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Zometool - 4-Polytopes - 120-Cell

⇒ GALLERY, ⇒ Stereo Pictures for Cross-Eyed Viewing, ⇒ Sub size (B0, Y0, R0, R00), ⇒ Regular size (B1, Y1, R1, R0), ⇒ Super size (B2, Y2, R2, R1)

Introduction

The 120-Cell is a convex regular 4-polytope (it is also called C120, Hecatonicosachoron, Dodecacontachoron and Hecantonicosahedroid) which boundary is composed of 120 dodecahedral cells with 4 meeting at each vertex.

It can be thought of as the 4-dimensional analog of the dodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the 120-Cell can be built up from 120 dodecahedra, with 3 around each edge. → wikipedia

Zometool Model

You can build a "3D-shadow" (cell-first parallel projection → eusebeia) of the 120-Cell with Zometool. This model is also commercially available as kit under the name Hyperdo (instructions (pdf)).

To construct the model (Hyperdo kit version) you need:
330
180 (B0)
200 (Y0)
120 (R0), 120 (R00)
Total parts count: 950

This model is rather compact (about 30 centimeters in diameter) and the inner structure is less visible. The inner structure is more visible by using the B1, Y1, R1 and R0 struts (see pictures in Gallery).

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

120-Cell - Sub size Model like the commercial Hyperdo kit

Construction Overview

In Ref [2] , the structural relation between the 4D 120-Cell and its 3D-shadow by cell-first parallel projection is described 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 4 layers. The B1, Y1, R1 and R0 struts will be used in this example. You can scale down (B0, Y0, R0 and R00 struts as in the Hyperdo kit), or scale up to a super size model using B2, Y2, R2 and R1 struts.

Layer Cell No. of cells
Layer 1 + 9* Regular Dodecahedron 1 + 1
Layer 2 + 8* Dodecahedron Type 1 12 + 12
Layer 3 + 7* Dodecahedron Type 2 20 + 20
Layer 4 + 6* Dodecahedron Type 3 12 + 12
Layer 5† Dodecahedron Type 4 30
    Total 120

*not build; †this layer is build as a separate model

Only 45 cells (Layer 1-4) have to be build to obtain the 3D-shadow of the 120-Cell with Zometool. The nodes and struts of Layer 5 coincide with the nodes and struts of Layer 3 and 4. This layer, which is the outer hull of the model, will be build separately. The 45 cells of layer 6-9 coincide with the first 4 layers. This coincidence is an artifact of the 4D to 3D projection.

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

Layer 1

A regular Dodecahedron is the only cell in the first layer.

To construct the Dodecahedron you need:
20
30 (B1)

If you are less experienced with Zometool you can add 20 yellow Y2 struts with nodes to a central node (see below) and construct the dodecahedron by connecting the nodes with blue B1 struts. The 20 yellow struts and the central node have to be removed afterwards.

First Layer - Dodecahedron

Layer 2

The second layer consists of 12 squished Dodecahedra Type 1 which are added to the 12 pentagonal sides of the regular Dodecahedron.

To add the 12 Type 1 Dodecahedra to the regular Dodecahedron you need:
110
60 (B1)
80 (Y1)
60 (R1)

Second Layer - Twelve  Dodecahedra Type 1 added

Layer 3

The third layer consists of 20 squished and skewed Dodecahedra Type 2 which fit into the 20 depressions obtained in Layer 2.

To add the 20 Dodecahedra Type 2 you need:
140
30 (B1)
120 (Y1)
60 (R1), 60 (R0)

Third Layer - Twenty  Dodecahedra Type 2 added

Layer 4 - Complete Model

The fourth layer consists of 12 Dodecahedra Type 3 which fit into the 12 pentagonal depressions in Layer 3.

To add the 12 Dodecahedra Type 3 you need:
60
60 (B1)
60 (R0)

Fourth Layer - Twelve  Dodecahedra Type 3 added

Layer 5 - Outer Hull

The Zometool model is actual complete in Layer 4.

The fifth layer, consisting of 30 Dodecahedra Type 4, coincides with the existing nodes and struts of layer 3 and 4.

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

The image shows the arrangement of the 30 Type 4 Dodecahedra, which forms the outer hull of the model.

To build this outer hull you need:
200
90 (B1)
120 (Y1)
60 (R1), 60 (R0)

120-Cell - Outer Hull

 

Gallery

Stereo Pictures for Cross-Eyed Viewing

Sub size (B0, Y0, R0, R00)

120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit
120-Cell - Sub size Model - Hyperdo kit

Regular size (B1, Y1, R1, R0)

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