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.

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.

ca = close approximation

J_{1} - Square Pyramid

J_{2} - Pentagonal Pyramid

J_{3} - Triangular Cupola

J_{4} - Square Cupola (ca)

J_{5} - Pentagonal Cupola

J_{6} - Pentagonal Rotunda

J_{7} - Elongated Triangular Pyramid (ca)

J_{8} - Elongated Square Pyramid (ca)

J_{9} - Elongated Pentagonal Pyramid (ca)

J_{11} - Gyroelongated Pentagonal Pyramid (Diminished Icosahedron)

J_{13} - Pentagonal Bipyramid (ca)

J_{15} - Elongated Square Bipyramid (ca)

J_{16} - Elongated Pentagonal Bipyramid (ca)

J_{18} - Elongated Triangular Cupola (ca)

J_{19} - Elongated Square Cupola (ca)

J_{20} - Elongated Pentagonal Cupola (ca)

J_{21} - Elongated Pentagonal Rotunda (ca)

J_{28} - Square Orthobicupola (ca)

J_{30} - Pentagonal Orthobicupola (ca)

J_{32} - Pentagonal Orthocupolarotunda

J_{33} - Pentagonal Gyrocupolarotunda (ca)

J_{34} - Pentagonal Orthobirotunda (ca)

J_{38} - Elongated Pentagonal Orthobicupola (ca)

J_{39} - Elongated Pentagonal Gyrobicupola (ca)

J_{40} - Elongated Pentagonal Orthocupolarotunda (ca)

J_{41} - Elongated Pentagonal Gyrocupolarotunda (ca)

J_{42} - Elongated Pentagonal Orthobirotunda (ca)

J_{43} - Elongated Pentagonal Gyrobirotunda (ca)

J_{58} - Augmented Dodecahedron (ca)

J_{59} - Parabiaugmented Dodecahedron (ca)

J_{60} - Metabiaugmented Dodecahedron (ca)

J_{61} - Triaugmented Dodecahedron (ca)

J_{62} - Metabidiminished Icosahedron

J_{63} - Tridiminished Icosahedron

J_{64} - Augmented Tridiminished Icosahedron (ca)

J_{65} - Augmented Truncated Tetrahedron

J_{66} - Augmented Truncated Cube (ca)

J_{67} - Biaugmented Truncated Cube (ca)

J_{76} - Diminished Rhombicosidodecahedron

J_{80} - Parabidiminished Rhombicosidodecahedron

J_{81} - Metabidiminished Rhombicosidodecahedron

J_{83} - Tridiminished Rhombicosidodecahedron

J_{86} - Sphenocorona (ca)

J_{91} - Bilunabirotunda

J_{92} - Triangular Hebesphenorotunda

Derived Johnson Solids

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)

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)

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)

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)

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)

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)

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)

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)

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)

The Gyroelongated Pentagonal Pyramid (Diminished Icosahedron) 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)

The Pentagonal Bipyramid has 7 vertices, 15 edges and 10 faces: 10 triangles.

To construct the Pentagonal Bipyramid, you need:

7

10 (B0, B1 or B2 size)

5 (R0, R1 or R2 size)

The Elongated Square Bipyramid has 10 vertices, 20 edges and 12 faces: 8 triangles, 4 squares.

To construct the Elongated Square Bipyramid, you need:

10

4 (B0, B1 or B2 size)

16 (G0, G1 or G2 size)

The Elongated Pentagonal Bipyramid has 12 vertices, 25 edges and 15 faces: 10 triangles, 5 squares.

To construct the Elongated Pentagonal Bipyramid, you need:

12

15 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Elongated Triangular Cupola has 15 vertices, 27 edges and 14 faces: 4 triangles, 9 squares, 1 hexagon.

To construct the Elongated Triangular Cupola, you need:

15

6 (Y0, Y1 or Y2 size)

21 (G0, G1 or G2 size)

The Elongated Square Cupola has 20 vertices, 36 edges and x faces: 4 triangles, 13 squares, 1 octagon.

To construct the Elongated square cupola, you need:

20

18 (B0, B1 or B2 size)

16 (G0, G1 or G2 size)

The Elongated Pentagonal Cupola has 25 vertices, 45 edges and 22 faces: 5 triangles, 15 squares, 1 pentagon, 1 decagon.

To construct the Elongated Pentagonal Cupola, you need:

25

35 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Elongated Pentagonal Rotunda has 30 vertices, 55 edges and 27 faces: 10 triangles, 10 squares, 6 pentagons, 1 decagon.

To construct the Elongated Pentagonal Rotunda, you need:

30

45 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Square Orthobicupola has 16 vertices, 32 edges and 18 faces: 8 triangles, 10 squares.

To construct the Square Orthobicupola, you need:

16

12 (B0, B1 or B2 size)

20 (G0, G1 or G2 size)

The Pentagonal Orthobicupola has 20 vertices, 40 edges and 22 faces: 10 triangles, 10 squares, 2 pentagons.

To construct the Pentagonal Orthobicupola, you need:

20

30 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Pentagonal Orthocupolarotunda has 25 vertices, 50 edges and 30 faces: 15 triangles, 5 squares, 7 pentagons.

To construct the Pentagonal Orthocupolarotunda, you need:

25

50 (B0, B1 or B2 size)

The Pentagonal Gyrocupolarotunda has 25 vertices, 50 edges and 30 faces: 15 triangles, 5 squares, 10 pentagons.

To construct the Pentagonal Gyrocupolarotunda, you need:

25

40 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Pentagonal Orthobirotunda has 30 vertices, 60 edges and 32 faces: 20 triangles, 12 pentagons.

To construct the Pentagonal Orthobirotunda, you need:

x

40 (B0, B1 or B2 size)

10 (Y0, Y1 or Y2 size)

10 (R0, R1 or R2 size)

The Elongated Pentagonal Orthobicupola has 30 vertices, 60 edges and 32 faces: 10 triangles, 20 squares, 2 pentagons.

To construct the Elongated Pentagonal Orthobicupola, you need:

30

40 (B0, B1 or B2 size)

20 (R0, R1 or R2 size)

The Elongated Pentagonal Gyrobicupola has 30 vertices, 60 edges and 32 faces: 10 triangles, 20 squares, 2 pentagons.

To construct the Elongated Pentagonal Gyrobicupola, you need:

30

50 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Elongated Pentagonal Orthocupolarotunda has 35 vertices, 70 edges and 37 faces: 15 triangles, 15 squares, 7 pentagons.

To construct the Elongated Pentagonal Orthocupolarotunda, you need:

35

60 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Elongated Pentagonal Gyrocupolarotunda has 35 vertices, 70 edges and 37 faces: 15 triangles, 15 squares, 7 pentagons.

To construct the Elongated Pentagonal Gyrocupolarotunda, you need:

35

50 (B0, B1 or B2 size)

20 (R0, R1 or R2 size)

The Elongated Pentagonal Orthobirotunda has 40 vertices, 80 edges and 42 faces: 20 triangles, 10 squares, 12 pentagons.

To construct the Elongated Pentagonal Orthobirotunda, you need:

40

50 (B0, B1 or B2 size)

10 (Y0, Y1 or Y2 size)

20 (R0, R1 or R2 size)

The Elongated Pentagonal Gyrobirotunda has 40 vertices, 80 edges and 42 faces: 20 triangles, 10 squares, 12 pentagons.

To construct the Elongated Pentagonal Gyrobirotunda, you need:

40

70 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Augmented Dodecahedron has 21 vertices, 35 edges and 16 faces: 5 triangles, 11 pentagons.

To construct the Augmented Dodecahedron, you need:

21

30 (B0, B1 or B2 size)

5 (R0, R1 or R2 size)

The Parabiaugmented Dodecahedron has 22 vertices, 40 edges and 20 faces: 10 triangles, 10 pentagons.

To construct the Parabiaugmented Dodecahedron, you need:

22

30 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Metabiaugmented Dodecahedron has 22 vertices, 40 edges and 20 faces: 10 triangles, 10 pentagons.

To construct the Metabiaugmented Dodecahedron, you need:

22

30 (B0, B1 or B2 size)

10 (R0, R1 or R2 size)

The Triaugmented Dodecahedron has 23 vertices, 45 edges and 24 faces: 15 triangles, 9 pentagons.

To construct the Triaugmented Dodecahedron, you need:

23

30 (B0, B1 or B2 size)

15 (R0, R1 or R2 size)

The Metabidiminished Icosahedron has 10 vertices, 20 edges and 12 faces: 10 triangles, 2 pentagons.

To construct the Metabidiminished Icosahedron, you need:

10

20 (B0, B1 or B2 size)

The Tridiminished Icosahedron has 9 vertices, 15 edges and 8 faces: 5 triangles, 3 pentagons.

To construct the Tridiminished Icosahedron, you need:

9

15 (B0, B1 or B2 size)

The Augmented Tridiminished Icosahedron has 10 vertices, 18 edges and 10 faces: 7 triangles, 3 pentagons.

To construct the Augmented Tridiminished Icosahedron, you need:

10

15 (B0, B1 or B2 size)

3 (Y0, Y1 or Y2 size)

The Augmented Truncated Tetrahedron has 15 vertices, 27 edges and 14 faces: 8 triangles, 3 squares, 3 hexagons.

To construct the Augmented Truncated Tetrahedron, you need:

15

27 (G0, G1 or G2 size)

The Augmented Truncated Cube has 28 vertices, 48 edges and 22 faces: 12 triangles, 5 squares, 5 octagons.

To construct the Augmented Truncated Cube, you need:

28

16 (B0, B1 or B2 size)

32 (G0, G1 or G2 size)

The Biaugmented Truncated Cube has 32 vertices, 60 edges and 30 faces: 16 triangles, 10 squares, 4 octagons.

To construct the Biaugmented Truncated Cube, you need:

32

20 (B0, B1 or B2 size)

40 (G0, G1 or G2 size)

The Diminished Rhombicosidodecahedron has 55 vertices, 105 edges and 52 faces: 15 triangles, 25 squares, 11 pentagons, 1 decagon.

To construct the Diminished Rhombicosidodecahedron, you need:

55x

105 (B0, B1 or B2 size)

The Parabidiminished Rhombicosidodecahedron has 50 vertices, 90 edges and 42 faces: 10 triangles, 20 squares, 10 pentagons, 2 decagons.

To construct the Parabidiminished Rhombicosidodecahedron, you need:

50

90 (B0, B1 or B2 size)

The Metabidiminished Rhombicosidodecahedron has 50 vertices, 90 edges and 42 faces: 10 triangles, 20 squares, 10 pentagons, 2 decagons.

To construct the Metabidiminished Rhombicosidodecahedron, you need:

50

90 (B0, B1 or B2 size)

The Tridiminished Rhombicosidodecahedron has 45 vertices, 75 edges and 32 faces: 5 triangles, 15 squares, 9 pentagons, 3 decagons.

To construct the Tridiminished Rhombicosidodecahedron, you need:

45

75 (B0, B1 or B2 size)

The Sphenocorona has 10 vertices, 22 edges and 14 faces: 12 triangles, 2 squares.

To construct the Sphenocorona, you need:

10

18 (B0, B1 or B2 size)

6 (Y0, Y1 or Y2 size)

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)

The Triangular Hebesphenorotunda has 18 vertices, 36 edges and 20 faces: 13 triangles, 3 squares, 3 pentagons, 1 hexagon.

To construct the Triangular Hebesphenorotunda, you need:

18

36 (B0, B1 or B2 size)

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)

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