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Zometool - 4-Polytopes - Rectified 120-Cell

Introduction

The Rectified 120-Cell or rectified hecatonicosachoron is a convex uniform 4-polytope composed of 600 regular tetrahedra and 120 icosidodecahedra cells. Its vertex figure is a triangular prism, with three icosidodecahedra and two tetrahedra meeting at each vertex. → wikipedia

It can be thought of as the 4-dimensional analog of the archimedic solid Icosidodecahedron.

Zometool Model

You can build a "3D-shadow" (parallel projection centered on an Icosidodecahedron → eusebeia) of the Rectified 120-Cell with Zometool

To construct the model you need:
640
480 (B1)
600 (Y1)
360 (R1), 360 (R0)

The images were generated with ZomePad. Click on the image to enlarge the view.

Rectified 120-Cell

Construction Overview

Eusebeia describes the structural relation between the 4D Rectified 120-Cell and its 3D-shadow by 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 9 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(s) No. of cells
Layer 1 + 19* Regular Icosidodecahedron 1 + 1
Layer 2 + 18* Tetrahedron Type 1 20 + 20
Layer 3 + 17* Icosidodecahedron Type 1
Tetrahedron Type 2
12 + 12
20 +20
Layer 4 + 16* Tetrahedron Type 3 30 +30
Layer 5 + 15* Tetrahedron Type 4 60 + 60
Layer 6 + 14* Icosidodecahedron Type 2 20 + 20
Layer 7 + 13* Icosidodecahedron Type 3
Tetrahedron Type 5
12 + 12
60 + 60
Layer 8 + 12* Tetrahedron Type 6 60 + 60
Layer 9 + 11* Tetrahedron Type 7 20 + 20
Layer 10† Icosidodecahedron Type 4
Tetrahedron Type 8
30
60
Total 720

*not build as both layers coincide
†this layer is also build as a separate model

 

Only 315 cells (Layer 1-9) have to be build to obtain the 3D-shadow of the Rectified 120-Cell with Zometool. The nodes and struts of Layer 10 coincide with the nodes and struts of Layer 9 and below. Layer 10, which forms the outer hull of the model, will be build separately. The 315 cells of layer 11-19 coincide with the first 9 layers. This coincidence is an artifact of the 4D to 3D projection.

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

Construction - Layer 1

A uniform Icosidodecahedron is the only cell in the first layer.

To construct the Icosidodecahedron you need:
30
60 (B1)

First Layer - Uniform Icosidodecahedro

Construction - Layer 2

The second layer consists of 20 Tetrahedra Type 1 which are added to the 20 triangular sides of the regular Icosidodecahedron.

To add the 20 Type 1 Tetrahedra to the uniform Icosidodecahedron you need:
20
60 (R1)

Second Layer - Twenty Tetrahedra Type 1 added

Construction - Layer 3

The third layer consists of 12 squished Icosidodecahedra Type 1 which fit into the 12 depressions obtained in Layer 2. By adding these 12 Icosidodecahedra, also 20 Tetrahedra of Type 2 are introduced.

To add the 12 Icosidodecahedra Type 1 you need:
180
150 (B1)
180 (Y1)
120 (R1)

Third Layer - Twelve Icosidodecahedron Type 1 added

Construction - Layer 4

The fourth layer consists of 30 Tetrahedra Type 3 which fit into the 20 depressions obtained in Layer 3.

To add the 30 Tetrahedra Type 3 you need:
30 (B1)

Fourth Layer - 30 Tetrahedra Type 3 added

Construction - Layer 5

The fifth layer consists of 60 Tetrahedra Type 4 which fit on the triangular face of the Icosidodecahedron in Layer 4.

To add the 60 Tetrahedra Type 4 you need:
60
120 (Y1)
60 (R0)

Fifth Layer - 60 Tetrahedra Type 4 added

Construction - Layer 6

The third layer consists of 20 squished Icosidodecahedra Type 2 which fit into the 12 depressions obtained in Layer 5.

To add the 20 Icosidodecahedra Type 2 you need:
210
120 (B1)
180 (Y1)
120 (R1), 120x (R0)

Sixth Layer - Twenty Icosidodecahedron Type 2 added

Construction - Layer 7

The third layer consists of 12 squished Icosidodecahedra Type 3 which fit into the 12 depressions obtained in Layer 6. By adding these 12 Icosidodecahedra, also 60 Tetrahedra of Type 5 are introduced.

To add the 12 Icosidodecahedra Type 3 you need:
120
120 (B1)
120 (Y1)
120 (R0)

Seventh Layer - Twelve Icosidodecahedron Type 3

Construction - Layer 8

The fifth layer consists of 60 Tetrahedra Type 6 which fit on the triangular face of the Icosidodecahedron in Layer 7.

To add the 60 Tetrahedra Type 6 you need:
60 (R1)

Eight Layer - 60 Tetrahedra Type 5 added

Construction - Layer 9 - Complete Zometool Model

The ninth layer consists of 20 Tetrahedra Type 7 which fit on the triangular face of the Icosidodecahedron in Layer 8.

The Zometool model of the Rectified 120-cell is now complete.

To add the 20 Tetrahedra Type 7 you need:
20
60 (R0)

Ninth Layer - 20 Tetrahedra Type 7 added

Construction - Layer 10 - Outer Hull

The Zometool model is actual complete in Layer 9.

The tenth layer, consisting of 30 Icosidodecahedra Type 4 and 60 Tetrahedra Type 8, coincides with the existing nodes and struts of layer 8 and 9.

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

The image shows the arrangement of the 30 Type 4 Icosidodecahedra and 60 Type 8 Tetrahedra, which form the outer hull of the model.

To build this outer hull you need:
350
180 (B1)
240 (Y1)
180 (R1), 180 (R0)

Rectified 120-Cell - Outer Hull

 

Gallery

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

Normal size (B1, Y1, R1, R0)

Rectified 120-Cell - Normal size Model
Rectified 120-Cell - Normal size Model
Rectified 120-Cell - Normal size Model

Super size (B2, Y2, R2, R1)

120-Cell - Super size Model
120-Cell - Super size Model
120-Cell - Super size Model
120-Cell - Super size Model
120-Cell - Super size Model
120-Cell - Super size Model

Parallel projection of the 3D Zometool model

120-Cell - Super size Model
120-Cell - Super size Model
120-Cell - Super size Model