## Abstract

In spite of their great importance and numerous applications in many civil, aerospace and biological systems, our understanding of tensegrity structures is still quite preliminary, fragmented and incomplete. Here we establish a unified closed-form analytical solution for the necessary and sufficient condition that ensures the existence of self-equilibrated and super-stable states for truncated regular polyhedral tensegrity structures, including truncated tetrahedral, cubic, octahedral, dodecahedral and icosahedral tensegrities.

## 1. Introduction

Tensegrities refer to a class of light-weight and reticulated structures made of a discontinuous set of axial compressive elements (bars) and a continuous set of axial tensile elements (strings) (Juan & Tur 2008; Feng *et al*. 2010). Owing to their unique features and properties (Skelton *et al*. 2001), tensegrity structures hold promises for a wide variety of technologically important applications, ranging from architectural designs (Motro 2003; Rhode-Barbarigos *et al*. 2010), deployable aerospace structures (Tibert & Pellegrino 2002; Sultan 2009), smart actuators and sensors (Ali & Smith 2010; Moored *et al*. 2011), advanced materials engineering (Luo & Bewley 2005; Fraternali *et al*. 2012), to molecular and cellular biomechanics (Ingber 1993; Luo *et al*. 2008; Pirentis & Lazopoulos 2010). In particular, truncated polyhedral tensegrities, a subclass of tensegrity structures with topology based on polyhedrons with truncated vertices (Pugh 1976; Murakami & Nishimura 2001; Li *et al*. 2010*b*), have been adopted in man-made geodesic domes (Fu 2005; Yuan *et al*. 2007) and models of cytoskeleton and other biological structures (Stamenovic & Ingber 2009; Ingber 2010). A key step in the design of a tensegrity structure is a self-equilibrium analysis to determine, once the overall topology has been specified, conditions under which the structure will be self-equilibrated in the absence of external loads. Various methods have been proposed for the self-equilibrium analysis based on the concept of force density, defined as the ratio of internal force to the current length of an element (Tibert & Pellegrino 2003). While analytical approaches based on the nodal force equilibrium have been developed for tensegrities with a relatively small number of elements or high symmetry (Connelly & Terrell 1995; Sultan *et al*. 2001; Zhang *et al*. 2010), numerical methods are often needed for larger or more complex structures (Estrada *et al*. 2006; Zhang & Ohsaki 2006; Tran & Lee 2010).

Following the self-equilibrium analysis, a stability analysis can be performed to determine the conditions under which a self-equilibrated tensegrity structure is stable. A stable structure always tends to return to its equilibrated configuration when subject to an infinitesimal and conservative disturbance. In the present paper, we consider only the static stability of tensegrities, excluding the instability problems under non-conservative follower forces (Langthjem & Sugiyama 2000). The stability of a tensegrity can be guaranteed by the positive definiteness of its tangent stiffness matrix, which is defined as the derivative of the external force vector with respect to the nodal displacement vector (Schenk *et al*. 2007; Zhang & Ohsaki 2007). Furthermore, the structure is said to be super-stable if it is stable for any level of force densities satisfying the self-equilibrium conditions without material failure (Connelly & Back 1998; Juan & Tur 2008). The relationships of different states of tensegrities are shown in figure 1. For many important applications, tensegrities are required to be not only self-equilibrated and stable but also super-stable.

During the past decade, much effort has been directed towards the self-equilibrium analysis, some also including structural stability, of truncated regular polyhedral tensegrities. Based on the nodal force equilibrium conditions and symmetric congruent operations, Murakami & Nishimura (2001) analysed the self-equilibrated states of truncated regular dodecahedral and icosahedral tensegrities. Tibert & Pellegrino (2003) obtained an analytical self-equilibrium solution for truncated tetrahedral tensegrities expressed in terms of force densities. Pandia Raj & Guest (2006) solved the self-equilibrated states of truncated tetrahedral tensegrities based on a group representation theory, and discussed the super-stability of such structures by means of a symmetry-adapted force density matrix. Estrada *et al*. (2006) developed a multi-parameter numerical procedure to determine the self-equilibrated states and configurations of truncated regular tetrahedral and icosahedral tensegrities by iteratively calculating both the equilibrium geometry and the force densities. Recently, Li *et al*. (2010*a*) proposed a Monte Carlo form-finding method for truncated regular tetrahedral, dodecahedral and icosahedral tensegrities. This method can be used to determine both the self-equilibrated configurations and the associated force densities in these structures.

In spite of the above-mentioned progress on both analytical and numerical methods for specific truncated regular polyhedral tensegrities, the existing analytical solutions are fragmented and lack a unified treatment. In addition, the stability of these structures remains a major concern that needs further investigations. The present study aimed to develop a unified solution for the self-equilibrated and super-stable states of all truncated regular polyhedral tensegrity structures.

The paper is organized as follows. Section 2 reviews some basic concepts including regular polyhedra, truncated regular polyhedra and truncated regular polyhedral tensegrity structures. Section 3 presents the theoretical framework for analysing self-equilibrium and super-stability. Section 4 derives the self-equilibrium solutions of truncated regular tetrahedral, cubic, octahedral, dodecahedral and icosahedral tensegrities. Section 5 provides a unified solution for the self-equilibrated states of all truncated regular polyhedral tensegrities, and §6 considers the necessary and sufficient condition for the super-stability of the solution. Section 7 summarizes the main contributions of the present study.

## 2. Truncated regular polyhedral tensegrity structures

Truncated regular polyhedral tensegrity structures can be constructed from truncated regular polyhedra (Murakami & Nishimura 2001; Li *et al*. 2010*b*) following a few basic steps discussed below.

### (a) Regular polyhedra

Regular polyhedra have faces that are congruent regular polygons assembled in the same way around each vertex. In this paper, our attention will be focused on convex regular polyhedra to be used in constructing tensegrity structures via a polyhedral truncation scheme. As shown in figure 2*a*, there are five types of convex regular polyhedra, sometimes referred to as Platonic solids (Cromwell 1997). To facilitate subsequent discussions, we adopt the following definitions of Schläfli symbol and duality for polyhedra (Coxeter 1973):

### Definition 2.1

A regular polyhedron can be uniquely identified by the Schläfli symbol {*n*,*m*}, where n is the number of edges of each face and m is the number of faces around each vertex.

### Definition 2.2

Polyhedron {*n*,*m*} is said to be dual to polyhedron {*m*,*n*}.

Based on definitions (2.1) and (2.2), the five regular polyhedra in figure 2*a* can be denoted as {3,3} for tetrahedron, {4,3} for cube, {3,4} for octahedron, {5,3} for dodecahedron and {3,5} for icosahedron. One can see that a cube and an octahedron, and a dodecahedron and an icosahedron, are dual-pairs, while a tetrahedron is self-dual.

### (b) Truncated regular polyhedra

A truncated regular polyhedron can be obtained from a regular polyhedron by truncating each vertex with a plane perpendicular to the radius that connects the polyhedral centre to the vertex. There are five types of truncated regular polyhedra, as shown in figure 2*b*. The truncated original faces and edges are referred to as *remaining faces* and *remaining edges*, and the newly produced faces and edges are called *truncating faces* and *truncating edges*, respectively. Each remaining edge is shared by two remaining faces, and each truncating edge by one remaining face and one truncating face. Two special states of truncated regular polyhedra are defined as follows:

### Definition 2.3

A critical truncated polyhedron is the truncated polyhedron whose remaining edges are of zero length.

### Definition 2.4

A hyper-truncated polyhedron is the truncated polyhedron whose remaining faces are of zero area.

For the truncated regular polyhedra in figure 2*b*, the corresponding critical truncated polyhedra are shown in figure 2*c*, where each vertex is located at the midpoint of an edge of the corresponding regular polyhedra in figure 2*a*. It is seen that a regular polyhedron and its dual counterpart have the same critical truncated polyhedron. Figure 2*d* illustrates the five hyper-truncated polyhedra, each of which has the same shape as the dual of the original regular polyhedron in figure 2*a*.

### (c) Truncated regular polyhedral tensegrities

A truncated regular polyhedral tensegrity structure can be constructed as follows. Let the edges and vertices of a truncated regular polyhedron correspond to strings and nodes. The strings along the remaining edges are denoted as *remaining-strings*, and those along the truncating edges referred to as *truncating-strings*. A polygon formed by truncating-strings is a *truncating polygon*. In such a structure, the nodes of each truncating *v*-polygon (*v* is the number of edges) are connected to the nodes of *v* adjacent truncating polygons by *v* bars that are added into the structure following the rules of *Z*-shaped elementary cells, referred to as a *Z*-based tensegrity (Pugh 1976; Li *et al*. 2010*b*). For example, figure 3*a* shows the edges and vertices of a truncated tetrahedron, and figure 3*b* illustrates the bars, strings and nodes of the corresponding truncated tetrahedral tensegrity, where a *Z*-shaped cell consisting of one bar (connecting nodes 1 and 4), one remaining-string (connecting nodes 2 and 3) and two truncating-strings (connecting nodes 1 and 2 and nodes 3 and 4) is highlighted. In the following, we will investigate the self-equilibrium and super-stability of all five types of truncated regular polyhedral tensegrities.

## 3. Self-equilibrium and super-stability of tensegrity structures

The self-equilibrium and super-stability of a given tensegrity structure are usually investigated via the eigenvalues of its force density matrix, which is discussed for truncated regular polyhedral tensegrities in this section.

### (a) Force density matrix

The force density of element *e* connecting nodes *i* and *j* is defined as
3.1which is sometimes also referred to as the stress or tension coefficient (Vassart & Motro 1999; Guest 2006; Schenk *et al*. 2007).

As discussed in §2*c*, the elements in a truncated regular polyhedral tensegrity can be classified into three kinds: truncating-strings, remaining-strings and bars. For each kind of elements, their force densities are assigned a specified value. In addition, the force density of all truncating-strings is set as unity because the self-equilibrium and super-stability conditions of a tensegrity structure depend only on the normalized force densities. Let *q*_{s} and *q*_{b} denote, respectively, the dimensionless force densities of the remaining-strings and bars normalized by the force density of the truncating-strings.

The self-equilibrium conditions of a tensegrity structure based on the force density matrix can be expressed as (Zhang & Ohsaki 2006)
3.2where **p**_{x}, **p**_{y} and **p**_{z} are the nodal coordinate vectors in the *x*, *y* and *z* directions, respectively; **D** is the force density matrix of dimension *n*_{n}×*n*_{n}, *n*_{n} being the total number of nodes in the structure.

The force density matrix **D** can be calculated according to the following scheme (Schenk *et al*. 2007):
3.3where for the truncated regular polyhedral tensegrities *q*_{e(ij)} should be taken as 1, *q*_{s}, *q*_{b} for truncating-strings, remaining-strings, bars, respectively. Alternatively, **D** can also be obtained from the following multiplication of matrices (Tibert & Pellegrino 2003):
3.4where **Q**=diag(⋯ ,*q*_{e(ij)},⋯ ) is the diagonal matrix consisting of the force densities of all elements in the structure and **C** is the connectivity matrix defined as
3.5The nodal sequences in each element can be arbitrarily chosen, but once this is done, they should remain fixed throughout the analysis.

### (b) Self-equilibrium conditions

If a tensegrity structure is to have a self-equilibrated state, the rank of its force density matrix should satisfy the following inequality (Tibert & Pellegrino 2003; Schenk *et al*. 2007):
3.6or equivalently,
3.7where rank(·) and null(·) denote the rank and the nullity of a matrix, respectively; *d* is the dimension of the structure: *d*=3 for the three-dimensional truncated regular polyhedral tensegrities under investigation.

The self-equilibrium analysis aims to determine the conditions that allow equation (3.7) to be satisfied. The eigenvalues of the force density matrix **D** can be determined from its characteristic polynomial:
3.8where **I** is the unit matrix, *λ* is an eigenvalue of **D** and *P*_{α}=*P*_{α}(*q*_{s},*q*_{b}) is a polynomial function of the force densities. If the inequality (3.7) with *d*=3 holds, equation (3.8) can be rewritten as
3.9Comparing equations (3.8) and (3.9), we obtain the following equivalent form of equation (3.7) with *d*=3:
3.10Note that there are no restrictions on the coefficients *P*_{α} for *α*≥4. If null(**D**)=4, we will have *P*_{4}≠0, and if null(**D**)=*n*_{D}>4, more coefficients would vanish, i.e. *P*_{nD}−1=⋯=*P*_{4}=0.

For a self-equilibrated tensegrity structure without any nodes fixed, the sum of all components in each row or column of the force density matrix should vanish, as can be seen from equation (3.3). Therefore, the force density matrix is always singular, indicating that *P*_{0}=0 is always true. Therefore, we only need to ensure
3.11for the *self-equilibrium* of a three-dimensional tensegrity, where *P*_{1}, *P*_{2} and *P*_{3} are polynomials of the force densities in the structural elements. In what follows, these conditions will be investigated for the truncated regular polyhedral tensegrities. Note that equation (3.11) can only guarantee the existence of self-equilibrated states; the conditions for super-stability of these structure will be further investigated.

### (c) Super-stability conditions

A tensegrity structure is said to be super-stable if it is stable for any level of force densities satisfying self-equilibrium without inducing material failure (Connelly & Back 1998; Juan & Tur 2008). For a super-stable tensegrity, increasing the force densities tends to stiffen and stabilize the structure. Here, we consider the super-stability of truncated regular polyhedral tensegrities. According to Connelly (1999) and Zhang & Ohsaki (2007), the *conditions of super-stability* for a truncated regular polyhedral tensegrity structure are:

— (i) the strings have positive force densities, and the bars have negative force densities;

— (ii) the force density matrix is positive semi-definite;

— (iii) the nullity of the force density matrix is exactly 4; and

— (iv) there are no affine (infinitesimal) flexes of the structure, or equivalently, the rank of the structural geometry matrix is 6.

The above conditions of super-stability can be understood as follows. The stability of a tensegrity structure requires that its tangent stiffness matrix **K** be positive definite (Schenk *et al*. 2007). According to Zhang & Ohsaki (2007), **K** can be decomposed into a sum of the linear stiffness matrix **K**_{M} and the geometrical stiffness matrix **K**_{G}=**D**⊗**I**^{3}, i.e. **K**=**K**_{M}+**K**_{G}, where ⊗ is the Kronecker product symbol (Williamson *et al*. 2003) and **I**^{3} is the third-rank identity matrix. The linear stiffness matrix **K**_{M} is positive semi-definite for all tensegrities consisting of conventional elements with positive axial stiffness, or in other words, **d**^{T}·**K**_{M}·**d**≥0 for any nodal displacement vector **d**.

While condition (i) is just the definition of tensegrity, condition (ii) is necessary for super-stability of the structure. If the force density matrix **D** is not positive semi-definite, the geometrical stiffness matrix **K**_{G}=**D**⊗**I**^{3} will have negative eigenvalues. In this case, the geometrical stiffness matrix **K**_{G} will become dominant over the linear stiffness matrix **K**_{M} at sufficiently large force densities in the elements, in which case the tangent stiffness matrix **K** ceases to be positive definite and the structure loses its stability.

Condition (iii) guarantees that the solution satisfies the self-equilibrium conditions and is also stable. If the nullity of the force density matrix is larger than 4, there will be multiple equilibrated configurations that render the structure unstable (Schenk *et al*. 2007). Conditions (iii) and (iv) together ensure that there exists no nodal displacement vector **d** under which **d**^{T}·**K**_{M}·**d**=0 and **d**^{T}·**K**_{G}·**d**=0 are simultaneously satisfied. Thus, the positive definiteness of the tangent stiffness matrix, **K**=**K**_{M}+**K**_{G}, can be guaranteed even if both **K**_{M} and **K**_{G} are positive semi-definite.

Among conditions (i)–(iv), (i) can be satisfied by setting *q*_{s}>0 and *q*_{b}<0. Conditions (ii) and (iii) can be checked by the sign of the minimum eigenvalue and the total number of zero-eigenvalues of the force density matrix, respectively. Condition (iv) will be examined by the rank of the structural geometry matrix defined by Zhang & Ohsaki (2007). As we will further show in §6, for the truncated regular polyhedral tensegrities under study, condition (iv) is automatically satisfied once conditions (i)–(iii) are met.

## 4. Self-equilibrium analysis

In this section, we analyse the self-equilibrium of truncated regular tetrahedral, cubic, octahedral, dodecahedral and icosahedral tensegrities based on equation (3.11). The self-equilibrium conditions will be expressed as polynomials of *q*_{s} and *q*_{b}, and those satisfying the super-stability conditions will be identified and used as a basis to construct a unified solution for the self-equilibrated and super-stable truncated regular polyhedral tensegrities.

### (a) Truncated regular tetrahedral tensegrity

A truncated regular tetrahedral tensegrity structure has 12 truncating-strings, six remaining-strings, six bars and 12 nodes, accompanied by a 12×12 force density matrix whose components can be obtained from equation (3.3) or (3.4). Here and in the sequel, the detailed expressions for the components of the force density matrix are omitted for the sake of simplicity. Substituting the obtained force density matrix into equation (3.8), we find the following expressions of *P*_{1}, *P*_{2} and *P*_{3} for the truncated tetrahedral tensegrities:
4.1
4.2
and
4.3where *P*_{2,4} and *P*_{3,4} are lengthy polynomials that do not affect self-equilibrium and are omitted in the paper.

Equations (4.1)–(4.3) suggest that the self-equilibrium conditions in equation (3.11) are reduced to just one condition 4.4for truncated tetrahedral tensegrities. This relation has been derived by Tibert & Pellegrino (2003) and Pandia Raj & Guest (2006) using different methods.

### (b) Truncated regular cubic tensegrity

A truncated regular cubic tensegrity structure has 24 truncating-strings, 12 remaining-strings, 12 bars and 24 nodes, with a 24×24 force density matrix given by equation (3.3) or (3.4). In this case, the expressions of *P*_{1}, *P*_{2} and *P*_{3} in equation (3.8) are found to be
4.5
4.6
and
4.7where *P*_{2,6} and *P*_{3,6} are lengthy polynomials that do not affect self-equilibrium.

Combining equations (4.5)–(4.7) with (3.11) results in the following two equations,
4.8and
4.9for the self-equilibrated states of a truncated regular cubic tensegrity. As we will further show in §6, a certain range of force densities satisfying equation (4.9) can ensure both self-equilibrium and super-stability. In contrast, it can be shown that no solution of equation (4.8) can be super-stable. To see this, let us rewrite equation (4.8) as
4.10Super-stability requires that the strings have positive force densities, and the bars have negative force densities, i.e. *q*_{s}>0 and *q*_{b}<0. Equation (4.10) would demand *q*_{b} to decrease from 0 to −3/2 as *q*_{s} increases from 0 to infinity. In this case, one can numerically determine the minimum eigenvalue of the force density matrix as always negative, as shown in figure 4*a*. Therefore, the solution from equation (4.8) cannot satisfy the positive semi-definite condition of super-stability.

### (c) Truncated regular octahedral tensegrity

A truncated regular octahedral tensegrity structure has 24 truncating-strings, 12 remaining-strings, 12 bars and 24 nodes, with a 24×24 force density matrix. In this case, the coefficients *P*_{1}, *P*_{2} and *P*_{3} in equation (3.8) are
4.11
4.12
and
4.13where *P*_{2,8} and *P*_{3,8} are lengthy polynomials that are irrelevant for the present analysis on self-equilibrium.

Combining equations (4.11)–(4.13) with (3.11) leads to two equations:
4.14and
4.15which ensure the self-equilibrium of truncated regular octahedral tensegrities. While a certain range of force densities from equation (4.15) can satisfy both self-equilibrium and super-stability, no solution of equation (4.14) can meet the positive semi-definite condition of super-stability, as shown in figure 4*b*.

### (d) Truncated regular dodecahedral tensegrity

A truncated regular dodecahedral tensegrity structure has 60 truncating-strings, 30 remaining-strings, 30 bars and 60 nodes, with a 60×60 force density matrix. The corresponding coefficients *P*_{1}, *P*_{2} and *P*_{3} in equation (3.8) are
4.16
4.17
and
4.18where *P*_{2,12} and *P*_{3,12} are lengthy polynomials that are irrelevant for the present analysis.

Substituting equations (4.16)–(4.18) into (3.11) leads to the following five equations:
4.19
4.20
4.21
4.22
and
4.23It can be shown that, while a certain range of force densities from equation (4.23) can satisfy both self-equilibrium and super-stability, no solutions from equations (4.19)–(4.22) can satisfy super-stability for the following reasons. Equations (4.19)–(4.21) lead to force density matrices with at least five zero-eigenvalues and thus do not meet the condition of nullity equal to 4, and the minimum eigenvalue of the force density matrix from equation (4.22) is always negative, as shown in figure 4*c*.

### (e) Truncated regular icosahedral tensegrity

A truncated regular icosahedral tensegrity structure has 60 truncating-strings, 30 remaining-strings, 30 bars and 60 nodes, with a 60×60 force density matrix. The corresponding coefficients *P*_{1}, *P*_{2} and *P*_{3} are
4.24
4.25
and
4.26The *P*_{2,20} and *P*_{3,20} are lengthy polynomials that are irrelevant for the present analysis.

Substituting equations (4.24)–(4.26) into (3.11) leads to the following five equations:
4.27
4.28
4.29
4.30
and
4.31While a certain range of force densities from equation (4.31) are super-stable, no solutions of equations (4.27)–(4.30) can meet all conditions associated with super-stability: The force density matrix from equations (4.27)–(4.29) has at least five zero-eigenvalues while the minimum eigenvalue of that associated with equation (4.30) is always negative, as shown in figure 4*d*.

We note that the self-equilibrium conditions of truncated regular cubic, octahedral, dodecahedral and icosahedral tensegrities have been previously analysed by Nishimura (2000) and Murakami & Nishimura (2001, 2003), using icosahedral group graphs and a reduced equilibrium matrix. They considered structure symmetries to simplify the calculations of nodal force equilibrium and expressed their final results in three coupled equations. It can be shown that the solutions of Nishimura (2000) and Murakami & Nishimura (2001, 2003) are fully consistent with those in equations (4.9), (4.15), (4.23) and (4.31).

## 5. Unified solution for self-equilibrated and super-stable states

Inspired by the solutions derived in §4, a unified and simple solution for the self-equilibrated and super-stable states of all truncated regular polyhedral tensegrities will be conjectured in this section, with coefficients determined from three special equilibrated states by considering their geometric features. The unified solution will be verified by comparison with all solutions in §4.

### (a) Conjecture of unified solution

In §4, we have individually derived the self-equilibrated states of truncated regular tetrahedral, cubic, octahedral, dodecahedral and icosahedral tensegrities by directly solving the self-equilibrium conditions. We have also identified that the solutions possibly satisfying super-stability are given in equations (4.4), (4.9), (4.15), (4.23) and (4.31). Inspired by these results, we further attempt to establish a unified solution for the self-equilibrated and super-stable states of all truncated regular polyhedral tensegrities.

### Conjecture 5.1

For the self-equilibrated and super-stable states of all truncated regular polyhedral tensegrities, the force densities of elements must satisfy the following relation:
5.1where *q*_{s} and *q*_{b} are the dimensionless force densities of the remaining-strings and bars, respectively, normalized by the force density of the truncating-strings, and the coefficients *A*_{1}, *A*_{2} and *A*_{3} are constants that depend on the Schläfli symbol {*n*,*m*} of the corresponding regular polyhedron, i.e.
5.2

It can be immediately seen that equation (5.1) has a similar form as the individual solutions given in equations (4.4), (4.9), (4.15), (4.23) and (4.31). Furthermore, the basic characters of truncated regular polyhedral tensegrities require that the unified solution should meet the following conditions:

— In accordance with the self-equilibrium solutions for all truncated regular polyhedral tensegrities discussed in the previous section, the unified solution should also be in a polynomial form of the normalized force densities of the remaining-strings and bars, i.e.

*q*_{s}and*q*_{b}.— In the unified solution,

*q*_{s}and*q*_{b}should be commutative because all truncated polyhedral tensegrity structures are*Z*-based tensegrities (Li*et al*. 2010*b*). In other words, if all remaining-strings and bars in such a structure are interchanged, the new structure will maintain the same topology as the original one.— There should be no constant term in the solution for the following reason. The force density of an element, defined as the ratio of its internal force over length, is infinite when the element is of zero length. Since the truncated regular polyhedral tensegrity structures belong to

*Z*-based tensegrities (Li*et al*. 2010*b*), they should have a special self-equilibrated state in which all truncating-strings have a zero length, corresponding to the un-truncated polyhedra shown in figure 2*a*. In this case, the force density in the truncating-strings is infinite and the normalized force densities in the bars and remaining-strings (i.e.*q*_{b}and*q*_{s}) are both zero. An un-truncated tetrahedron is shown as an example in figure 5*a*, and the corresponding tensegrity with truncating-strings of zero length is illustrated in figure 5*b*. Clearly, in order for the unified solution to capture the state*q*_{s}=*q*_{b}=0, it cannot have a constant term.

To determine the expressions of *A*_{1}, *A*_{2} and *A*_{3} for a truncated regular polyhedron with given Schläfli index {*n*,*m*}, we consider three special self-equilibrated states of truncated regular polyhedral tensegrities, corresponding to one asymptotic line and two special points, I (*q*_{s}=0) and II (*q*_{s}=*q*_{b}), on the curves from equations (4.4), (4.9), (4.15), (4.23) and (4.31), as shown in figure 6.

### (b) Determination of coefficients in conjecture 5.1

#### (i) Coefficient *A*_{1}

First, the critical truncated polyhedral tensegrity in which all remaining-strings have a zero length is considered to determine the coefficient *A*_{1}. This special self-equilibrated state corresponds to the *critical truncated polyhedron* in definition 2.3, as shown in figure 2*c*, in which case the force density of remaining-strings approaches infinity, corresponding to the asymptotic line of in the *q*_{b}−*q*_{s} curve.

Rewriting equation (5.1) as 5.3and then letting yield 5.4where denotes the force density of bars in the critical truncated polyhedral tensegrity as . The problem is now reduced to calculating .

There are three types of critical truncated polyhedral tensegrities, as shown in figure 7. Consider the elements connecting to a specific node, say node R highlighted by a green solid ball in figure 7, where all strings (magenta) are drawn but only those bars (blue) connected to node R are shown. Here, all strings are truncating-ones and have force density equal to unity. The force density in the bars can be calculated from force equilibrium. As shown in figure 7, the four nodes connecting to R via strings are numbered from 1 to 4, and the two nodes joining with R via bars are numbered 5 and 6. Clearly, force equilibrium in the *x*-direction at node R requires
5.5where *x*_{R} and *x*_{i} are the *x*-coordinates of R and connecting node *i*, respectively.

Substituting the nodal coordinates of critical truncated polyhedra into equation (5.5) leads to
5.6It follows from equation (5.4) that the coefficient *A*_{1} is
5.7where is the maximum total number of edges of truncating polygons in the critical truncated polyhedron, {*n*,*m*} being the Schläfli symbol of the corresponding regular polyhedron. It can be shown that *p*=3 and *A*_{1}=1 for the truncated tetrahedral tensegrities, *p*=4 and *A*_{1}=2/3 for the truncated cubic and octahedral tensegrities, and *p*=5 and for the truncated dodecahedral and icosahedral tensegrities. These results are tabulated in table 1. Note that *A*_{1} has the same value for dual tensegrities such as the truncated dodecahedron and icosahedron (see definition 2.2).

#### (ii) Coefficient *A*_{3}

Next we proceed to determine coefficient *A*_{3}. Consider the truncated polyhedral tensegrity in which all remaining-strings are of zero force density and can thus be eliminated without affecting self-equilibrium (Li *et al*. 2010*a*). Once all remaining-strings are removed, the resulting structure corresponds to the special point I of *q*_{s}=0 in the *q*_{b}−*q*_{s} curve. As an example, a self-equilibrated configuration of truncated tetrahedral tensegrity with all remaining-strings removed is shown in figure 8*a*.

Substituting *q*_{s}=0 into equation (5.1) gives
5.8where denotes the force density of bars in the truncated polyhedral tensegrity with *q*_{s}=0. The problem is now reduced to finding .

In the special state with all remaining-strings removed, there are only two truncating-strings and one bar connected to each node. In this case, force equilibrium requires that the three elements must lie in the same plane (Motro 2003), as shown in figure 8*a*. Because of the structural symmetry, the solid centre of the structure, the centres of all truncating polygons and the midpoints of all bars must be overlapped. Therefore, a specific node (e.g. node R denoted as a solid ball in figure 8*b*) and its three connecting nodes must lie on a circle with centre located at the midpoint of a bar and diameter equal to the length of the bar, as shown in figure 8*b*. Based on these considerations, the nodal coordinates and force density of bars can be determined.

As shown in figure 8*b*, nodes 1 and 2 are connected to the specified node R via two strings, and node 3 to R via one bar. In the local Cartesian coordinate system (*x*-*O*-*y*), where the origin *O* is located at the centre of the circle, the *x*-axis and *y*-axis are along and perpendicular to the bar, respectively, the force equilibrium condition of node R in the *x*-direction is
5.9In writing equation (5.9), we have used the fact that all truncating-strings have force density equal to unity. Taking the radius of the circle as unit of length, we find *x*_{R}=1, , ) and *x*_{3}=−1, where *m* as one of the Schläfli parameters in {*n*,*m*} is the total number of edges in each truncating polygon. Substituting these nodal coordinates into equation (5.9) leads to
5.10Combining equations (5.7) and (5.10) with (5.8) results in
5.11where *m*=3 for the truncated tetrahedral, cubic and dodecahedral tensegrities, *m*=4 for the truncated octahedral tensegrities, and *m*=5 for the truncated icosahedral tensegrities. The values of *A*_{3} calculated from equation (5.11) are tabulated in table 1.

#### (iii) Coefficient *A*_{2}

Since the truncated regular polyhedral tensegrity structures belong to *Z*-based tensegrities (Li *et al*. 2010*b*), we take hyper-truncated polyhedral tensegrities, in which the four elements of each *Z*-shaped elementary cell are in a line, as the third special self-equilibrated state to determine the coefficient *A*_{2}. This special state corresponds to the *hyper-truncated polyhedron* in definition 2.4, as shown in figure 2*d*. For example, figure 9 shows the hyper-truncated tetrahedron and its corresponding hyper-truncated tetrahedral tensegrity.

In a hyper-truncated polyhedral tensegrity structure, the force densities in all remaining-strings and bars are identical and force equilibrium along the element direction shows
5.12Note that the strings are allowed to bear compressive forces in the hyper-truncated polyhedral tensegrity only for the sake of calculations. In this special case, the force densities correspond to the special point II in the *q*_{b}−*q*_{s} curve, which is at the same location (−1,−1) for all truncated regular polyhedral tensegrities.

Substituting equation (5.12) into (5.1) leads to 5.13It follows from equations (5.7) and (5.11) that 5.14

### (c) Unified solution

With the determination of coefficients *A*_{1}, *A*_{2} and *A*_{3} in equation (5.1), we can establish the following necessary condition for the self-equilibrated and super-stable states of all truncated regular polyhedral tensegrities:

### Theorem 5.2

*For the self-equilibrated and super-stable states of all truncated regular polyhedral tensegrities, the dimensionless force densities of the remaining-strings (q*_{s}*) and bars (q*_{b}*) normalized by the force density of the truncating-strings must satisfy the following relation:
*5.15*where
*5.16*with parameters m and* *following the definition of Schläfli symbol {n,m}for the corresponding regular polyhedron.*

Table 1 lists the self-equilibrium solutions of truncated regular tetrahedral, cubic, octahedral, dodecahedral and icosahedral tensegrities given by equations (5.15) and (5.16). It can be verified that the earlier-mentioned unified solution covers all super-stable solutions derived in §4. It is a necessary condition for the self-equilibrium and super-stability of all truncated regular polyhedral tensegrity structures. The sufficient condition will be determined in §6.

## 6. Super-stability analysis

In order to determine the necessary and sufficient condition for the self-equilibrated and super-stable states, we must find the super-stable solutions among the self-equilibrated ones defined by the unified solution in §5.

We first solve equation (5.15) for the force density *q*_{b} of bars as a function of the force density *q*_{s} of remaining-strings. Rewriting equation (5.15) in the quadratic form
6.1leads to two possible solutions
6.2and
6.3Using parameter values listed in table 1, it can be verified that the following inequality holds for all truncated regular polyhedral tensegrities:
6.4indicating that the solution (*q*_{s},*q*_{b}) exists in the entire range of . Thus all self-equilibrated states, expressed in terms of the force densities (*q*_{s}, *q*_{b}), can be obtained from equations (6.2) and (6.3). For each type of truncated regular polyhedral tensegrities, the self-equilibrated states captured by the unified solution form three *q*_{b}−*q*_{s} curves, as shown in figure 6. Curves 1 and 3 are from equation (6.2), which is singular and discontinuous at *q*_{s}=−*A*_{1}, while curve 2 from equation (6.3) is continuous throughout . Once the self-equilibrated state (*q*_{s}, *q*_{b}) has been found, the corresponding force density matrix can be determined from equation (3.3) or (3.4).

As discussed in §3*c*, the super-stability of a tensegrity structure first requires that the strings have positive (*q*_{s}>0) and the bars negative force densities (*q*_{b}<0). Also, the force density matrix must be positive semi-definite with exactly four zero-eigenvalues. Our calculations show that all force density matrices corresponding to the solid part of curve 3 satisfy the earlier two conditions, while the force density matrices associated with curves 1 and 2 (the dashed lines) has at least one negative eigenvalue and the dash-dotted part of curve 3 are not in the region *q*_{s}>0 and *q*_{b}<0. In addition, our calculations further show that all geometry matrices corresponding to the solid part of curve 3 exactly have a rank of 6, satisfying the super-stability condition (iv) in §3*c*. Therefore, all super-stable states of truncated regular polyhedral tensegrities will be located in the solid part of curve 3, and other states in the three curves are not super-stable. Some representative self-equilibrated configurations of truncated regular polyhedral tensegrities determined by the unified solution as *q*_{s}=1 are plotted in figure 10. The configurations in the first column (figure 10*a*) are super-stable, with *q*_{b} solved from equation (6.2), while those in the second column (figure 10*b*) from equation (6.3) are not super-stable. For clearer observation of the three-dimensional self-equilibrated and super-stable configurations in figure 10*a*, the reader may refer to the website (Connelly & Terrell 2008).

Therefore, we have obtained the following necessary and sufficient condition for the self-equilibrated and super-stable states of all truncated regular polyhedral tensegrities:

### Theorem 6.1

*A truncated regular polyhedral tensegrity is self-equilibrated and super-stable if and only if its force densities satisfy the following relation:
*6.5*where q*_{s}*>0, and A*_{1}*, A*_{2}*, A*_{3} *have been given in equation (5.16).*

## 7. Conclusions

In this paper, we have performed a detailed theoretical analysis of self-equilibrium and super-stability properties of truncated regular polyhedral tensegrity structures. The most important result of the work is that we have found the necessary and sufficient condition, expressed in a simple and explicit form in equation (6.5), for a self-equilibrated and super-stable truncated regular polyhedral tensegrity structure. We hope the present analysis will stimulate further theoretical studies in the community on super-stable tensegrity structures.

## Acknowledgements

Supports from the National Natural Science Foundation of China (grant nos 10972121 and 10732050), Tsinghua University (2009THZ02122) and the 973 Program of MOST (2010CB631005) are acknowledged.

- Received April 28, 2012.
- Accepted June 25, 2012.

- This journal is © 2012 The Royal Society