This paper proposes a new take on the geometrical design of structurally engaged foldable mechanisms.The flexibility of transformable structures and the continuity of folded surfaces hold benefits for many technical applications, for instance, self-supporting adaptive architectural envelopes, adjustable wings or prostheses.Current ideas for rigid-foldable constructions (e.g. [3], [8]) are essentially based on the theory that emerges from technical origami. This discipline is principally in pursuit of flat foldability; one state equals the planarity of the paper sheet in full deployment, and in the majority of cases the state of maximum compaction is also bound to become flat folded. While in a non-flat folded state, rigid-foldable origami patterns may have advantageous structural properties, even while they are still transformable.In general, load-bearing structures must maintain consistent structural integrity at all states from the fully compacted to the fully deployed state (e.g. provide minimal effective depth). Also, physical transforming mechanisms must avoid dead-centre positions. Therefore, any possible (approximately) flat states are to be circumvented in order to create transformable folding structures. This paper induces a novel approach in the geometrical design of folding mechanisms which are not flat-foldable.