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Zometool - Johnson Solids

Introduction

A Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform. It is not necessary that each face must be the same polygon, or that the same polygons join around each vertex.

In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers [1]. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

For more information on the Johnson solids visit Wikipedia - Johnson Solid.

Zometool Models

You cannot construct all Johnson solids with Zometool but most of them. When the model contains only struts of one size and color, the model is exact. If the model contains struts of different colors and sizes the model is a close approximation of the Johnson Solid.

The images were rendered in ZomePad. Click on the image to enlarge the view. The ZomePad files, which also include the building instructions, are available for download.

This list of possible zometool models is not complete. This page will be regularly updated and new models will be added (Last update: 14 July 2015).

Quick Links

ca = close approximation

J1 - Square Pyramid
J2 - Pentagonal Pyramid
J3 - Triangular Cupola
J4 - Square Cupola (ca)
J5 - Pentagonal Cupola
J6 - Pentagonal Rotunda
J7 - Elongated Triangular Pyramid (ca)
J8 - Elongated Square Pyramid (ca)
J9 - Elongated Pentagonal Pyramid (ca)
J11 - Gyroelongated Pentagonal Pyramid
J91 - Bilunabirotunda

Derived Johnson Solids

6J91 - "6 Bilunabirotundae around a cube"

J1 - Square Pyramid

Bilunabirotunda

The Square Pyramid has 5 vertices, 8 edges and 5 faces: 4 triangles, 1 square.

To construct the Square Pyramid, you need:
5
8 (G0, G1 or G2 size)

Download ZomePad file

J2 - Pentagonal Pyramid

Pentagonal Pyramid

The Pentagonal Pyramid has 6 vertices, 10 edges and 6 faces: 5 triangles, 1 pentagon.

To construct the Pentagonal Pyramid, you need:
6
10 (B0, B1 or B2 size)

Download ZomePad file

J3 - Triangular Cupola

Triangular Cupola

The Triangular Cupola has 9 vertices, 15 edges and 8 faces: 4 triangles, 3 squares, 1 hexagon.

To construct the Triangular Cupola, you need:
9
15 (G0, G1 or G2 size)

Download ZomePad file

J4 - Square Cupola

(close approximation)

Square Cupola

The Square Cupola has 12 vertices, 20 edges and 10 faces: 4 triangles, 5 squares, 1 octagon.

To construct the Square Cupola as close approximation, you need:
12
8 (B2)
12 (G1)

Download ZomePad file

J5 - Pentagonal Cupola

Pentagonal Cupola

The Pentagonal Cupola has 15 vertices, 25 edges and 12 faces: 5 triangles, 5 squares, 1 pentagon, 1 decagon.

To construct the Pentagonal Cupola, you need:
15
25 (B0, B1 or B2 size)

Download ZomePad file

J6 - Pentagonal Rotunda

Pentagonal Rotunda

The Pentagonal Rotunda has 20 vertices, 35 edges and 17 faces: 10 triangles, 6 pentagons, 1 decagon.

To construct the Pentagonal Rotunda, you need:
20
35 (B0, B1 or B2 size)

Download ZomePad file

J7 - Elongated Triangular Pyramid

(close approximation)

Elongated Triangular Pyramid

The Elongated Triangular Pyramid has 7 vertices, 12 edges and 7 faces: 4 triangles, 3 squares.

To construct the Elongated Triangular Pyramid as close approximation, you need:
7
3 (Y2)
9 (G1)

Download ZomePad file

J8 - Elongated Square Pyramid

(close approximation)

Elongated Square Pyramid

The Elongated Square Pyramid has 9 vertices, 16 edges and 9 faces: 4 triangles, 5 squares.

To construct the Elongated Square Pyramid as close approximation, you need:
9
4 (B1)
12 (G1)

Download ZomePad file

J9 - Elongated Pentagonal Pyramid

(close approximation)

Elongated Pentagonal Pyramid

The Elongated Pentagonal Pyramid has 11 vertices, 20 edges and 11 faces: 5 triangles, 5 squares, 1 pentagon.

To construct the Elongated Pentagonal Pyramid as close approximation, you need:
11
15 (B1)
5 (R1)

Download ZomePad file

J11 - Gyroelongated Pentagonal Pyramid

Gyroelongated Pentagonal Pyramid

The Gyroelongated Pentagonal Pyramid has 11 vertices, 25 edges and 16 faces: 15 triangles, 1 pentagon.

To construct the Gyroelongated Pentagonal Pyramid, you need:
11
25 (B0, B1 or B2 size)

Download ZomePad file

J91 - Bilunabirotunda

Bilunabirotunda

The Bilunabirotunda has 14 vertices, 26 edges and 14 faces: 8 triangles, 2 squares, 4 pentagons. See this YouTube video for the construction and animated views of the Bilunabirotunda.

To construct the Bilunabirotunda, you need:
14
26 (B0, B1 or B2 size)

Download ZomePad file

Bilunabirotundae - Parallel Projection
Bilunabirotundae - Parallel Projection

Derived Johnson Solids

6J91 - "6 Bilunabirotundae around a cube"

6 Bilunabirotundae around a cube

The "6 Bilunabirotundae around a cube" has 56 vertices, 120 edges and 54 faces: 24 triangles, 6 squares, 24 pentagons. See this YouTube video for the construction and animated views of the 6 Bilunabirotundae model.

To construct the "6 Bilunabirotundae around a cube", you need:
56
120 (B0, B1 or B2 size)

Download ZomePad file

6 Bilunabirotundae around a cube - Parallel Projection
6 Bilunabirotundae around a cube - Parallel Projection

References

[1] Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169-200