  In a series of our three previous papers, we presented several constructions
of positive and negative 3D gadgets in origami extrusions which create with two
simple outgoing pleats a top face parallel to the ambient paper and two side
faces sharing a ridge, where a 3D gadget is said to be positive (resp.
negative) if the top face of the resulting gadget seen from the front side lies
above (resp. below) the ambient paper. For any possible set of angle
parameters, we obtained an infinite number of positive 3D gadgets in our second
paper, while we obtained a unique negative 3D gadget by our third construction
in our third paper. In this paper we present a geometric (ruler and compass)
construction of our third negative 3D gadgets, while the construction presented
in our third paper was a numerical one using a rather complicated formula.
Also, we prove that there exists a unique positive 3D gadget corresponding to
each of our third negative ones. Thus we obtain a canonical pair of a positive
and a negative 3D gadget. The proof is based on a geometric redefinition of the
critical angles which we introduced in constructing our positive 3D gadgets.
This redefinition also enables us to give a simplified proof of the existence
theorem of our positive 3D gadgets in our second paper. As an application, we
can construct a positive and a negative extrusion from a common crease pattern
by using the canonical counterparts, as long as there arise no interferences.
