Dual-chirality flexagon linkages with infinite eversion and surface reconfigurability

The flexagon, a classical kirigami structure capable of revealing hidden faces through eversion, has long captivated researchers. However, its potential for robust engineering applications has been limited by inherent structural discontinuities and unclear kinematic mechanisms. Here, we introduce a cyclic graph model and an idealized dual-chirality flexagon linkage with infinite eversion to explain the underlying mechanics of this eversion. We demonstrate that the eversion exhibits topological periodicity and multiple symmetries, while its kinematics correspond to a cyclic permutation of congruent axis sets induced by bifurcated motion. By tuning topological parameters, we construct a comprehensive atlas of the flexagon family. To address challenges in physical implementation, we propose a linkage convertibility strategy that eliminates mechanical interference and enables the design of reconfigurable, deployable networked structures. Furthermore, we develop an interchangeable-chirality flexagon that achieves an exponential expansion in accessible surface states, where the incremental states added in each expansion follow a geometric progression, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>g</mml:mi> </mml:msup> </mml:mrow> </mml:math> (where p denotes the eversion period and g represents the recursive generation index). This work bridges the gap between abstract topological concepts and physical realizations, offering a pathway to transform classical kirigami into advanced engineering linkages. It also provides foundational insights for related eversion systems, including Möbius strips, kaleidocycles, and other cyclic topological structures.