Construction and mobility analysis of networks of deployable even-sided equilateral polygons

Deployable planar mechanisms represent a classical category of deployable systems with broad applications in civil engineering, space exploration, metamaterials, and beyond — either as standalone structures or as compositional units. Despite significant progress in recent decades, designing such mechanisms remains challenging due to their overconstrained nature. This paper addresses the construction and mobility analysis of networks of deployable even-sided equilateral polygons (NDEEPs). Two approaches are proposed: the link-connection method and the joint-connection approach. Three classes of one degree-of-freedom (DOF) NDEEPs and one class of 2-DOF NDEEPs are introduced. Since conventional formulas for the mobility analysis are not applicable to NDEEPs derived via the link-connection method, a mobility analysis method based on a minimal set of constraint equations is proposed. The geometric characteristics of these NDEEPs are also identified. Two classes of NDEEPs derived from tessellations differ fundamentally from hinged tessellations. Additionally, two novel tessellations composed of three types of semi-regular equilateral polygons are discovered as a by-product. This work complements existing research on expanding polygon arrays, hinged tessellations, and deployable networks, contributing to the broader study of overconstrained mechanisms.