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Zometool - 4-Polytopes - 24-Cell

⇒ Cell-First Projection ⇒ Vertex-First Projection ⇒ GALLERY

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.

24-Cell

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)

First Layer - regular Octahedron

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)

Second Layer - Eight Octahedra Type 1 added

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

24-Cell - Outer Hull

 

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.

24-Cell

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)

First Layer - 6 Octahedra T3

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.

First Layer - 6 Octahedra T3

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

24-Cell - Outer Hull

 

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

 

Vertex-First Projection