Universal Hinge Patterns for Folding Strips Efficiently into Any Grid Polyhedron
Name
1611.03187.pdf
Description
Submitted version
Size
5.55 MB
Format
Adobe PDF
Checksum (MD5)
6aa4d026a74edadc18855188fb34470b
Author(s)
Benbernou, Nadia M.
Demaine, Erik D.
Demaine, Martin L.
Lubiw, Anna
Date Issued
2017
Publisher
Springer Nature
Citation
Benbernou, Nadia M., Demaine, Erik D., Demaine, Martin L. and Lubiw, Anna. 2017. "Universal Hinge Patterns for Folding Strips Efficiently into Any Grid Polyhedron."
Version
Original manuscript
Abstract
© Springer International Publishing AG 2017. We present two universal hinge patterns that enable a strip of material to fold into any connected surface made up of unit squares on the 3D cube grid—for example, the surface of any polycube. The folding is efficient: for target surfaces topologically equivalent to a sphere, the strip needs to have only twice the target surface area, and the folding stacks at most two layers of material anywhere. These geometric results offer a new way to build programmable matter that is substantially more efficient than what is possible with a square N × N sheet of material, which can fold into all polycubes only of surface area O(N) and may stack Θ(N2) layers at one point. We also show how our strip foldings can be executed by a rigid motion without collisions (albeit assuming zero thickness), which is not possible in general with 2D sheet folding. To achieve these results, we develop new approximation algorithms for milling the surface of a grid polyhedron, which simultaneously give a 2-approximation in tour length and an 8/3-approximation in the number of turns. Both length and turns consume area when folding a strip, so we build on past approximation algorithms for these two objectives from 2D milling.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/978-3-319-62127-2_10
