# Different Configurations of the Non-Minimal Prismatic Tensegrities
## 1 Introduction  
Tensegrities are a subclass of spatial trusses known from their intrinsic features such as their ability to be prestressed and infinitesimal mechanisms. Currently the research on these systems is expanding, especially in fields such as space exploration [1, 2, 3, 4], robotics [5, 6, 7, 8, 9], deployable structures [10, 11, 12, 13, 14, 15, 16] and civil engineering [10, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Please note that these fields interfere. Tensegrities are considered lightweight and material-efficient [26]. Some basic information and current trends on tensegrities can be found in [27, 28, 29]. The minimal and regular triplex, presented in Figure 0(b), is the first invented and patented tensegrity structure [30, 31, 32]. Minimal refers to possessing the least number of members required to obtain a stable configuration [33], while regular refers to the fact that members of the same type are of an equal length. For example, structures presented in Figure 0(b)-d are minimal, while the structure presented in Figure 0(e) is non-minimal. All of the structures presented in the figure are regular. The triplex belongs to a group of structures called prismatic tensegrities and is the one that has the least number of bars, i.e., three [34]. Prismatic tensegrities systems include the triplex (\(t=3\)), the quarex (\(t=4\)), the pentex (\(t=5\)) and so on--see Figure 0(b)-d, respectively. Prismatic tensegrities (Figure 0(b)-d) are built from bars (denoted as thick, black lines), cross cables (thin gray lines connecting upper nodes with lower ones) and horizontal cables (thin gray lines connecting upper nodes with themselves, as well as lower nodes with themselves). Horizontal cables create two parallel bases. It is characteristic that there is a defined angle of rotation between the upper and lower bases--here denoted as \(\alpha\). Angle \(\alpha\) is calculated from a theoretical situation in which the bars are vertical, and the upper base starts to twist counterclockwise looking at the structure from top view (see Figure 0(a), the structure presented in this figureis unstable). Please note that here the form-finding process is not considered, since it is assumed in advance that some members are bars and some cables. The angle is only a function of the number of bars--denoted as \(t\). Exemplary values of \(\alpha\) angles at which prismatic tensegrities remain stable are equal to: the triplex--\(\alpha=150^{\circ}\) (Figure 1b), the quarter--\(\alpha=135^{\circ}\) (Figure 1c) and the pentex--\(\alpha=126^{\circ}\) (Figure 1d). The first studies on prismatic tensegrities were performed by [35], to be followed by [36] and others [37; 38]. In [39] the connectivity, incidence and node matrices for all prismatic tensegrities in an analytical form were given. Ref. [40] studied methods of the morphogenesis of icosahedral and prismatic tensegrities, which was also practically used to analyze a tensegrity footbridge. The stiffness of prismatic tensegrities was studied in [41]. These are general studies on \(t\)-bar systems; however, a lot of individual studies were performed on triplex or quarter prisms, as well as on towers created based on these modules, just to mention the latest [42; 43; 44; 45; 46; 47]. Prismatic tensegrities are categorized as \"abstract tensegrities\" by the classification presented in [48] and expanded in [49], which is based on the mechanics of trusses. Since the classification is important in the presented study, a short explanation is given here. A total of six features need to be defined in order to classify whether the structure is or is not a tensegrity.