Convex hulls having regular diamonds

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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