Fig. 2: Geometric modification.

a Initial configuration of modular origami structures of a sphere generated by volumetric mapping and volumetric shrinkage \(\left(3\times 3\times 3\right)\) of a cubic unit cell. Note that modules can be built by selective tubular extrusion of shrunken polyhedrons. b Configuration after geometric modification. Equations (4) and (5) produce the constraints for the rectangular tubes, i.e., \({d}_{1}={d}_{2}={d}_{3}={d}_{4}\) and \({\theta }_{1}={\theta }_{2}={\theta }_{3}={\theta }_{4}\). c A tube with intersecting hinges, where \({\theta }_{1}\) is the angle between the projected \({{{{{{\bf{v}}}}}}}_{4}-{{{{{{\bf{v}}}}}}}_{1}\) and \({{{{{{\bf{v}}}}}}}_{3}-{{{{{{\bf{v}}}}}}}_{2}\), and \({\theta }_{2}\) is the angle between \({{{{{{\bf{v}}}}}}}_{3}-{{{{{{\bf{v}}}}}}}_{2}\) and \({{{{{{\bf{v}}}}}}}_{1}-{{{{{{\bf{v}}}}}}}_{6}\). \({\theta }_{3}\) and \({\theta }_{4}\) are similarly obtained. The extended lines of these four hinges (\({{{{{{\bf{v}}}}}}}_{1}-{{{{{{\bf{v}}}}}}}_{4},{{{{{{\bf{v}}}}}}}_{2}-{{{{{{\bf{v}}}}}}}_{3},{{{{{{\bf{v}}}}}}}_{6}-{{{{{{\bf{v}}}}}}}_{7},{{{{{{\bf{v}}}}}}}_{5}-{{{{{{\bf{v}}}}}}}_{8}\)) intersect at a point. d A tube with parallel hinges having limited foldability in the initial configuration. e A tube with intersecting hinges having flat-foldability in the modified configuration with a planar constraint of \({{{{{{\bf{p}}}}}}}_{3}\bullet \left({{{{{{\bf{p}}}}}}}_{1}\times {{{{{{\bf{p}}}}}}}_{2}\right)=0\), where vectors \({{{{{{\bf{p}}}}}}}_{1},\,{{{{{{\bf{p}}}}}}}_{2},\) and \({{{{{{\bf{p}}}}}}}_{3}\) are on the edges of a face.