# Biomimetic Origami: Planar Single-Vertex Multi-Crease Mechanism Design and Optimization
## 2 Single-Vertex Multi-Crease Origami Pattern Design
### Determination of the Number of Origami Units  
When designing a complete crease pattern starting with \(\Omega\)-shaped units, several factors require consideration for mathematical analysis and optimization. The number of \(\Omega\)-shaped units is a critical aspect. Evaluating this number involves assessing the folding ratio, manufacturing complexity, and structural reliability. The choice of unit count directly influences deployment performance and must take priority in the design as shown in Figure 6.  
To simplify analysis, assume each unit transforms from a plane into a truncated polyhedron during folding. At least three \(\Omega\)-shaped units are needed to create a continuous closed-loop structure. In the engineering application of space antennas, besides forming a closed-loop structure, the folding ratio is also an important reference factor. Folding ratio is a key metric for measuring the antenna's foldability and portability. It is defined as the ratio of the antenna's radial dimensions in the folded state to those in the deployed state. A lower folding ratio means the antenna takes up less space when folded, making transportation easier and reducing the cost. Calculations show that the folding ratio increases with the number of units and approaches 0.25 as the unit count approaches infinity. The design must also consider the impact of the smallest facet angle on manufacturing precision. Extremely small angles increase production and assembly challenges, raising costs and reducing reliability. Designs should avoid excessively small facet angles. Based on the relationships between unit count, folding ratio, and minimum facet angle, results  
Figure 5: Unit division of origami pattern; (**a**) origami pattern divided into 6 \(\Omega\)-shaped folding units and 24 facets, (**b**) \(\Omega\)-shaped crease divided into four smaller sub-facets \(\alpha_{1}\). Figure 6: Effect of different unit numbers on the folded shape when the sector angle ratio is constant. indicate that the unit count should remain between 3 and 10. Fewer than 3 units are insufficient to form a continuous closed-loop structure, while more than 10 units greatly increase manufacturing and assembly challenges without significant gains in folding ratio performance. Additionally, the number of facets significantly affects complexity and reliability. Adding more facets introduces additional hinges and panels, increasing structural complexity and internal gaps while reducing reliability. Assuming each crease connects through two hinges, a relationship between unit count and hinge count emerges. Analysis of these factors leads to several conclusions. The foldability ratio decreases with increasing unit count, with the rate of change slowing around seven units. Similarly, changes in the smallest facet angle also slow at this point. Furthermore, component count increases linearly with unit count. Table 1 illustrates these trends for different unit numbers. The final determination of the unit number is affected by several factors. Each of these factors has a varying level of impact. To address this complexity, the Analytic Hierarchy Process (AHP) is introduced as a comprehensive evaluation system. This system will assess the influence weight of each factor on the final number of units. In the hierarchical analysis, we constructed a two-by-two comparison matrix for each of the three indicators and determined the relative weights of each item according to the importance of expert scores or design requirements. Specifically, a judgment matrix (a table comparing factors in pairs) is constructed using the 1-9 scale (e.g., 1 = equal importance, 9 = extreme dominance of one factor over another). The importance of each factor is determined according to the design requirements set by the launch vehicle's payload constraints. Pairwise comparisons are used to create the evaluation matrix \(A\), as follows:  
\[A= \begin{bmatrix}1&4&4\\ \frac{1}{4}&1&\frac{1}{2}\\ \frac{1}{4}&2&1\end{bmatrix}