, which is the outer hull of the model, will also be build separately. The 6 cells of layer 3 coincide with the first 2 layers. This coincidence is an artifact of the 4D to 3D projection.
You will see that the form of the octahedra will change going from the centre to the outer layers. The octahedra will be squished and skewed. This is also an artifact from the 4D to 3D projection.
Construction - Layer 1
There are 6 octahedra (of Type 3, slightly flattened by the projection) packed around it in a cubic symmetry.
To construct these 6 Octahedra, you need: 15 12 (B1) 32 (Y1)
Construction - Layer 2 - Complete Zometool Model
The second layer consists of 8 squished Octahedra Type 4 which coincide with the nodes and struts of Layer 1.
Construction - Layer 3 - Outer Hull
The Zometool model is actual complete in Layer 1.
The second layer, consisting of 8 Octahedra Type 4, coincides with the existing nodes and struts of layer 1.
Because of the projection, Octahedron Type 4 appears to be flat, but in reality (4D) it is a perfect regular Octahedron.
The image shows the arrangement of the 8 Type 2 Octahedra, which forms the outer hull of the Zometool model.
To build this outer hull you need: 14 12 (B1) 24 (Y1))
Total parts count: 66
Gallery
Stereo Pictures for Cross-Eyed Viewing
If you don't know this technique, here a small tutorial (YouTube): 3D without glasses, Cross-Eye HD
Cell-First Projection
