Blooming (geometry)

Blooming of platonic solid

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. A blooming, reversed to go from the flat net to a polyhedron, can be thought of intuitively as a way to fold the polyhedron from a paper net without bending the paper except at its designated creases.

An early work on blooming by Biedl, Lubiw, and Sun from 1999 showed that some nets for non-convex but topologically spherical polyhedra have no blooming. ^[1]

The question of whether every convex polyhedron admits a net with a blooming was posed by Robert Connelly, and came to be known as Connelly’s blooming conjecture. ^[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. This was proven in 2009 by Demaine et al., who showed in addition that every convex polyhedral net whose polygons are connected in a single path has a blooming, and that every net can be refined to a path-connected net. ^[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. ^[2]

Unsolved problem in mathematics

Does every net of a convex polyhedron have a blooming?

More unsolved problems in mathematics

Because it is unknown whether every convex polyhedron has a net that cuts only edges of the polyhedron, and not across its faces ("Dürer's conjecture"), it is also unknown whether every convex polyhedron has a blooming that cuts only edges. In an unpublished manuscript from 2009, Igor Pak and Rom Pinchasi have claimed that this is indeed possible for every Archimedean solid. ^[4]

The problem of finding a blooming for a polyhedral net has also been approached computationally, as a problem in motion planning. ^{[5] [6] [7]}

References

1. Biedl, Therese; Lubiw, Anna; Sun, Julie (2005), "When can a net fold to a polyhedron?", Computational Geometry , 31 (3): 207–218, doi: 10.1016/j.comgeo.2004.12.004 , MR   2143321 . Announced at the Canadian Conference on Computational Geometry, 1999.
2. Miller, Ezra; Pak, Igor (2008), "Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings", Discrete & Computational Geometry , 39 (1–3): 339–388, doi: 10.1007/s00454-008-9052-3 , MR   2383765 . Announced in 2003.
3. Demaine, Erik D.; Demaine, Martin L.; Hart, Vi; Iacono, John; Langerman, Stefan; O'Rourke, Joseph (2011), "Continuous blooming of convex polyhedra", Graphs and Combinatorics , 27 (3): 363–376, doi:10.1007/s00373-011-1024-3, hdl: 1721.1/67481 , MR   2787423, S2CID   82408 . Announced at the Japan Conference on Computational Geometry and Graphs, 2009.
4. Pak, Igor; Pinchasi, Rom (2009), How to cut out a convex polyhedron (PDF), archived from the original (PDF) on 2021-01-20, retrieved 2021-06-21. As cited by Demaine et al. (2011).
5. Song, Guang; Amato, N. M. (February 2004), "A motion-planning approach to folding: From paper craft to protein folding", IEEE Transactions on Robotics and Automation, 20 (1): 60–71, doi:10.1109/tra.2003.820926, S2CID   9636
6. Xi, Zhonghua; Lien, Jyh-Ming (September 2015), "Continuous unfolding of polyhedra – a motion planning approach", 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, pp. 3249–3254, doi:10.1109/iros.2015.7353828, ISBN   978-1-4799-9994-1, S2CID   14376277
7. Hao, Yue; Kim, Yun-hyeong; Lien, Jyh-Ming (June 2018), "Synthesis of fast and collision-free folding of polyhedral nets", Proceedings of the 2nd ACM Symposium on Computational Fabrication, ACM, pp. 1–10, doi: 10.1145/3213512.3213517 , ISBN   978-1-4503-5854-5