After designing the X-Module Double Helix, I got interested in building a tensegrity torus with same-sense X-Modules instead of the alternating sense modules I'd used before. I was curious how the not-quite-three period of the double helix would translate into a torus. In addition, using same-sense modules allowed the building of a torus with an odd number of stages instead of the even number of stages which using alternating-sense modules constrains one to.

The result is the design you see above. I guess I should have at least expected a triangular looking torus since nine modules make make three periods, but what I got was a structure which exhibits two-fold symmetry. I had to differentially weight the inner and outer tendons to get a structure without interference problems. I did this carefully to avoid introducing additional asymmetries. With this structure, the minimum strut-tendon centerline-to-centerline clearance is 0.151 model units and the minimum strut-strut centerline-to-centerline clearance is 0.199.

The struts are colored to emphasize the double helix except there is a discontinuity at the bottom due to the fact that the design encompasses three periods which is an odd number. Because of the location of the discontinuity, the two-fold symmetry transform preserves the colors as well as the geometry.

The table below enumerates the critical statistics by member. The "Weight" value is the weight that was applied to tendons in the objective function. This is not applicable to the struts since they were all constrained to be of length 2.0 model units. It is also not applicable to the girdling tendons which mark the boundary between stages (pink in the diagram above) since they were all constrained to be of length 1.0 model units. The "Weight" value is of no interest to model builders. Only people who are interested in the mathematical programming problem that was used to design the structure will find it of interest. When the value appears as an expression, "^" indicates exponentiation, "*" indicates multiplication and "/" indicates division. The exponent is applied before any division or multiplication is done.

LabelEnd Points LengthForceWeight
st0103p  pt01A  pt03a   2.00000  -5.73504    N/A
st0103q pt01Bpt03b 2.00000-3.88959  N/A
side01ap  pt01Apt03b 1.84781  1.84781  1.00000
side01bp pt11apt03b 1.59298  0.972316  1.0/2.45^0.550929
side01cp pt01Apt02B 1.08355  1.77521   1.0*2.45^0.550929
side01aq pt01Bpt03a 1.84781  1.84781  1.00000
side01bq pt11bpt03a 1.08355  1.77521   1.0*2.45^0.550929
side01cq pt01Bpt02A 1.59298  0.9723161.0/2.45^0.550929
gird01ap pt02Apt03a 1.00000  0.961236N/A
gird01bp pt02Apt03b 1.00000  2.17191  N/A
gird01aq pt02Bpt03b 1.00000  0.574425N/A
gird01bq pt02Bpt03a 1.00000  3.01893  N/A
st0204p pt02Apt04a 2.00000-4.19561   N/A
st0204q pt02Bpt04b 2.00000-6.01463  N/A
side02ap pt02Apt04b 1.81772  0.9036411.0/2.45^0.779954
side02bp pt04apt03b 1.11558  1.23002   1.0*2.45^0.108978
side02cp pt02Apt03A 1.86981  0.7888191.0/2.45^0.963138
side02aq pt02Bpt04a 1.35013  2.71586   1.0*2.45^0.779954
side02bq pt04bpt03a 1.16002  1.05210   1.0/2.45^0.108978
side02cq pt02Bpt03B 1.04442  2.47569   1.0*2.45^0.963138
gird02ap pt03Bpt04a 1.00000  0.110541N/A
gird02bp pt03Apt04a 1.00000  3.88965  N/A
gird02aq pt03Apt04b 1.00000  1.04863  N/A
gird02bq pt03Bpt04b 1.00000  3.88965  N/A
st0305p pt03Apt05a 2.00000-6.01463   N/A
st0305q pt03Bpt05b 2.00000-4.19561  N/A
side03ap pt03Apt05b 1.81772  0.9036411.0/2.45^0.779954
side03bp pt05apt04a 1.04442  2.47569   1.0*2.45^0.963138
side03cp pt03Apt04B 1.16002  1.05210   1.0/2.45^0.108978
side03aq pt03Bpt05a 1.35013  2.71586   1.0*2.45^0.779954
side03bq pt05bpt04b 1.86981  0.7888191.0/2.45^0.963138
side03cq pt03Bpt04A 1.11558  1.23002   1.0*2.45^0.108978
gird03ap pt04Bpt05a 1.00000  3.01893  N/A
gird03bp pt04Apt05a 1.00000  0.574425N/A
gird03aq pt04Apt05b 1.00000  2.17191  N/A
gird03bq pt04Bpt05b 1.00000  0.961236N/A
st0406p pt04Apt06a 2.00000-3.88959   N/A
st0406q pt04Bpt06b 2.00000-5.73504  N/A
side04ap pt04Apt06b 1.84781  1.84781  1.00000
side04bp pt06apt05b 1.59298  0.9723161.0/2.45^0.550929
side04cp pt04Apt05A 1.59298  0.9723161.0/2.45^0.550929
side04aq pt04Bpt06a 1.84781  1.84781  1.00000
side04bq pt06bpt05a 1.08355  1.77521   1.0*2.45^0.550929
side04cq pt04Bpt05B 1.08355  1.77521   1.0*2.45^0.550929
gird04ap pt05Bpt06a 1.00000  0.574425N/A
gird04bp pt05Apt06a 1.00000  2.17191  N/A
gird04aq pt05Apt06b 1.00000  0.961236N/A
gird04bq pt05Bpt06b 1.00000  3.01893  N/A
st0507p pt05Apt07a 2.00000-4.19561   N/A
st0507q pt05Bpt07b 2.00000-6.01463  N/A
side05ap pt05Apt07b 1.81772  0.9036411.0/2.45^0.779954
side05bp pt07apt06a 1.11558  1.23002   1.0*2.45^0.108978
side05cp pt05Apt06B 1.86981  0.7888191.0/2.45^0.963138
side05aq pt05Bpt07a 1.35013  2.71586   1.0*2.45^0.779954
side05bq pt07bpt06b 1.16002  1.05210   1.0/2.45^0.108978
side05cq pt05Bpt06A 1.04442  2.47569   1.0*2.45^0.963138
gird05ap pt06Bpt07a 1.00000  3.88965  N/A
gird05bp pt06Apt07a 1.00000  0.110541N/A
gird05aq pt06Apt07b 1.00000  3.88965  N/A
gird05bq pt06Bpt07b 1.00000  1.04863  N/A
st0608p pt06Apt08a 2.00000-4.19561   N/A
st0608q pt06Bpt08b 2.00000-6.01463  N/A
side06ap pt06Apt08b 1.35013  2.71586   1.0*2.45^0.779954
side06bp pt08apt07b 1.86981  0.788819 1.0/2.45^0.963138
side06cp pt06Apt07A 1.11558  1.23002   1.0*2.45^0.108978
side06aq pt06Bpt08a 1.81772  0.9036411.0/2.45^0.779954
side06bq pt08bpt07a 1.04442  2.47569   1.0*2.45^0.963138
side06cq pt06Bpt07B 1.16002  1.05210   1.0/2.45^0.108978
gird06ap pt07Bpt08a 1.00000  0.961236N/A
gird06bp pt07Apt08a 1.00000  2.17191  N/A
gird06aq pt07Apt08b 1.00000  0.574425N/A
gird06bq pt07Bpt08b 1.00000  3.01893  N/A
st0709p pt07Apt09a 2.00000-3.88959   N/A
st0709q pt07Bpt09b 2.00000-5.73504  N/A
side07ap pt07Apt09b 1.84781  1.84781  1.00000
side07bp pt09apt08a 1.59298  0.9723161.0/2.45^0.550929
side07cp pt07Apt08B 1.59298  0.9723161.0/2.45^0.550929
side07aq pt07Bpt09a 1.84781  1.84781  1.00000
side07bq pt09bpt08b 1.08355  1.77521   1.0*2.45^0.550929
side07cq pt07Bpt08A 1.08355  1.77521   1.0*2.45^0.550929
gird07ap pt08Bpt09a 1.00000  2.17191  N/A
gird07bp pt08Apt09a 1.00000  0.574425N/A
gird07aq pt08Apt09b 1.00000  3.01893  N/A
gird07bq pt08Bpt09b 1.00000  0.961236N/A
st0810p pt08Apt10a 2.00000-6.01463   N/A
st0810q pt08Bpt10b 2.00000-4.19561  N/A
side08ap pt08Apt10b 1.35013  2.71586   1.0*2.45^0.779954
side08bp pt10apt09b 1.16002  1.05210   1.0/2.45^0.108978
side08cp pt08Apt09A 1.04442  2.47569   1.0*2.45^0.963138
side08aq pt08Bpt10a 1.81772  0.9036411.0/2.45^0.779954
side08bq pt10bpt09a 1.11558  1.23002   1.0*2.45^0.108978
side08cq pt08Bpt09B 1.86981  0.7888191.0/2.45^0.963138
gird08ap pt09Bpt10a 1.00000  1.04863  N/A
gird08bp pt09Apt10a 1.00000  3.88965  N/A
gird08aq pt09Apt10b 1.00000  0.110541N/A
gird08bq pt09Bpt10b 1.00000  3.88965  N/A
st0911p pt09Apt11a 2.00000-4.19561   N/A
st0911q pt09Bpt11b 2.00000-6.01463  N/A
side09ap pt09Apt11b 1.35013  2.71586   1.0*2.45^0.779954
side09bp pt11apt10a 1.86981  0.7888191.0/2.45^0.963138
side09cp pt09Apt01B 1.11558  1.23002   1.0*2.45^0.108978
side09aq pt09Bpt11a 1.81772  0.9036411.0/2.45^0.779954
side09bq pt11bpt10b 1.04442  2.47569   1.0*2.45^0.963138
side09cq pt09Bpt01A 1.16002  1.05210   1.0/2.45^0.108978
gird09ap pt01Bpt11a 1.00000  2.17191  N/A
gird09bp pt01Apt11a 1.00000  0.961236N/A
gird09aq pt01Apt11b 1.00000  3.01893  N/A
gird09bq pt01Bpt11b 1.00000  0.574425N/A

The design for this structure was completed September 21, 2004.