Blooming (geometry)

In the geometry of convex polyhedra, blooming or continuous blooming is a continuous three-dimensional motion of the surface of the polyhedron, cut to form a polyhedral net, from the polyhedron into a flat and non-self-overlapping placement of the net in a plane.

As in rigid origami, the polygons of the net must remain individually flat throughout the motion, and are not allowed to intersect or cross through each other.

[2] More specifically, Miller and Pak suggested in 2003 that the source unfolding, a net that cuts the polyhedral surface at points with more than one shortest geodesic to a designated source point (including cuts across faces of the polyhedron), always has a blooming.

[3] It is unknown whether every net of a convex polyhedron has a blooming, and Miller and Pak were unwilling to make a conjecture in either direction on this question.

In an unpublished manuscript from 2009, Igor Pak and Rom Pinchasi have claimed that this is indeed possible for every Archimedean solid.