]> A Practical Guide to Tensegrity Design: 6.2 A Procedure for Designing Double-Layer Tensegrity Domes

6.2 Double-Layer Tensegrity Domes: An Example

The following steps implement the design of a double-layer dome like that described in Section 6.1:

Step 1 Solve the tensegrity programming problem for the sphere.
Step 2 Implement the topological changes required by the truncation.
Step 3 Adjust the base points (the points of the truncation polylateral as they manifest themselves on the inner tendon network) so they lie evenly-spaced on a circle which approximates as closely as possible their unadjusted positions in the original sphere.
Step 4 Add guys.
Step 5 Using the coordinate values from the sphere as initial values, solve the tensegrity programming problem for the dome.
Step 6 Make necessary adjustments to fix member force and interference problems.

To illustrate this method for truncating double-layer spheres, the tensegrity based on the 6ν octahedron is useful. It has a low-enough frequency to be pedagogically tractable and a high-enough frequency that the appearance of higher-frequency structures can be anticipated in studying it.

6.2.1 Dome Step 1: Compute the sphere

Figures 6.3 and 6.4 diagram the basic triangle network for the 6ν double-layer tensegrity octahedron sphere and a coordinate system for its analysis in the same manner as Figures 5.3 and 5.5 did for the 4ν version of the sphere in Section 5.3. The main difference is that, with the higher frequency, there is more of everything. For example, now the struts in Figure 6.4 are clustered about three basic t-tripods instead of two as in Figure 5.5.

Table 6.1 enumerates the members of this 6ν version of the double-layer sphere. The anomalous value of 1.5 for the length of Member #33 in Table 6.1 is chosen in light of experience with the 4ν structure.1

The weights for the inner and outer binding tendons in the objective function are derived using the formula k b1 +b2 2 b1 b2 2 where the values used for k are 1.2 and 0.5 respectively for the inner and outer binding tendons. b1 and b2 represent the spherical excess corresponding to the initial values of the two end points of the tendon.2 The spherical excess is the amount the sphere radius exceeds the distance of the unprojected point from the center of the octahedron. This number is calculated as a ratio and is always greater than or equal to 1.0. It is equal to 1.0 at the vertexes of the octahedron. Giving a smaller weight to the tendons distant from the vertexes of the basis octahedron allows them to be longer than they would otherwise be. This allows the octahedral faces to bulge out more than they would otherwise and gives the structure a more spherical, less faceted, look. The objective-function weights for the primary and secondary interlayer tendons are 2.0 and 1.4 respectively independent of any spherical excess values.

schematic diagram showing pentagonal grid overlaying alternating-triangle grid with annotations
Figure 6.3: 6ν T-Octahedron Sphere: Symmetry Regions


schematic diagram showing schematic struts overlaying alternating-triangle grid with outer-convergence triangles noted
Figure 6.4: 6ν T-Octahedron Sphere: Truss Members


 Member 
#
 End Points   Weight   Constrained 
Length
 Comments 
1
2
3
4
5
6
7
8
9
P1' P11
P2' P7
P3' P10
P4' P12
P9' P15
P8' P2
P7' P8
P5' P3
P14' P6
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 3.0 
 Struts 
10
11
12
13
14
15
16
17
18
P2' P11
P3' P7
P1' P10
P9' P12
P8' P15
P4' P2
P13' P8
P6' P3
P7' P6
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 2.0 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 Primary Interlayer Tendons 
19
20
21
22
23
24
25
26
27
P1' P2
P2' P3
P3' P1
P4' P9
P9' P8
P8' P4
P7' P13
P5' P6
P14' P7
 1.4 
 1.4 
 1.4 
 1.4 
 1.4 
 1.4 
 1.4 
 1.4 
 1.4 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 Secondary Interlayer Tendons 
28
29
30
31
32
33
34
35
36
P2 P11
P3 P7
P1 P10
P9 P12
P8 P15
P4 P2
P13 P8
P6 P3
P7 P6
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.5 
 1.0 
 1.0 
 1.0 
 Inner Convergence Tendons 
37
38
39
40
41
42
43
44
45
P1' P2'
P2' P3'
P3' P1'
P4' P9'
P9' P8'
P8' P4'
P7' P13'
P5' P6'
P14' P7'
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 1.0 
 Outer Convergence Tendons 
46
47
48
49
50
51
52
53
54
P2' P11'
P3' P7'
P1' P10'
P9' P12'
P8' P15'
P4' P2'
P13' P8'
P6' P3'
P7' P6'
 0.5000 
 0.3065 
 0.4196 
 0.3065 
 0.2692 
 0.4196 
 0.3065 
 0.3065 
 0.2692 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 Outer Binding Tendons 
55
56
57
58
59
60
61
62
63
P1 P2
P2 P3
P3 P1
P4 P9
P9 P8
P8 P4
P7 P13
P5 P6
P14 P7
 1.2000 
 1.0069 
 1.0069 
 0.7356 
 0.6462 
 0.7356 
 0.7356 
 0.7356 
 0.6462 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 N/A 
 Inner Binding Tendons 
 
Table 6.1: 6ν T-Octahedron Sphere: Truss Members

As with the 4ν version of this sphere, the derivation of the initial point values is facilitated by the use of the geodesic breakdown. Kenner's tables3 are used to generate initial point coordinates. Again, Kenner's table has to be expanded by rotating all the points about the z axis by 90°. Table 6.2 outlines the correspondence between the basic points and his coordinate system. (Rotated points are indicated with an asterisk.)

 Point 
 Kenner's Label 
 Coordinates 
θ φ
  P1  P1'  
  P2  P2'  
  P3  P3'  
  P4  P4'  
  P5  P5'  
  P6  P6'  
  P7  P7'  
  P8  P8'  
  P9  P9'  
1,0
1,1
2,1
2,1*
3,0
3,1
3,2
3,1*
3,2*
 0.0000   11.3099 
 90.0000   11.3099 
 45.0000   19.4712 
 135.0000   19.4712 
 0.0000   45.0000 
 26.5651   36.6992 
 63.4349   36.6992 
 116.5651   36.6992 
 153.4349   36.6992 
 
Table 6.2: 6ν T-Octahedron: Angular Point Coordinates

The initial coordinate values for inner and outer realizations of these points are summarized in Table 6.3. These are derived from the angular values in Table 6.2 with inner and outer radiuses applied. The inner radius (3.15) is chosen so the triangle tendon lengths average approximately 1 (0.995729). The outer radius (5.15) is chosen so strut lengths in the double-layer versions of the structure would initially average approximately 3 (2.94314). The implied initial lengths are summarized in Table 6.4.

 Point 
 Coordinates 
x y z
P1
P2
P3
P4
P5
P6
P7
P8
P9
 0.6178   0.0000   3.0888 
 0.0000   0.6178   3.0888 
 0.7425   0.7425   2.9698 
-0.7425   0.7425   2.9698 
2.2274   0.0000   2.2274 
1.6837   0.8419   2.5256 
0.8419   1.6837   2.5256 
-0.8419   1.6837   2.5256 
-1.6837   0.8419   2.5256 
P1'
P2'
P3'
P4'
P5'
P6'
P7'
P8'
P9'
 1.0100   0.0000   5.0500 
 0.0000   1.0100   5.0500 
 1.2139   1.2139   4.8555 
-1.2139   1.2139   4.8555 
3.6416   0.0000   3.6416 
2.7528   1.3764   4.1292 
1.3764   2.7528   4.1292 
-1.3764   2.7528   4.1292 
-2.7528   1.3764   4.1292 
 
Table 6.3: 6ν T-Octahedron Sphere: Initial Basic Point Coordinates


 Member 
#
 Length   Member 
#
 Length   Member 
#
 Length 
1  2.5487  2  2.7450  3  2.7577 
4  3.0672  5  3.3095  6  2.7450 
7  2.9385  8  3.0672  9  3.3095 
10  2.2908  11  2.4057  12  2.2248 
13  2.4057  14  2.5135  15  2.2248 
16  2.4057  17  2.4057  18  2.5135 
19  2.2908  20  2.2248  21  2.2248 
22  2.4057  23  2.5135  24  2.4057 
25  2.4057  26  2.4057  27  2.5135 
28  0.8737  29  1.0456  30  0.7622 
31  1.0456  32  1.1906  33  0.7622 
34  1.0456  35  1.0456  36  1.1906 
37  1.4284  38  1.2461  39  1.2461 
40  1.7094  41  1.9465  42  1.7094 
43  1.7094  44  1.7094  45  1.9465 
46  1.4284  47  1.7094  48  1.2461 
49  1.7094  50  1.9465  51  1.2461 
52  1.7094  53  1.7094  54  1.9465 
55  0.8737  56  0.7622  57  0.7622 
58  1.0456  59  1.1906  60  1.0456 
61  1.0456  62  1.0456  63  1.1906 
 
Table 6.4: 6ν T-Octahedron Sphere: Initial Member Lengths

The derivation of the symmetry points from the basic points is shown in Table 6.5. The symmetry transforms on which this table is based are enumerated in Table 5.5. Outer points follow the same symmetries as inner points.

 Point 
Coordinates
x y z
 Basic 
Point
 Transform 
Number
P10
P11
P12
P13
P14
P15
-x4 -y4 z4
-x1 -y1 z1
-x5 -y5 z5
y5 z5 x5
y6 z6 x6
-y9 z9 -x9
P4 4
P1 4
P5 4
P5 2
P6 2
P9 5
 
Table 6.5: 6ν T-Octahedron Sphere: Symmetry Point Correspondences

The structure is computed by minimizing a weighted combination of the interlayer and binding tendons subject to constraints on the struts and convergence tendons. Two initial iterations are done using the penalty formulation μ= 105 in conjunction with Fletcher-Reeves to bring the initial points into approximate conformance with the constraints. After this five iterations are done with the exact formulation in conjunction with Fletcher-Reeves to bring the values to convergence. The derivatives of the objective function with respect to the independent coordinate values are all less than 10 -3.

Tables 6.6 and 6.7 show the values for the final lengths and relative forces.4 Table 6.8 shows the final values for the coordinates of the basic points. Figure 6.5 shows how the final version of the spherical structure appears as viewed from outside one of the octahedral vertices. For clarity, interlayer tendons have been excluded and members in the background have been eliminated by truncation. For reference, selected points in Figure 6.5 are labeled.

 Member 
#
 Length   Member 
#
 Length   Member 
#
 Length 
1  3.0000  2  3.0000  3  3.0000 
4  3.0000  5  3.0000  6  3.0000 
7  3.0000  8  3.0000  9  3.0000 
10  2.3545  11  2.3871  12  2.4881 
13  2.2793  14  2.2883  15  2.3153 
16  2.2212  17  2.2209  18  2.2354 
19  2.1286  20  2.0833  21  2.1669 
22  2.0342  23  2.0334  24  1.6827 
25  2.0342  26  2.0454  27  2.0516 
28  1.0000  29  1.0000  30  1.0000 
31  1.0000  32  1.0000  33  1.5000 
34  1.0000  35  1.0000  36  1.0000 
37  1.0000  38  1.0000  39  1.0000 
40  1.0000  41  1.0000  42  1.0000 
43  1.0000  44  1.0000  45  1.0000 
46  1.8502  47  2.4709  48  2.1613 
49  2.6524  50  2.7414  51  2.4735 
52  2.7549  53  2.6798  54  2.6482 
55  1.2081  56  1.2653  57  1.2626 
58  1.3406  59  1.6730  60  1.2480 
61  1.9008  62  1.8434  63  1.9693 
 
Table 6.6: 6ν T-Octahedron Sphere: Final Member Lengths


 Member 
#
 Relative 
 Force 
 Member 
#
 Relative 
 Force 
 Member 
#
 Relative 
 Force 
1 -11.294  2 -9.788  3 -10.052 
4 -10.125  5 -10.019  6 -9.925 
7 -10.052  8 -10.064  9 -9.870 
10 4.709  11 4.774  12 4.976 
13 4.559  14 4.577  15 4.631 
16 4.442  17 4.442  18 4.471 
19 2.980  20 2.917  21 3.033 
22 2.848  23 2.847  24 2.356 
25 2.848  26 2.863  27 2.872 
28 4.945  29 4.580  30 3.811 
31 5.009  32 5.092  33 4.947 
34 4.958  35 5.258  36 5.163 
37 4.144  38 4.887  39 4.040 
40 4.865  41 4.867  42 5.214 
43 4.815  44 5.083  45 5.547 
46 0.925  47 0.757  48 0.907 
49 0.813  50 0.738  51 1.038 
52 0.844  53 0.821  54 0.713 
55 1.450  56 1.274  57 1.271 
58 0.986  59 1.081  60 0.918 
61 1.398  62 1.356  63 1.273 
 
Table 6.7: 6ν T-Octahedron Sphere: Final Member Forces


 Point 
 Coordinates 
x y z
P1
P2
P3
P4
P5
P6
P7
P8
P9
 1.0378   -0.2360   3.5592 
 -0.0640   0.2053   3.7844 
 0.8711   1.0149   3.5173 
-1.0998   1.2224   3.4065 
2.9400   -0.4191   2.3400 
1.7538   0.7511   3.1285 
1.3434   1.6303   2.8864 
-1.4134   2.2919   2.8451 
-2.3233   0.8934   2.9682 
P1'
P2'
P3'
P4'
P5'
P6'
P7'
P8'
P9'
 1.3525   0.2829   5.3714 
 0.3628   0.4068   5.4440 
 0.9467   1.1801   5.1968 
-1.5610   1.7442   4.6513 
3.7764   0.4745   3.0005 
3.0842   0.8558   3.6132 
1.4390   2.9290   3.5221 
-2.1735   2.3144   4.1038 
-2.4411   1.3815   4.3450 
 
Table 6.8: 6ν T-Octahedron: Final Basic Point Coordinates


line drawing of dowel-and-fishing-line of near half of 6v sphere with some point labels
Figure 6.5: 6ν T-Octahedron Sphere: Vertex View

1 See Section 8.2.3 for details on this exception as it is introduced to the 4ν structure.

2 b stands for bulge.

3 Kenner76, "Octahedron Class I Coordinates: Frequencies 12, 6, 3", column 6ν, p. 126.

4 See Section 7.2 for the method of computing relative forces.