As for Mr. Bonnie M. Stewart's previous work on the subject, here is an excerpt from the book: "Search projects We dignify the following exercises with the title "search projects", since, if the explorer wishes, each proposal may become quite an ambitious undertaking and each one is open-end in character with adventurous trails to follow in many directions. Ex. 160 Let (R') indicate that every face of a polyhedron P is either regular or is a rhombus <> of unit edge with a 60° angle, i.e., a compound of two regular triangles."

all the examples below have the pink faces called regular diamonds or more specifically known as Vesisa Piscis.

Models made with Great Stella

cutting list: D 3 4 5 6 8 10

volume area Vertex Faces Edges

1 2 2 1 * * * * v 0.35355     a 3.59808     V 6     F 5    E 9 J1T1 .stel     .wrl

2 3 4 * * * * * v 0.58926     a 4.33013    V 7    F 7     E 12 S3TT .stel     .wrl

3 6 * * * * * * v 0.70711     a 5.19615    V 8     F 6    E 12 S3TTBB .stel     .wrl

4* 4 * 2 * * * * v 0.70711     a 5.46410    V 8   F 6    E 12 Y4Y3Y4 .stel     .wrl

5*** 2 4 2 * * * * v 0.70711    a 5.46410    V 8   F 8    E 14 P3S1 .stel     .wrl

6 2 * 4 * * * * v 0.86603     a 4.86603    V 8    F 6     E 12 P3S1 .stel     .wrl

7** 2 4 3 * * * * v 1.10173    a 6.46410    V 9   F 9     E 16 P3T1 .stel     .wrl

8** 2 8 2 * * * * v 1.33743    a 7.19615    V 10   F 12    E 20 P3T1B1 .stel     .wrl

9** 2 8 2 * * * * v 1.33743    a 7.19615    V 10    F 12     E 20 P3T1B2 .stel     .wrl

10*** 3 2 2 * 1 * * v 1.41421     a 8.25592    V 10    F 8     E 16 J3T1 .stel     .wrl

11*** 4 * 2 * 2 * * v 2.12132     a 5.46410    V 12   F 8     E 18 J3Y3Y3Y4 .stel     .wrl

12 4 4 5 * * * * v 2.59272     a 10.19615    V 13    F 13    E 24 B4TT .stel     .wrl

13*** 3 6 5 * * * * v 2.59272     a 10.19615    V 13    F 14     E 25 J27T1 .stel     .wrl

14*** 6 4 4 * * * * v 2.82843     a 10.92820    V 14    F 14     E 26 J27T1B2 .stel     .wrl

15 8 * 4 * * * * v 2.82843     a 10.92820    V 18    F 12     E 28 B4TTBB .stel     .wrl

16*** 2 9 2 3 * * * v 3.39522     a 12.79061    V 15    F 14     E 27 SXJ62 .stel     .wrl

17*** 2 9 2 3 * * * v 3.39522     a 12.79061    V 15    F 14     E 27 J91T1 .stel     .wrl

18*** 4 5 2 3 * * * v 3.39522     a 12.79061    V 15    F 14     E 27 J91SIDE .stel     .wrl

19*** 4 10 2 2 * * * v 3.69672     a 13.23519    V 16   F 18     E 32 J91T1B2 .stel     .wrl

20*** 4 10 2 2 * * * v 3.69672     a 13.23519    V 22    F 17     E 37 J91T1T2 .stel     .wrl

21 3 2 3 * 3 * * v 3.88909     a 14.83956    V 15    F 11     E 24 T3TT .stel     .wrl

22*** 3 7 3 3 1 * * v 5.10875    a 19.01361    V 21    F 16     E 35 J92T1 .stel     .wrl

23* 3 12 3 2 1 * * v 5.41025    a 21.07371    V 19   F 21     E 38 J92T1 .stel     .wrl

24*** 4 * 4 * 4 * * v 5.65685    a 7.46410    V 18    F 12     E 28 T3Y4Y4Y3 .stel     .wrl

25*** 4 7 * 5 * * 1 v 7.21927     a 22.79180    V 18   F 17     E 33 J6T1 .stel     .wrl

26 5 5 * 5 * * 1 v 7.21927     a 22.79180    V 15    F 16     E 29 J6TT .stel     .wrl

27*** 8 4 * 4 * * 1 v 7.52077    a 23.23638    V 21    F 17    E 36 J6T1T3 .stel     .wrl

28*** 4 12 5 6 * * * v 9.54331    a 23.98313    V 26   F 27    E 51 J32T1 .stel     .wrl

29*** 4 12 5 6 * * * v 9.54331    a 23.98313    V 26    F 27     E 51 J33T1 .stel     .wrl

30 5 10 5 6 * * * v 9.54331     a 23.98313    V 26    F 26     E 50 J32TT .stel     .wrl

31* 5 10 5 6 * * * v 9.54331     a 23.98313    V 26    F 26     E 50 J33TT .stel     .wrl

32*** 8 9 5 5 * * * v 9.62839     a 24.42772    V 26   F 27     E 51 J32T1T3 .stel     .wrl

33*** 8 9 5 5 * * * v 9.62839     a 24.42772    V 27   F 27     E 52 J33T1T3 .stel     .wrl

34* 5 25 * 5 * * 1 v 13.9585     a 31.45025    V 31    F 36     E 65 J25TT .stel     .wrl

35*** 4 17 * 11 * * * v 14.137     a 29.75060.    V 31    F 32     E 61 J34T1 .stel     .wrl

36 5 15 * 11 * * * v 14.137    a 30.19518    V 32    F 31    E 61 B5TT .stel     .wrl

37 5 15 * 11 * * * v 14.137     a 29.75060    V 31    F 31     E 60 J34TT .stel     .wrl

38*** 9 12 * 10 * * * v 14.4385    a     V    F 31    E J34T1B1 .stel     .wrl

39*** 8 14 * 10 * * * v 14.4385     a 30.19518    V 32   F 32     E 62 J34T1B2 .stel     .wrl

40*** 8 14 * 10 * * * v 14.4385     a 30.19518    V 32    F 32     E 62 J34T1B3 .stel     .wrl

41*** 8 14 * 10 * * * v 14.4385    a 30.19518    V 32   F 32     E 62 J34T1T3 .stel     .wrl

42 10 10 * 10 * * * v 14.4385     a 30.19518    V 27    F 30     E 55 B5TTBB .stel     .wrl

43 5 10 15 6 * * * v 14.4385     a 33.98313    V 36    F 36     E 70 J34TTBB .stel     .wrl

44 10 10 * 10 * * * v 14.4385     a 30.19518    V 27    F 30     E 55 B5TTB1 .stel     .wrl

45*** 12 11 * 9 * * * v 14.74     a 30.63976    V 33    F 32     E 63 J34T1T3B2 .stel     .wrl

46*** 12 11 * 9 * * * v 14.74     a 30.63976    V 33    F 32     E 63 J34T1T3B4 .stel     .wrl

47*** 13 9 * 9 * * * v 14.74     a 29.75060    V    F 31    E J34TTB1B3 .stel     .wrl

48 15 5 * 9 * * * v 14.74    a 30.63976    V 33    F 29     E 60 B5TTB1B3 .stel     .wrl

49* 5 5 10 5 * * 1 v 14.9135    a32.79180    V 31    F 26    E 55 J21TT .stel     .wrl	50*** 16 8 * 8 * * * v 15.0415    a 31.08435    V 34    F 32     E 64 J34T1T3B2B4 .stel     .wrl	51 4 4 5 * 5 * * v 15.5425     a 34.33830    V 28    F 18    E 44 T4TT .stel     .wrl

52* 5 30 5 6 * * * v 16.2926    a 32.64399    V 36    F 46    E 80 J47TT .stel     .wrl

53* 5 10 5 6 * * * v 17.2375     a 23.98313    V 26    F 26     E 50 J40TT .stel     .wrl

54* 5 10 15 6 * * * v 17.2375     a 33.98313    V 36    F 36     E 70 J41TT .stel     .wrl

55*** 4 8 10 * * 4 * v 17.5843     a 36.24192    V 32    F 26     E 56 J66BB .stel     .wrl

56* 5 35 * 11 * * * v 20.8863    a 38.41085    V 41    F 51     E 90 J48TT .stel     .wrl

57* 10 30 * 10 * * * v 21.1878     a 38.85544    V 42    F 50    E90 J48TTBB .stel     .wrl

58* 5 15 10 11 * * * v 21.8312     a 36.24192    V    F 46     E J43TT .stel     .wrl

59* 5 15 10 11 * * * v 21.8312    a 39.75060    V 41   F 41     E 80 J42TT .stel     .wrl

60 8 * 10 * * 4 * v 22.1327    a 36.24192    V 32    F 22     E 52 T4TTBB .stel     .wrl

61* 10 10 10 10 * * * v 22.1327     a 40.19518    V 42   F 40     E 80 J43TTBB .stel     .wrl

62* 10 10 10 10 * * * v 22.1327     a 40.19518    V 27    F 40     E 65 J42TTBB .stel     .wrl

63 5 15 5 1 * * 11 v 87.3637    a 112.18211    V 65    F 37     E 100 T5TT .stel     .wrl

64 10 10 10 2 * * 10 v 89.6878     a 103.37344    V 70    F 42     E 110 T5TTBB .stel     .wrl

65 10 10 10 2 * * 10 v 89.6878     a 103.37344    V 70    F 47     E 110 T5TTB1 .stel     .wrl

66*** 5 20 10 2 * * 10 v 89.6878     a 103.37344    V 70    F 47     E 115 J68BB .stel     .wrl

67*** 5 20 10 2 * * 10 v 89.6878     a 103.37344    V 70    F 47     E 115 J68B1 .stel     .wrl

68 15 5 15 3 * * 9 v 92.0118     a 104.56477    V 75    F 47     E 120 T5TTB1B3 .stel     .wrl

69*** 10 15 15 3 * * 9 v 92.0118     a 104.56477    V 75    F 52     E 125 J68B1B3 .stel     .wrl

70*** 5 25 15 3 * * 9 v 92.0118     a 104.56477    V 75    F 57    E 130 J70B3 .stel     .wrl

71** 4 10 4 2 2 * * v 7.12377     a     V 22    F 22    E 42 stel    wrl

Alex Doskey's page showing these same 70 grouped by base used for augmentation

Special thanks to Alex Doskey for making all the graphics files available on this page.

B. M. Stewart's original 21 entries are referenced as the integer only ones above

Credit for additions is denoted by the end of each numbered entry...

*    Steve Waterman April 2006 **  Roger Kaufman April 2006 *** Alex Doskey April 2006

An e-mail received from Norman Johnson in March 2006 -

Dear Steve,

While I was the first to describe the 92 nonuniform convex polyhedra with regular faces, I did not prove that my list was complete. That was done shortly afterward by a team led by Viktor Zalgaller in Leningrad (now St. Petersburg). The method they used was to consider all combinatorially possible cases and then exclude any that cannot be realized as regular-faced Euclidean solids. Their work was published in Russian in 1967 and in English in 1969 as a monograph "Convex Polyhedra with Regular Faces" (Consultants Bureau, New York).

If you can find a copy of Zalgaller's monograph, you may be able to identify the figures the would allow two adjacent triangles to be merged as a "regular diamond." As to which of these can actually be constructed, that might be a harder problem. At any rate, I do not think this question has been addressed before, so that you are free to investigate it if you want.

Norman

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