]> A Practical Guide to Tensegrity Design: 6.2.2 Dome Step 2: Implement the truncation

6.2.2 Dome Step 2: Implement the truncation

Figure 6.6 diagrams the four "great"-circle truncation possibilities for this structure as they fall on its reference octahedron. Figure 6.7 shows the same four truncation boundaries as they fall on the inner layer of the sphere. None of these boundaries corresponds to a true great circle. The true great circle lies at the center of their range and is not usable as a truncation at this frequency. In a higher-frequency structure there would be still more of these circles available. All of them are possibilities as truncation definitions, although the ones farther away from the true great circle would probably require greater adjustments to work well. At this frequency, the two middle truncations are equally far from the true great circle, and so neither has an advantage as far as adjustments required. This being the case, the one that allows more volume is selected.

alternating-triangle grid arranged as octahedron with latitudinal circular girdles and annotations
Figure 6.6: 6ν Octahedron: Unprojected Truncation Boundaries


alternating-triangle grid arranged as sphere with latitudinal circular girdles and annotations
Figure 6.7: 6ν Octahedron: Projected Truncation Boundaries

Figures 6.8 and 6.9 diagram the basic triangle network for the truncated structure and a coordinate system for its analysis. These figures are in roughly the same style as the corresponding figures for the 4ν and 6ν spheres. Figure 6.8 is more complex than for those earlier structures since it attempts to diagram the correspondence between the symmetry regions of the 6ν sphere and those of the dome.

The boundaries of the symmetry regions for the sphere are outlined with dotted lines and labeled with small numbers in circles. The numbers correspond to the symmetry transformations listed in Table 5.5. The boundaries for the dome are outlined by a hollow dotted line. The dome's symmetry regions are enumerated with larger numbers in circles. These numbers also correspond to the symmetry transformations listed in Table 5.5, although, due to the loss of symmetry, only the first three entries in the table are possibilities. A grasp of the correspondence between the symmetry regions for the sphere and those for the dome is useful for generating initial points for the dome calculations from the final values of the sphere calculations. These correspondences are used in Tables 6.16 and 6.17. As noted below, these correspondences are altered slightly for the inner points at the base of the dome.

schematic diagram showing pentagonal grid overlaying alternating-triangle grid with annotations
Figure 6.8: 6ν T-Octahedron Dome: Symmetry Regions


schematic diagram showing schematic struts overlaying alternating-triangle grid with outer-convergence triangles noted
Figure 6.9: 6ν T-Octahedron Dome: Truss Members

Tables 6.9 through 6.16 enumerate the members of the truncated structure. The anomalous members which have a length of 1.5 correspond to Member #33 in Table 6.1 for the sphere. For the most part, the weights for this structure are mapped from the weights used for the corresponding members in the spherical version of the structure. The exceptions are the weights for members #160, #161, #163, #164, #166 and #167. The reasons for these exceptions are discussed below. In addition to the members enumerated in Tables 6.9 through 6.16, guys are introduced in Step 4.

Due to the loss of symmetry induced by the truncation, the tables for the dome are much larger than for the sphere, and the computations required are correspondingly more massive. The dome is composed of three symmetrical parts whereas the same area on the sphere is composed of about eight symmetrical parts. The net result is that the tables for the dome are over twice as large as those for the sphere.

Decisions must be made in the neighborhood of the truncation on how to reroute the struts whose inner terminal points lay on the set of triangles which are excluded. The best procedure seems to be to connect them to the inner binding triangle which underlies their tripod. To make this work, the weights are multiplied by 512 for inner binding triangle tendons which touch the base. These are the members mentioned above whose weights do not equal those of the corresponding members in the spherical structure (#160, #161, #163, #164, #166, #167). This reduction in the weights allows the final dome to achieve a height which approximates the height of its initial configuration. With unaltered weights, it would turn out more squat.

The secondary interlayer tendons at these positions, #64, #67 and #70, disappear since there are struts at those same positions in this configuration. Also, the tendons generated by the truncation are eliminated from the model. The inner truncation tendons are redundant since they connect the base points which are fixed. The outer truncation tendons are not necessary for structural integrity and detract from the appearance of the structure.

In Tables 6.9 through 6.16, members re-routed to a new inner point (as compared with the configuration of their corresponding member in the sphere) due to the truncation are marked with †. Members which are excluded (although for completeness they are included in the tables) are marked with ‡.

 Member 
#
 End Points   Weight   Constrained 
Length
Sphere
 Member 
1
2
3
4
5
6
P2' P7
P29' P4
P25' P1
P5' P9
P6' P2
P4' P11
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
7
8
9
1
2
3
7
8
9
10
11
12
P8' P17
P13' P26
P7' P6
P10' P19
P14' P8
P9' P5
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
4
5
6
2
3
1
13
14
15
16†
17
18
P12' P21
P15' P10
P11' P29
P16' P22
P28' P20
P27' P13
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
5
6
4
6
4
5
19†
20
21
22†
23
24
P18' P23
P23' P16
P17' P14
P20' P24
P24' P18
P19' P15
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
9
7
8
8
9
7
 
Table 6.9: 6ν T-Octahedron Dome: Struts


 Member 
#
 End Points   Weight   Constrained 
Length
Sphere
 Member 
25
26
27
28
29
30
P3' P7
P1' P4
P2' P1
P6' P9
P4' P2
P5' P11
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
16
17
18
10
11
12
31
32
33
34
35
36
P13' P17
P7' P26
P8' P6
P14' P19
P9' P8
P10' P5
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
13
14
15
11
12
10
37
38
39
40†
41
42
P15' P21
P11' P10
P12' P29
P22' P22
P21' P20
P16' P13
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
14
15
13
15
13
14
43†
44
45
46†
47
48
P23' P23
P17' P16
P18' P14
P24' P24
P19' P18
P20' P15
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
18
16
17
17
18
16
 
Table 6.10: 6ν T-Octahedron Dome: Primary Interlayer Tendons


 Member 
#
 End Points   Weight   Constrained 
Length
Sphere
 Member 
49
50
51
52
53
54
P2' P3
P29' P1
P25' P2
P5' P6
P6' P4
P4' P5
 1.4 
 1.4 
 1.4 
 1.4 
 1.4 
 1.4 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
25
26
27
19
20
21
55
56
57
58
59
60
P8' P13
P13' P7
P7' P8
P10' P14
P14' P9
P9' P10
 1.4 
 1.4 
 1.4 
 1.4 
 1.4 
 1.4 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
22
23
24
20
21
19
61
62
63
64‡
65
66
P12' P15
P15' P11
P11' P12
P16' P22
P28' P21
P27' P16
 1.4 
 1.4 
 1.4 
 N/A 
 1.4 
 1.4 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
23
24
22
24
22
23
67‡
68
69
70‡
71
72
P18' P23
P23' P17
P17' P18
P20' P24
P24' P19
P19' P20
 N/A 
 1.4 
 1.4 
 N/A 
 1.4 
 1.4 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
27
25
26
26
27
25
 
Table 6.11: 6ν T-Octahedron Dome: Secondary Interlayer Tendons


 Member 
#
 End Points   Weight   Constrained 
Length
Sphere
 Member 
73
74
75
76
77
78
P3 P7
P1 P4
P2 P1
P6 P9
P4 P2
P5 P11
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
34
35
36
28
29
30
79
80
81
82
83
84
P13 P17
P7 P26
P8 P6
P14 P19
P9 P8
P10 P5
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.0 
 1.5 
 1.0 
 1.0 
 1.0 
31
32
33
29
30
28
85
86
87
88‡
89
90
P15 P21
P11 P10
P12 P29
P22 P22
P21 P20
P16 P13
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.5 
 1.0 
 N/A 
 1.0 
 1.0 
32
33
31
33
31
32
91‡
92
93
94‡
95
96
P23 P23
P17 P16
P18 P14
P24 P24
P19 P18
P20 P15
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.0 
 N/A 
 1.0 
 1.0 
36
34
35
35
36
34
 
Table 6.12: 6ν T-Octahedron Dome: Inner Convergence Tendons


 Member 
#
 End Points   Weight   Constrained 
Length
Sphere
 Member 
97
98
99
100
101
102
P2' P3'
P29' P1'
P25' P2'
P5' P6'
P6' P4'
P4' P5'
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
43
44
45
37
38
39
103
104
105
106
107
108
P8' P13'
P13' P7'
P7' P8'
P10' P14'
P14' P9'
P9' P10'
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
40
41
42
38
39
37
109
110
111
112
113
114
P12' P15'
P15' P11'
P11' P12'
P16' P22'
P28' P21'
P27' P16'
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
41
42
40
42
40
41
115
116
117
118
119
120
P18' P23'
P23' P17'
P17' P18'
P20' P24'
P24' P19'
P19' P20'
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
45
43
44
44
45
43
 
Table 6.13: 6ν T-Octahedron Dome: Outer Convergence Tendons


 Member 
#
 End Points   Weight   Constrained 
Length
Sphere
 Member 
121
122
123
124
125
126
P3' P7'
P1' P4'
P2' P1'
P6' P9'
P4' P2'
P5' P11'
 0.3065 
 0.3065 
 0.2692 
 0.5000 
 0.3065 
 0.4196 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
52
53
54
46
47
48
127
128
129
130
131
132
P13' P17'
P7' P26'
P8' P6'
P14' P19'
P9' P8'
P10' P5'
 0.3065 
 0.2692 
 0.4196 
 0.3065 
 0.4196 
 0.5000 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
49
50
51
47
48
46
133
134
135
136‡
137
138
P15' P21'
P11' P10'
P12' P29'
P22' P22'
P21' P20'
P16' P13'
 0.2692 
 0.4196 
 0.3065 
 N/A 
 0.3065 
 0.2692 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
50
51
49
51
49
50
139‡
140
141
142‡
143
144
P23' P23'
P17' P16'
P18' P14'
P24' P24'
P19' P18'
P20' P15'
 N/A 
 0.3065 
 0.3065 
 N/A 
 0.2692 
 0.3065 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
54
52
53
53
54
52
 
Table 6.14: 6ν T-Octahedron Dome: Outer Binding Tendons


 Member 
#
 End Points   Weight   Constrained 
Length
Sphere
 Member 
145
146
147
148
149
150
P2 P3
P29 P1
P25 P2
P5 P6
P6 P4
P4 P5
 0.7356 
 0.7356 
 0.6462 
 1.2000 
 1.0069 
 1.0069 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
61
62
63
55
56
57
151
152
153
154
155
156
P8 P13
P13 P7
P7 P8
P10 P14
P14 P9
P9 P10
 0.7356 
 0.6462 
 0.7356 
 1.0069 
 1.0069 
 1.2000 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
58
59
60
56
57
55
157
158
159
160
161
162
P12 P15
P15 P11
P11 P12
P16 P22
P28 P21
P27 P16
 0.6462 
 0.7356 
 0.7356 
 0.3758 
 0.3758 
 0.6462 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
59
60
58
60
58
59
163
164
165
166
167
168
P18 P23
P23 P17
P17 P18
P20 P24
P24 P19
P19 P20
 0.2692 
 0.3065 
 0.7356 
 0.3462 
 0.3065 
 0.7356 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
63
61
62
62
63
61
 
Table 6.15: 6ν T-Octahedron Dome: Inner Binding Tendons


 Member 
#
 End Points 
169‡
170‡
171‡
P22 P23
P23 P24
P24 P28
172‡
173‡
174‡
P22' P23'
P23' P24'
P24' P28'
 
Table 6.16: 6ν T-Octahedron Dome: Truncation Tendons

The basic points and their initial coordinate values (as derived from the final values for the corresponding points in the sphere) are summarized in Tables 6.17 and 6.18. The applicable transforms are listed in Table 5.5. The coordinates of P23 and P24 do not correspond exactly to the values of the corresponding points in the sphere. This is due to the Step 3 adjustment.

P22, P23 and P24 of the truncated sphere map from P22, P6 and P3 respectively of the complete sphere, rather than P4, P7 and P6 as would be expected from an unaltered symmetry mapping. This alteration is made so that, even with the change in topology, the initial positions and lengths of the struts in the dome correspond to their final positions in the sphere computations. Since the spherical excesses of P2 and P3 differ from P4 and P6, the initial weights for members #160, #161, #166 and #167, also differ from the values for the corresponding members of the spherical structure. In addition, as mentioned above, the actual weights used for these members, as well as those for #163 and #164, are 512 times the weights corresponding to the sphere.

 Point 
Coordinates
x y z
 Sphere 
Point
 Transform 
Number
P1
P2
P3
P4
P5
P6
 1.7538   0.7511   3.1285 
 1.3434   1.6303   2.8864 
 -0.4191   2.3400   2.9400 
 0.8711   1.0149   3.5173 
 1.0378   -0.2360   3.5592 
 -0.0640   0.2053   3.7844 
P6 1
P7 1
P5 2
P3 1
P1 1
P2 1
P7
P8
P9
P10
P11
P12
 -1.4134   2.2919   2.8451 
 -1.0998   1.2224   3.4065 
 -1.0378   0.2360   3.5592 
 0.0640   -0.2053   3.7844 
 1.0998   -1.2224   3.4065 
 2.3233   -0.8934   2.9682 
P8 1
P4 1
P1 4
P2 4
P4 4
P9 4
P13
P14
P15
P16
P17
P18
 -2.3233   0.8934   2.9682 
 -0.8711   -1.0149   3.5173 
 1.4134   -2.2919   2.8451 
 -2.8451   1.4134   2.2919 
 -2.9400   0.4191   2.3400 
 -1.7538   -0.7511   3.1285 
P9 1
P3 4
P8 4
P8 9
P5 4
P6 4
P19
P20
P21
P22
P23
P24
 -1.3434   -1.6303   2.8864 
 0.4191   -2.3400   2.9400 
 0.8934   -2.9682   2.3233 
 -3.7309   -0.0860   0.0478 
 -3.1023   -1.7622   1.0954 
 -1.6100   -3.2017   1.0425 
P7 4
P5 11
P9 8
P2 9
P6 9
P3 11
 
Table 6.17: 6ν T-Octahedron Dome: Initial Inner Coordinate Values


 Point 
Coordinates
x y z
 Sphere 
Point
 Transform 
Number
P1 '
P2 '
P3 '
P4 '
P5 '
P6 '
 3.0842   0.8558   3.6132 
 1.4390   2.9290   3.5221 
 0.4745   3.0005   3.7764 
 0.9467   1.1801   5.1968 
 1.3525   0.2829   5.3714 
 0.3628   0.4068   5.4440 
P6 ' 1
P7 ' 1
P5 ' 2
P3 ' 1
P1 ' 1
P2 ' 1
P7 '
P8 '
P9 '
P10 '
P11 '
P12 '
 -2.1735   2.3144   4.1038 
 -1.5610   1.7442   4.6513 
 -1.3525   -0.2829   5.3714 
 -0.3628   -0.4068   5.4440 
 1.5610   -1.7442   4.6513 
 2.4411   -1.3815   4.3450 
P8 ' 1
P4 ' 1
P1 ' 4
P2 ' 4
P4 ' 4
P9 ' 4
P13 '
P14 '
P15 '
P16 '
P17 '
P18 '
 -2.4411   1.3815   4.3450 
 -0.9467   -1.1801   5.1968 
 2.1735   -2.3144   4.1038 
 -4.1038   2.1735   2.3144 
 -3.7764   -0.4745   3.0005 
 -3.0842   -0.8558   3.6132 
P9 ' 1
P3 ' 4
P8 ' 4
P8 ' 9
P5 ' 4
P6 ' 4
P19 '
P20 '
P21 '
P22 '
P23 '
P24 '
 -1.4390   -2.9290   3.5221 
 -0.4745   -3.0005   3.7764 
 1.3815   -4.3450   2.4411 
 -4.6513   1.5610   1.7442 
 -3.5221   -1.4390   2.9290 
 -0.8558   -3.6132   3.0842 
P7 ' 4
P5 ' 11
P9 ' 8
P4 ' 9
P7 ' 9
P6 ' 11
 
Table 6.18: 6ν T-Octahedron Dome: Initial Outer Coordinate Values

The derivation of the symmetry points from the basic points is shown in Table 6.19. Outer points follow the same symmetries as inner points. As mentioned above, due to the loss of symmetry as a result of the truncation, only the first three entries of Table 5.5 are possibilities here.

 Point 
Coordinates
x y z
 Basic 
Point
 Transform 
Number
P25
P26
P27
P28
P29
y1 z1 x1
y12 z12 x12
y21 z21 x21
z22 x22 y22
z3 x3 y3
P1 2
P12 2
P21 2
P22 3
P3 3
 
Table 6.19: 6ν T-Octahedron Dome: Symmetry Point Correspondences