Tensegrity

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Donald E. Ingber and Misia Landau (2012), Scholarpedia, 7(2):8344. doi:10.4249/scholarpedia.8344 revision #122885 [link to/cite this article]
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1.00 - Donald E. Ingber

Tensegrity is a design principle that applies when a discontinuous set of compression elements is opposed and balanced by a continuous tensile force, thereby creating an internal prestress that stabilizes the entire structure.

Contents

Introduction

Tensegrity is a term with a rich and sprawling history. It was coined by Buckminster Fuller, the iconoclastic architect, engineer, and poet, to describe his vision of a new kind of architecture, one that looked like it was built by nature instead of by humans. In contrast to the pyramids, columns, and brick-on-brick buildings of the past, which pile solid elements compressively, one on top of the other, Fuller imagined a world full of unconventional structures that maintain their stability, or integrity, through a pervasive tensional force, hence the term tensegrity.

Figure 1: Illustration of Snelson's first tensegrity sculpture.

Fuller began developing his vision in the 1920s, at a time when many were exploring new directions in design and architecture. But it was his student, the sculptor Kenneth Snelson, who, in 1949, created the first structure to be defined as a "tensegrity" (Fig. 1). Using two X-shaped wooden struts suspended in air by a taut nylon cable, Snelson captured the defining features of tensegrity:

  1. Pervasive tension and a separation of rigid elements. In Snelson's now iconic structure, the compression-resistant struts do not touch but instead are individually lifted, each embraced and interconnected by a system of continuously tensed cables, a condition that Snelson and Fuller called "continuous tension, discontinuous compression."
  2. Stable. Though ethereal in appearance—its wooden Xs appear almost to float—Snelson's sculpture is remarkably stable, despite its minimal use of rigid elements. This stability is due to the fact that tensile and compressive components are, at all times, in mechanical equilibrium.
  3. Prestressed. This mechanical equilibrium results from the way the compression and tensile components interact to bring out each other's essential nature: the cables pull in on both ends of the struts, while the struts push out and stretch the cables. The result is that each element in a tensegrity structure is already stressed—the compression elements are already compressed, the tensile elements already tensed—and they are stressed by each other, a condition known as "self-stress" or "prestress."
  4. Resilient. While they are stabilized by prestress, tensegrity structures are also exquisitely responsive to outside perturbation. Their components immediately reorient when the structure is deformed, and they do so reversibly and without breaking.
  5. Globally Integrated. Because the components are so intimately interconnected, what is felt by one is felt by all, producing a truly holistic structure.
  6. Modular. Though complete on its own, a tensegrity structure can combine with other such structures to form a larger tensegrity system. In these systems, individual tensegrity units can be disrupted without compromising overall system integrity.
  7. Hierarchical. In fact, smaller tensegrity structures may function as compressive or tensile components in a larger tensegrity system, which in turn may perform a similar function in still larger systems (Fig. 2).

For much of history, architecture had been preoccupied with making things stable but Snelson's X-structure unlocked a world in which structures could be flexible and firm, holistic and hierarchical. Over the past 60 years, artists, engineers, and architects have used the lessons of tensegrity to build previously impossible structures—space frames, deployable moon-base shelters, as well as sky-piercing sculptures—helping to realize Fuller's vision of a universe filled with man-made tensegrity structures.

Figure 2: Snelson's Needle Tower sculpture.

Snelson would later argue that tensegrity is a principle that is realized only through man-made objects. But Fuller's vision rested on the conviction that nature builds using tensegrity. Indeed, the human frame with its many tensile muscles, ligaments, and tendons pulling up on the rigid bones of the body, thereby stabilizing and supporting them against the force of gravity, is a prime example of tensegrity at work. In the last few decades, scientists have shown that tensegrity is a fundamental design principle of nature, operating at the level of organs, tissues, cells, and even molecules (Ingber, 1998). Their discoveries are leading to a whole new array of man-made tensegrity structures, this time at the micro- and even the nano-scales.

Tensegrity is not an easy concept to grasp. It is best seen and felt, and authors often suggest that readers build their own tensegrity structures. Another way in is through history. What follows is essentially a story: the rise of tensegrity from a concept known to an esoteric few to become a well-recognized design principle of nature, one that is leading to radically new solutions to age-old problems in medicine, engineering and beyond.

Emergence of the idea: 1920 to 1950

Though tensegrity was first realized in the mid 20th century, some have seen hints in a 1920 sculpture by the Russian constructivist artist, Karl Ioganson (Emmerich, 1988; Motro, 2003). Yet Ioganson's purpose was to show how a tensile structure could be deformed rather than made stable. Fuller, on the other hand, appears to have been obsessed by the idea of stability even as a child. Nearly blind from birth, he developed a tactile feel for geometrical shapes, in particular triangles and tetrahedra, which impressed him as the most stable shapes in nature.

Fuller perceived that nature's forms were the result of matter being acted upon by force and, in 1917, proposed that nature itself is a finite energy system consisting of the forces of tension and compression acting synergetically, a theory he would later term Energetic-Synergetic Geometry (Fuller, 1961). Though compression had been considered dominant and, for this reason, had been favored by architects and builders, Fuller was finding tension to be the stronger and more versatile force. For example, he saw that if a compression-bearing element such as a rod or tube were too long, it would buckle and finally break when compressed at both ends, while a tension-bearing element such as a cable or rope might be pulled with great force over virtually unlimited distances and still not tear apart.

In 1927, he became captivated by the possibility of building a suspension bridge without end and turned to nature for models of how tension operates on a grand scale. Looking to the sky, he viewed the planets as isolated compression elements held in place by the invisible but pervasive tensile force of gravity. He believed that this same arrangement—discontinuous compression, continuous tension—was mirrored in the atom, with its swirl of electrons orbiting around the nucleus, all bound together by attractive and repulsive forces operating at the subatomic level.

Searching for an example in the man-made world, he eventually fixed on the wire bicycle wheel. Fuller saw how the wheel's hub and rim acted as discontinuous compression elements, each resisting the deforming pull of the tension-bearing spokes, and was impressed by how the spokes could be made thinner and thinner without compromising the wheel's stability. Indeed, the wire bicycle wheel would become Fuller's template for thinking about tensional integrity for decades to come.

It was still his model in the summer of 1948, when he was invited to teach at Black Mountain College, an experimental arts school in North Carolina that had counted among its teachers such luminary artists as Martha Graham, Merce Cunningham, and Josef Albers. Though not yet world-famous, Fuller gained an immediate following. After arriving, he gave a three hour evening lecture on tensional integrity, complete with geometric models, that, according to later reports, captivated the students, none more so than Kenneth Snelson. Snelson, who had come to Black Mountain to study painting with Albers, had been invited to help Fuller set up his models before the lecture and was so excited by what he heard that he spent the night recreating Fuller's models. The two became close. The following autumn, Snelson, inspired by his summer studies with Fuller and Albers, completed a series of three sculptures, the last of which consisted of two X-shaped plywood modules, one floating miraculously above the other, held in place only by tensed nylon wires, anchored to a plywood base (Fig. 1).

Fuller was stunned when he saw Snelson's X-structure and would later write that it 'catalyzed' his thinking (Fuller, 1961). Snelson would recount the event differently. In a letter written years later, he described how Fuller studied the sculpture, turned it over in his hands, and then asked if he could take it home (Snelson, 1990). The next day, Fuller told Snelson he had gotten the configuration wrong, asked him to replace the X-shaped modules with tetrahedra, and then had his picture taken with the new structure. That photo would later appear in magazines without mention of Snelson.

Early explorations: 1950 to 1970

Figure 3: A patent on a Fuller geodesic dome design.

Using his newfound insight into what, in 1955, he termed 'tensegrity,' Fuller turned his attention to the building of towering masts and geodesic domes. He knew that the geodesic arrangement builds on the strength of the triangle and had long suspected that tensegrity was also at work. With Snelson's template in mind, he was now able to see how the dome, though built of non-flexible elements, exhibited tensegrity: he saw how the continuous transmission of tensional forces across the frame is balanced by a subset of struts that resist compression.

Having confirmed his hunch that tensegrity was key to the dome's stability, he began working out the mathematical proportions for a wide array of geodesic designs (Fig. 3). Many of these designs resulted in lucrative patents, popping up in such diverse projects as weather observatories, Cold War early warning radar-detecting stations, moon and satellite structures, the Sydney Opera House, and a World's Fair pavilion that won a Gold Medal award from the American Institute of Architects in 1970. Initially, Fuller acknowledged Snelson's contribution but eventually dropped mention of his name.

In 1959, Fuller was invited to exhibit his tensegrity structures, including a 30 foot mast, at the Museum of Modern Art in New York City. Snelson, who had become alienated from Fuller following their dispiriting encounter, heard about the upcoming show and contacted the curator, who arranged for Snelson to exhibit a recreation of his 'Early X-Structure,' among other sculptures (Snelson, 1990; Gough, 2009). His contribution finally acknowledged, Snelson embarked on what would become a brilliant career as a sculptor, producing a dazzling array of tensegrity structures. By experimenting with the arrangement of cables and struts—and he turned from wood to metal rods and tubes—he was able manipulate the prestress in the system to create structures of crystalline beauty in virtually any form. Like Fuller, he would explore the apparently limitless power of tensed cables to create structures of enormous length and height, including his famous Needle Tower, which he erected in 1969 outside the Hirshhorn Museum in Washington D.C. (Fig. 2). At 60 feet, it dwarfed Fuller's 30 foot mast.

Spanning the Art-Science Interface

Figure 4: Quasi-equivalence in the geodesic arrangement of a viral capsid.

Both men had hit upon a key feature of tensegrity, its modularity, but the conceptual schism between them grew. Though Snelson originally defined tensegrity as 'discontinuous compression, continuous tension,' he later narrowed his definition so that it appeared to include only the kind of strut-and-cable structures that he makes. He would claim that the principles of tensegrity are best realized in art and yet he would write that the vaster universe is the result of tensegrity (Snelson, 1990). Fuller, who began his quest to understand tensegrity by looking to the solar system, maintained throughout his life that tensegrity was a design principle of nature (Fuller, 1982). Yet neither he nor Snelson explored how, exactly, tensegrity worked in nature. (Snelson would later submit a patent describing a model of the atom based on tensegrity.)

It was the London-based artist John McHale who inspired the first foray into biology (Morgan, 2006). In 1959, he read that researchers had discovered that the outer coat (capsid) of a plant virus consisted of symmetrical icosahedrons but were puzzled as to how exactly they fit together. McHale arranged a meeting between Fuller and two of the researchers. Inspired by the meeting, one of the researchers, Aaron Klug, along with the American virologist Donald Caspar, began examining images of Fuller's tensegrity sphere to see how the viral icosahedrons were arranged to form the capsid. They came to realize that the icosahedral units could fit if they exhibited a certain amount of 'give,' or what they called 'quasi-equivalence,' in the way they bound each other (Fig. 4). They published their findings in 1962, though without referring to the concept of tensegrity. It was only decades later that Caspar would write about the role that tensegrity, and the building of Fuller-inspired models, played in their discovery (Caspar 1980). Klug won a Nobel prize in 1982 in part for discovering how viruses establish their shape stability using geodesic architecture.

Expanding into nature: 1970 to 1990

By the early 1970s, Fuller was practically a household name. His geodesic domes had become a symbol of the 1960s' counter-cultural movement, even as they were appearing in military camps and industrial parks. Civil engineers were using the principles of tensegrity to build pavilions, domes, and space frames—the girded structures that span the ceilings and walls of barns and other large buildings. Articles on tensegrity were beginning to appear and the first textbooks were being written (Pugh, 1976; Kenner, 1976). Snelson, too, was catapulted into the everyday world, his monumental floating steel and aluminum structures gracing a growing number of museums and plazas. His work, along with Fuller's, was taught in college courses.

Figure 5: Geodesic forms in the tensed cytoskeleton of a living human cell.

It was in the mid 1970s, in a sculpture class entitled "Three-Dimensional Design," that Donald Ingber, then a Yale undergraduate, encountered his first tensegrity structure, an odd-looking sculpture consisting of six wood dowels held together by elastics. When the instructor pressed on the tensegrity structure, it flattened; when he lifted his hand, the structure rounded and jumped into the air. Ingber was immediately struck. He had seen something similar just a few days earlier, while learning how to culture cells in a cancer lab. Cultured cells flatten as they adhere to the bottom of a petri dish. Ingber had learned how to remove the flattened cells by adding an enzyme that degrades their sticky adhesions. When dislodged, the cells rounded up and leapt off the dish, just like the tensegrity stick-and-string model. Ingber left the class with a new idea: cells are tensegrity structures.

Once thought of as membranous bags of gel, researchers had recently discovered that all cells contain a "cytoskeleton" (Fig. 5)—an internal framework made out of fibrous protein and consisting of three main components: microfilaments, intermediate filaments, and microtubules. Ingber thought that cells might use the newly-discovered cytoskeleton to control their shape, much like a tensegrity structure. Just as Snelson's X-structure would lose its shape if it were not attached to its plywood base, Ingber further reasoned, cells must use their substrate, the extracellular matrix, to anchor themselves.

Ingber's hunch—that the extracellular matrix plays a mechanical role in cell shape control—was strengthened when he learned about the work of Albert K. Harris, who was growing cells on a flexible rubber substrate (Ingber, 1998). Instead of flattening, as they do on a rigid glass or plastic petri dish, the cultured cells rounded and, at the same time, pulled up on the rubber substrate, causing it to form compression wrinkles. This pulling suggested that the cell was exerting tensional forces on the malleable substrate.

Figure 6: The spherical tensegrity cell model with a nucleus at top flattens and polarizes its nucleus to the base when it attaches and spreads on a substrate, just like living cells.

Ingber decided to reproduce the behavior of cultured cells by building a simple stick-and-string tensegrity model of his own. He attached it to a rigid wood substrate and saw that it flattened, just like cells grown on rigid plastic petri dishes. He then took the flattened model and anchored it to a flexible sheet of fabric. It spontaneously pulled the fabric into compression wrinkles, just like Harris's cells. But real cells have a nucleus. To replicate nature more closely, Ingber built a large strut-and-elastic string tensegrity model and this time placed a smaller tensegrity sphere at its center. He linked this smaller sphere to the surface of the model cell by adding tensile filaments—creating one of the first hierarchical tensegrity structures (Fig. 6). Ingber repeated his experiments using his hierarchical model and found that the cell and nucleus flattened in a coordinated manner. What's more, the flattened cell appeared to polarize, with the nucleus moving to the region of the base where the greatest pulling forces were being exerted. He later confirmed experimentally that living cells exhibit precisely these same behaviors (Wang et al., 1993, 2001; Kumar et al., 2006; Brangwynne et al., 2006), and other researchers showed how more complex cell shapes, such as the long extended processes of nerve cells, can be established using tensegrity (Joshi and Heidemann, 1985).

Ingber would incorporate his ideas about the cytoskeleton and extracellular matrix into an ambitious model of cellular tensegrity, one that has offered new insights and predictions into how cells and tissues form, how they function, and how cancers can arise (Ingber, 1985; 1993; 2003a). Working with Dimitrije Stamenovic, he would develop a mathematical model of tensegrity that predicts how cells from many different tissues behave mechanically (Stamenovic, 1996). Many of these models' predictions have been confirmed experimentally. For example, if the cell surface and nucleus are connected or 'hard-wired' by tensile filaments, as the cellular tensegrity model suggests, then forces exerted at the cell surface should produce immediate structural changes deep inside the nucleus, where the genes lie. By developing new methods that allow them to apply mechanical forces to specific receptors on the surface of a single living cell, Ingber and his team confirmed that the cell is, in fact, interconnected all the way down—from the extracellular matrix to the cytoskeleton to the chromosomes and genes (Maniotis et al., 1997).

Recent Developments: 1990 to the present

Fuller and Snelson used the principle of tensegrity to create structures that were flexible and firm, and engineers, architects, and artists would build on their work throughout the last century and into the present one. Over the past 20 years, there has been an explosion of activity in the areas of civil engineering and architecture, inspired in part by the writings of Rene Motro, Robert Skelton, and Robert Burkhardt, whose efforts to systematize, categorize, and develop algorithms for the building of a wide range of tensegrity structures have made tensegrity design accessible to a wider audience (Motro, 2003; Skelton and de Oliveira 2009; Burkhardt, 2008). Mathematicians such as Robert Connelly have been studying the underlying geometric principles of tensegrity structures, defining them as a system of points and vertices, in an effort to understand how various tensegrity structures maintain their stability and to define which structures are possible and which are not (Connelly and Back, 1998). Snelson and other artists have continued to employ the principles of tensegrity to create structures of great strength, resilience, and beauty.

From Form to Function

What the discoveries by Ingber and colleagues show is that nature has gone one step further: using tensegrity to build the cell—connecting it all the way down, from the extracellular matrix to the nucleus—nature created a structure that is firm, flexible, and responsive, in the true sense of the word (Ingber, 2006). Indeed, Ingber and his colleagues have shown that mechanical forces exerted at the surface of the cell can be transmitted to the nucleus, resulting in biochemical changes and ultimately, in genetic ones—the turning on and off of genes. This process, known as "mechanotransduction," is central to the way cells, and also tissues and organs, respond to physical force: the way bone and cartilage respond to compression; muscle and skin respond to tension; blood vessels to the pressure of blood flow (Ingber, 1996; 2003b). Through tensegrity, even the force of gravity may influence the growth of cells, tissues, and organs, as Ingber has shown (Ingber, 1999; Ingber, 2006). Over the past 20 years, mechanotransduction, and tensegrity more generally speaking, has attracted the attention not just of biologists but of physicians and other health care practitioners—chiropractors, acupuncturists, physical and massage therapists—who have long suspected that mechanical manipulations can exert a profound effect on the tissues and organs of the body (Ingber 2003c; Levin, 2002).

The principles of tensegrity, in particular its modular and hierarchical aspects, are also being used to understand how molecules self-assemble to form cells; how cells form into tissues; and how tissues join together to create organs with specialized shapes and functions. Years ago, Ingber showed that while the extracellular matrix plays a role in the tensegrity structure of individual cells—resisting the pull of the cell's cytoskeleton—it also acts as a structural element in a hierarchical tensegrity to influence growth patterns at the tissue level. Tissues are made up of layers of cells attached to extracellular matrix. These cells exert a tensile force, pulling up on the extracellular matrix. Normally, the extracellular matrix resists their pull, but during organ formation, local chemical signals can cause some areas of the matrix to thin and pull apart, like a run in a stocking. According to the cellular tensegrity model, cells attached to the thinned regions should stretch as well, which, in turn, should signal them to divide and grow while their undeformed neighbors remain quiescent. It turns out, extracellular matrix has also been observed to thin uncontrollably during tumor formation, which led Ingber to suggest that changes in the extracellular matrix might be stimulating the cell growth that drives cancer. Over the past 20 years, he and colleagues have gathered experimental evidence that supports the tensegrity model in both cases, normal development and cancer.

Researchers have also shown that the cytoskeletal frameworks of individual cells are linked to the matrix, and to one another, through mechanical connections that span the cell membrane and link to central nuclei, forming a continuous, prestressed structural network (Wang et al., 1993, 2009). These intercellular connections produce a kind of harmonic coupling, helping to coordinate the timing of activities of cells in far-flung corners of tissues, organs, and the entire body (Pienta and Coffey, 1991; Ingber 2006). In this way, tensegrity exerts its influence not just in space but also time. Although the term 'tensegrity' is not always used, the concept that tensional prestress is key to providing overall cell shape stability and cytoskeletal coherence has become widely accepted.

But the hierarchical and modular aspects of tensegrity do not stop there. Moving down the hierarchy, it now appears that the individual actomyosin filaments that make up the cytoskeleton, and which generate its tensile forces, are themselves built from smaller molecules that do not physically touch but instead gain their stability through attractive forces, creating a kind of molecular tensegrity. It turns out, many cellular proteins, and even DNA itself, gain their shape and stability through a similar arrangement, namely by intramolecular attractive forces coming into balance with regions of the molecule that resist being compressed (Ingber, 1998; Ingber, 2000; Morrison et al, 2011). Molecular modelers are drawing on these discoveries to simulate how proteins fold and take on specific shapes, efforts which could someday lead to the development of new therapeutics.

Inspired by these and other findings, including the observation that crystals and many other inorganic substances assume a geodesic form, Ingber recently proposed that the origin of life itself may have been driven by hierarchical self-assembly (Ingber, 2000). In this view, tensegrity—along with other fundamental natural design principles such as energy minimization, topological constraints, and self-emergence of auto-catalytic systems—provides an explanation not just for the origin of the first cells but also for the evolution of increasingly complex multi-modular structures such as tissues and organs, each exhibiting progressively more sophisticated functions.

From Macrocosm to Microcosm

As Fuller glimpsed nearly a century ago when he first spoke about Energetic Geometry, the key to tensegrity—what makes it such a superb mechanism for structural stability—is that it lets force and matter work together in the way they were designed to, by nature. The gossamer feel of Fuller's geodesic domes and the crystalline beauty of Snelson's strut-and-cable sculptures reflect aesthetic qualities found in organic forms—and that may be no coincidence. Gazing at them, one has the feeling of looking into the future but there is something primordial about them as well.

Fuller lived to see his vision of a world filled with man-made tensegrity structures come true on a grand scale, in towering masts, colossal domes, and sprawling space frames. His design concept, born in the field of art and architecture, has changed the world in which we live, and continues to do so, though in a direction even Fuller might not have anticipated—at the nanoscale. In 1985, two years after his death, researchers discovered an extraordinary new form of carbon, one that arranges its 60 carbon atoms in the form of a geodesic sphere, like a microscopic soccer ball. The scientists, who received the 1996 Nobel Prize, named these carbon compounds Buckminster Fullerenes, also called buckyballs (Kroto et al, 1985). Carbon fullerenes can also assume a cylindrical shape. These carbon nanotubes, or buckytubes, are extraordinarily efficient conductors of electricity and can carry electric current densities one thousand times higher than copper. They have a tensile strength fifty times greater than carbon steel, while displaying greater flexibility. Because of their superb mechanical and electrical properties, they are now being applied to various exciting uses, ranging from ultrafast microelectronics to bullet-proof fabrics. More recently, material scientists have been fabricating more complex prestressed tensegrities at the nanoscale. For example, Joanna Aizenbeg and colleagues have created tensegrities composed of arrays of stiff nanometer-sized metal bars connected by prestressed elastomeric hydrogels. By pulling the nanobars into different orientations, the hydrogels can change their own optical properties (Sidorenko et al., 2007).

Biologists, too, have been using the principles of tensegrity to build nanotubes and other nanodevices that might someday deliver drugs directly to cells or help regenerate tissues. Working with DNA instead of carbon, nanobiologists such as Ned Seeman, William Shih, and Tim Liedl are using tensegrity designs to build molecular structures composed of DNA struts held together by tensile chemical bonding forces (Seeman, 2010; Liedl et al, 2010). Some mimic the complex three-dimensional geodesic forms of viral capsids. Others can self assemble, growing from nanoscale components into near millimeter-sized crystals by leveraging the mechanical stability that comes from tensegrity design principles.

Meanwhile, bioengineers such as Hod Lipson, Radhika Nagpal, and others, are using tensegrity to build macroscale robots that can move individually and, at the same time, self-assemble into larger collectives that behave in a coordinated manner, much like tissues that form from groups of smaller cells (Paul et al 2006; Nagpal 2008).

One of Fuller's favorite sayings was that nature has no separate departments of mathematics, physics, chemistry, biology, art, or architecture. To fully explore tensegrity, one must span all these disciplines. Many have made this leap—their efforts can be seen in fields as diverse as nanotechnology, tissue engineering, architecture and space exploration. If the future is one in which solutions to our most pressing societal and environmental problems come from nature, then tensegrity will likely become even more central.

References

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Recommended Reading

  • Edmondson A. 2009. A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller. Emergent World LLC.

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