Zometool - Castellated Rhombicosidodecahedral Prism
Introduction
The Castellated Rhombicosidodecahedral Prism is a Convex Regular-Faced (CRF) polychoron bounded by 2 rhombicosidodecahedra, 24 pentagonal pyramids, 40 tetrahedra, and 30 bilunabirotundae.
This is the first known non-trivial CRF polychoron that contains the Johnson solid Bilunabirotunda (J91) as cells.→ eusebeia
Zometool Model - Centered on a Bilunabirotunda
You can build a "3D-shadow" (centered on a Bilunabirotunda → eusebeia) of the Castellated Rhombicosidodecahedral Prism with Zometool.
To construct the model you need:
92 
72 (B1)
72 (Y1)
52 (R1), 52 (R0)
Total parts count: 340
The images were generated with ZomePad.
Click on the image to enlarge the view.
Construction Overview
The structural relation between the 4D Castellated Rhombicosidodecahedral Prism and its 3D-shadow (centered on a Bilunabirotunda) is described in detail by → eusebeia.
| Layer | Cell | No. of cells |
|---|---|---|
| Layer 1 + 13* | Regular Bilunabirotunda | 1 + 1 |
| Layer 2 + 12* | Bilunabirotunda Type 1 Tetrahedron Type 1 |
4 + 4 4 + 4 |
| Layer 3 + 11* | Bilunabirotunda Type 2 Pentagonal Pyramid Type 1 |
4 + 4 4 + 4 |
| Layer 4 + 10* | Bilunabirotunda Type 3 Tetrahedron Type 2 |
4 + 4 8 + 8 |
| Layer 5 + 9* | Pentagonal Pyramid Type 2 | 4 + 4 |
| Layer 6 + 8* | Tetrahedron Type 3 | 4 + 4 |
| Layer 7† | Bilunabirotunda Type 4 Bilunabirotunda Type 5 Rhombicosidodecahedron Type 1 Tetrahedron Type 4 Pentagonal Pyramid Type 3 |
2 2 2 8 8 |
| Total 96 |
*not build as both layers coincide
†this layer is also build as a separate model
37 Cells (Layer 1-6) have to be build to obtain the 3D-shadow of the Castellated Rhombicosidodecahedral Prism with Zometool. The nodes and struts of Layer 7 coincide with the nodes and struts of Layer 6 and below. This layer, which is the outer hull of the model, will be build separately. Layer 8-13 coincide with the layers 1-6. This coincidence is an artifact of the 4D to 3D projection.
You will see that the form of the Bilunabirotunda will change going from the centre to the outer layers. The Bilunabirotunda will be squished and skewed. This is also an artifact from the 4D to 3D projection.
Construction - Layer 1
A regular Bilunabirotunda (Johnson solid (J91) is the only cell in the first layer.
To construct the Bilunabirotunda you need: 14 26 (B0)
Construction - Layer 2
The second layer consists of 4 Bilunabirotundae Type 1 and 4 Tetrahedra Type 1. The Tetrahedra are between the regular Bilunabirotunda and two of Type 1.
To add the Bilunabirotundae and Tetrahedra, you need: 28 18 (B0) 28 (Y0) 28 (R0)
Construction - Layer 3
The third layer consists of 4 Bilunabirotundae Type 2 and 4 Pentagonal Pyramid Type 1. The Pentagonal Pyramids are between the Bilunabirotundae Type 2 and the model of layer 2.
To add the Bilunabirotundae and Pentagonal Pyramids, you need: 30 4 (B0) 20 (Y0) 24 (R0), 28 (R00)
Construction - Layer 4
The fourth layer consists of 4 Bilunabirotundae Type 3 and 4 Tetrahedra Type 2. The Tetrahedra are between the Bilunabirotundae Type 3 and the model of layer 3.
To add the Bilunabirotundae and Tetrahedra, you need: 20 16 (B0) 24 (Y0) 24 (R00)
Construction - Layer 5
The fifth layer consists of 4 Pentagonal Pyramids Type 2.
To add the Pentagonal Pyramids, you need: 4 (B0)
Construction - Layer 6 - complete Zometool Model
The sixth layer consists of 4 Tetrahedra Type 3.
To add the Tetrahedra, you need: 4 (B0)
Construction - Layer 7 - Outer Hull
The Zometool model is actual complete in Layer 6.
The seventh layer, consisting of two Bilunabirotunda Type 4, two Bilunabirotunda Type 5, two Rhombicosidodecahedron Type 1, four Tetrahedron Type 4 and four Pentagonal Pyramid Type 3, coincides with the existing nodes and struts of layers 6 and below.
Because of the projection, both Bilunabirotunda Types, Rhombicosidodecahedron Type 1, Tetrahedron Type 4 and Pentagonal Pyramid Type 3 appear to be flat, but in reality (4D), they are regular polyhedra.
To build this outer hull you need: 82 38 (B1) 48 (Y0) 40 (R0), 48 (R00)
Total parts count: 256
Zometool Model - Centered on Rhombicosidodecahedron
You can build a "3D-shadow" (centered on Rhombicosidodecahedron → eusebeia) of the Castellated Rhombicosidodecahedral Prism with Zometool.
To construct the model you need: 92 120 (B1) 60 (R0), 60 (R00)
Total parts count: 332
The images were generated with ZomePad.
Click on the image to enlarge the view.
Construction Overview
| Layer | Cell | No. of cells |
|---|---|---|
| Layer 1 + 7* | regular Rhombicosidodecahedron | 1 + 1 |
| Layer 2 + 6* | Tetrahedron Type 5 | 20 + 20 |
| Layer 3 + 5* | Pentagonal Pyramid 4 | 12 + 12 |
| Layer 4† | Bilunabirotunda Type 6 | 30 |
| Total 96 |
*not build; †this layer is also build as a separate model
33 Cells (Layer 1-3) have to be build to obtain the 3D-shadow of the Castellated Rhombicosidodecahedral Prism (centered on Rhombicosidodecahedron) with Zometool. The nodes and struts of Layer 4 coincide with the nodes and struts of Layer 1 and 2.
Construction - Layer 1
A regular Rhombicosidodecahedron (an Archimedean solid) is the only cell in this layer.
To construct the regular Rhombicosidodecahedron, you need: 60 120 (B1)
Construction - Layer 2
The second layer consists of 20 squished Tetrahedra Type 5.
Construction - Layer 3 - complete Zometool Model
The second layer consists of 12 squished Pentagonal Pyramids Type 4.
Construction - Layer 4
The Zometool model is actual complete in Layer 3.
The fourth layer, consisting of 30 Bilunabirotundae Type 6, coincides (100%) with the existing nodes and struts of layer 1 and 2.
Because of the projection, Bilunabirotunda Type 6 appears to be flat, but in reality (4D) it is a perfect regular Bilunabirotunda.
The image shows the arrangement of the 8 Type 2 Octahedra, which forms the outer hull of the Zometool model.
Bilunabirotunda Type 6
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
Centered on Bilunabirotunda
