Fractal Tensegrity

"Can geometry deliver what the Greek root of its name [geo-] seemed to promise—truthful measurement, not only of cultivated fields along the Nile River but also of untamed Earth?"

Benoit Mandelbrot asked that question, and created the first-ever "theory of roughness." He saw that roughness in the shapes of mountains, coastlines and river basins. It lurked in the structures of plants, blood vessels and lungs. It danced in the splay of galaxies. His personal quest was to devise a mathematical formula to measure this overall "roughness," an aspect of nature large and small.

Roughness, he determined, is an assessment of the degree of fractal consistency across scales. The tree cartoons reproduced here are an example of a very smooth fractal iteration. The large tree has added more tinier trees, each generation adding the exact same shape, just iterated in ever tinier versions. Mandelbrot called these "bottomless wonders [that] spring from simple rules, which are repeated without end." 

Obviously, when we consider fractals we are considering a very natural phenomenon. This simple tree cartoon, dividing and dividing again, reveals to us a powerful repeating pattern that occurs in nature and indeed, in our own bodies. 

R. Buckminster Fuller applied a similar, simple iterative process in his work on tensegrity. Consider the typical, classic tensegrity structure. It is composed of struts and tendons that radically separate the work of compression and tension. Fuller imagined substituting for any individual strut in the tensegrity, another tensegrity, and so on.

In his words:

The tensegrity masts can be substituted for the individual (so-called solid) struts in the tensegrity spheres. In each one of the separate tensegrity masts, acting as struts, in the tensegrity spheres it can be seen that there are little (so-called) solid struts. A miniature tensegrity mast may be substituted for each of those solid struts. The subminiature tensegrity mast within the tensegrity struts of the tensegrity struts of the tensegrity sphere and a subsubminiature tensegrity mast may be substituted for each of those solid struts, and so on to subsubsubminiature tensegrities until we finally get down to the size of the atom and this becomes completely compatible with the atom for the atom is tensegrity and there are no "solids" left in the entire structural system. There are no solids in structures, ergo no solids in Universe. There is nothing incompatible with what we may see as solid at the visual level and what we are finding out to be the structural relationships in nuclear physics.

Above is the illustration Fuller commissioned for his two-volume work "Synergetics." Graham Scarr recently rendered this fractal tensegrity concept in an artifact, photographed and reproduced  below. 

I blog about fractal tensegrity because I recently found a team that realized such a  sub-, sub-, sub-miniaturized structure. A European team used the fractal techniques described here to build structures that maintain their strength despite weighing significantly less.  They created a hollow metal beam, then re-engineered said beam by changing its diameter and thickness such that the beam weighs as little as possible but is still able to withstand the weight or pressure it will be subjected to as a final structure.

Once that is achieved, label it generation-0. They then created a larger beam made out of generation-0 beams arranged using triangular subdivisions. This next beam is labeled a generation-1 beam. Generation-2 beams are proposed that replace all of the generation-1 beams with full scale versions of itself, only in a larger construct. Theoretically, the process can be repeated as many times as is desired or is possible, depending on scale. The resulting structures weigh less than those made with solid steal because they contain less of it – the more generations used, the lighter the structure becomes proportionally to one made of solid steel.

As an example the researchers suggest that a solar sail made with a solid metal 100 meter boom could be made to be 10,000 times lighter if a generation-3 beam were used instead.

Hmmm... What is the smallest possible tensegrity? Let's consider that in our next post.

Links:

Benoit Mandelbrot TED talk on roughness

Ultralight Fractal Structures from Hollow Tubes

Synergetics by Fuller: Miniaturization Approaches Atomic Structure

Abstract:

A fractal design is shown to be highly efficient both as a load bearing structure and as a general metamaterial. Through changing the hierarchical order of the structure, the scaling of material required for stability against loading can be manipulated. We show that the transition from solid to hollow beams changes the scaling in a manner analogous to increasing the hierarchical order by one. An example second order solid beam frame is constructed using rapid prototyping techniques. The optimal hierarchical order of the structure is found for different values of loading. Possible fabrication methods and applications are then discussed.

A fractal design is shown to be highly efficient both as a load bearing structure and as a general metamaterial. Through changing the hierarchical order of the structure, the scaling of material required for stability against loading can be manipulated. We show that the transition from solid to hollow beams changes the scaling in a manner analogous to increasing the hierarchical order by one. An example second order solid beam frame is constructed using rapid prototyping techniques. The optimal hierarchical order of the structure is found for different values of loading. Possible fabrication methods and applications are then discussed.

Read more at: http://phys.org/news/2012-12-team-fractal-geometry-lighter.html#jCp

A fractal design is shown to be highly efficient both as a load bearing structure and as a general metamaterial. Through changing the hierarchical order of the structure, the scaling of material required for stability against loading can be manipulated. We show that the transition from solid to hollow beams changes the scaling in a manner analogous to increasing the hierarchical order by one. An example second order solid beam frame is constructed using rapid prototyping techniques. The optimal hierarchical order of the structure is found for different values of loading. Possible fabrication methods and applications are then discussed.

Read more at: http://phys.org/news/2012-12-team-fractal-geometry-lighter.html#jCp

A fractal design is shown to be highly efficient both as a load bearing structure and as a general metamaterial. Through changing the hierarchical order of the structure, the scaling of material required for stability against loading can be manipulated. We show that the transition from solid to hollow beams changes the scaling in a manner analogous to increasing the hierarchical order by one. An example second order solid beam frame is constructed using rapid prototyping techniques. The optimal hierarchical order of the structure is found for different values of loading. Possible fabrication methods and applications are then discussed.

Read more at: http://phys.org/news/2012-12-team-fractal-geometry-lighter.html#jCp