The Johnson solids are the Convex Polyhedra having regular faces (with the exception of the completely regular Platonic Solids, the ``Semiregular'' Archimedean Solids, and the two infinite families of Prisms and Antiprisms). There are 28 simple (i.e., cannot be dissected into two other regular-faced polyhedra by a plane) regular-faced polyhedra in addition to the Prisms and Antiprisms (Zalgaller 1969), and Johnson (1966) proposed and Zalgaller (1969) proved that there exist exactly 92 Johnson solids in all.

A database of solids and Vertex Nets of these solids is maintained
on the Bell Laboratories Netlib server, but a few errors exist in
several entries. A concatenated and corrected version of the files is given by Weisstein, together with *Mathematica*
(Wolfram Research, Champaign, IL) code to display the solids and nets. The following table summarizes
the names of the Johnson solids and gives their images and nets.

1. Square pyramid

The number of constituent -gons () for each Johnson solid are given in the following table.

1 | 4 | 1 | 47 | 35 | 5 | 7 | |||||||

2 | 5 | 1 | 48 | 40 | 12 | ||||||||

3 | 4 | 3 | 1 | 49 | 6 | 2 | |||||||

4 | 4 | 5 | 1 | 50 | 10 | 1 | |||||||

5 | 5 | 5 | 1 | 1 | 51 | 14 | |||||||

6 | 10 | 6 | 1 | 52 | 4 | 4 | 2 | ||||||

7 | 4 | 3 | 53 | 8 | 3 | 2 | |||||||

8 | 4 | 5 | 54 | 4 | 5 | 2 | |||||||

9 | 5 | 5 | 1 | 55 | 8 | 4 | 2 | ||||||

10 | 12 | 1 | 56 | 8 | 4 | 2 | |||||||

11 | 15 | 1 | 57 | 12 | 3 | 2 | |||||||

12 | 6 | 58 | 5 | 11 | |||||||||

13 | 10 | 59 | 10 | 10 | |||||||||

14 | 6 | 3 | 60 | 10 | 10 | ||||||||

15 | 8 | 4 | 61 | 15 | 9 | ||||||||

16 | 10 | 5 | 62 | 10 | 2 | ||||||||

17 | 16 | 63 | 5 | 3 | |||||||||

18 | 4 | 9 | 1 | 64 | 7 | 3 | |||||||

19 | 4 | 13 | 1 | 65 | 8 | 3 | 3 | ||||||

20 | 5 | 15 | 1 | 1 | 66 | 12 | 5 | 5 | |||||

21 | 10 | 10 | 6 | 1 | 67 | 16 | 10 | 4 | |||||

22 | 16 | 3 | 1 | 68 | 25 | 5 | 1 | 11 | |||||

23 | 20 | 5 | 1 | 69 | 30 | 10 | 2 | 10 | |||||

24 | 25 | 5 | 1 | 1 | 70 | 30 | 10 | 2 | 10 | ||||

25 | 30 | 6 | 1 | 71 | 35 | 15 | 3 | 9 | |||||

26 | 4 | 4 | 72 | 20 | 30 | 12 | |||||||

27 | 8 | 6 | 73 | 20 | 30 | 12 | |||||||

28 | 8 | 10 | 74 | 20 | 30 | 12 | |||||||

29 | 8 | 10 | 75 | 20 | 30 | 12 | |||||||

30 | 10 | 10 | 2 | 76 | 15 | 25 | 11 | 1 | |||||

31 | 10 | 10 | 2 | 77 | 15 | 25 | 11 | 1 | |||||

32 | 15 | 5 | 7 | 78 | 15 | 25 | 11 | 1 | |||||

33 | 15 | 5 | 7 | 79 | 15 | 25 | 11 | 1 | |||||

34 | 20 | 12 | 80 | 10 | 20 | 10 | 2 | ||||||

35 | 8 | 12 | 81 | 10 | 20 | 10 | 2 | ||||||

36 | 8 | 12 | 82 | 10 | 20 | 10 | 2 | ||||||

37 | 8 | 18 | 83 | 5 | 15 | 9 | 3 | ||||||

38 | 10 | 20 | 2 | 84 | 12 | ||||||||

39 | 10 | 20 | 2 | 85 | 24 | 2 | |||||||

40 | 15 | 15 | 7 | 86 | 12 | 2 | |||||||

41 | 15 | 15 | 7 | 87 | 16 | 1 | |||||||

42 | 20 | 10 | 12 | 88 | 16 | 2 | |||||||

43 | 20 | 10 | 12 | 89 | 18 | 3 | |||||||

44 | 20 | 6 | 90 | 20 | 4 | ||||||||

45 | 24 | 10 | 91 | 8 | 2 | 4 | |||||||

46 | 30 | 10 | 2 | 92 | 13 | 3 | 3 | 1 |

**References**

Bell Laboratories. http://netlib.bell-labs.com/netlib/polyhedra/.

Bulatov, V. ``Johnson Solids.'' http://www.physics.orst.edu/~bulatov/polyhedra/johnson/.

Cromwell, P. R. *Polyhedra*. New York: Cambridge University Press, pp. 86-92, 1997.

Hart, G. W. ``NetLib Polyhedra DataBase.'' http://www.li.net/~george/virtual-polyhedra/netlib-info.html.

Holden, A. *Shapes, Space, and Symmetry.* New York: Dover, 1991.

Hume, A. *Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals.* Computer Science
Technical Report #130. Murray Hill, NJ: AT&T Bell Laboratories, 1986.

Johnson, N. W. ``Convex Polyhedra with Regular Faces.'' *Canad. J. Math.* **18**, 169-200, 1966.

Pugh, A. ``Further Convex Polyhedra with Regular Faces.'' Ch. 3 in *Polyhedra: A Visual Approach.*
Berkeley, CA: University of California Press, pp. 28-35, 1976.

Weisstein, E. W. ``Johnson Solids.'' Mathematica notebook JohnsonSolids.m.

Weisstein, E. W. ``Johnson Solid Netlib Database.'' Mathematica notebook JohnsonSolids.dat.

Zalgaller, V. *Convex Polyhedra with Regular Faces.* New York: Consultants Bureau, 1969.

© 1996-9

1999-05-25