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Zometool - 4-Polytopes - Omnitruncated 24-Cell

⇒ GALLERY, ⇒ Stereo Pictures for Cross-Eyed Viewing

Introduction

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.

Zometool Model

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.

Omnitruncated 24-Cell

Construction Overview

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 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; †this layer is 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.

Layer 1

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

To construct the Truncated Cuboctahedron you need:
48
24 (B1)
48 (G1)

First Layer - regular Truncated Cuboctahedron

Layer 2

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)

Second Layer - Eight Hexagonal Prisms Type 1 added

Layer 3

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)

Third Layer - Twelve Hexagonal Prisms Type 2 added

Layer 4

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)

Fourth Layer - Six Truncated Cuboctahedra Type 1 added

Layer 5

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)

Fifth Layer - Twelve Hexagonal Prisms Type 3 added

Layer 6

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)

Sixth Layer - Twenty-four Hexagonal Prisms Type 4 added

Layer 7

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)

Seventh Layer - Twenty-four Hexagonal Prisms Type 5 added

Layer 8 - Complete Model

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)

Eigth Layer - Six Truncated Cuboctahedra Type 2 added

Layer 9 - Outer Hull

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 model.

To build this outer hull you need:
336
144 (B0), 72 (B1)
144 (G1)
192 (Y1)
Total parts count: 888

Omnitruncated 24-Cell - Outer Hull

 

Gallery

Stereo Pictures for Cross-Eyed Viewing