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 as both layers coincide
†this layer is also 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.

Construction - 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.