Minimum Forcing Sets for Miura Folding Patterns
ACM-SIAM Symposium on Discrete Algorithms (SODA15), (2015), 136-147 We introduce the study of forcing sets in mathematical origami. The origami material folds flat along straight line segments called creases, each of which is assigned a folding direction of mountain or valley. A subset $F$ of crease...
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| creator | Ballinger, Brad Damian, Mirela Eppstein, David Flatland, Robin Ginepro, Jessica Hull, Thomas |
| description | ACM-SIAM Symposium on Discrete Algorithms (SODA15), (2015),
136-147 We introduce the study of forcing sets in mathematical origami. The origami
material folds flat along straight line segments called creases, each of which
is assigned a folding direction of mountain or valley. A subset $F$ of creases
is forcing if the global folding mountain/valley assignment can be deduced from
its restriction to $F$. In this paper we focus on one particular class of
foldable patterns called Miura-ori, which divide the plane into congruent
parallelograms using horizontal lines and zig-zag vertical lines. We develop
efficient algorithms for constructing a minimum forcing set of a Miura-ori map,
and for deciding whether a given set of creases is forcing or not. We also
provide tight bounds on the size of a forcing set, establishing that the
standard mountain-valley assignment for the Miura-ori is the one that requires
the most creases in its forcing sets. Additionally, given a partial
mountain/valley assignment to a subset of creases of a Miura-ori map, we
determine whether the assignment domain can be extended to a locally
flat-foldable pattern on all the creases. At the heart of our results is a
novel correspondence between flat-foldable Miura-ori maps and $3$-colorings of
grid graphs. |
| doi_str_mv | 10.48550/arxiv.1410.2231 |
| format | Article |
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136-147 We introduce the study of forcing sets in mathematical origami. The origami
material folds flat along straight line segments called creases, each of which
is assigned a folding direction of mountain or valley. A subset $F$ of creases
is forcing if the global folding mountain/valley assignment can be deduced from
its restriction to $F$. In this paper we focus on one particular class of
foldable patterns called Miura-ori, which divide the plane into congruent
parallelograms using horizontal lines and zig-zag vertical lines. We develop
efficient algorithms for constructing a minimum forcing set of a Miura-ori map,
and for deciding whether a given set of creases is forcing or not. We also
provide tight bounds on the size of a forcing set, establishing that the
standard mountain-valley assignment for the Miura-ori is the one that requires
the most creases in its forcing sets. Additionally, given a partial
mountain/valley assignment to a subset of creases of a Miura-ori map, we
determine whether the assignment domain can be extended to a locally
flat-foldable pattern on all the creases. At the heart of our results is a
novel correspondence between flat-foldable Miura-ori maps and $3$-colorings of
grid graphs.</description><identifier>DOI: 10.48550/arxiv.1410.2231</identifier><language>eng</language><subject>Computer Science - Data Structures and Algorithms ; Computer Science - Discrete Mathematics ; Mathematics - Combinatorics</subject><creationdate>2014-10</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1410.2231$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1410.2231$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Ballinger, Brad</creatorcontrib><creatorcontrib>Damian, Mirela</creatorcontrib><creatorcontrib>Eppstein, David</creatorcontrib><creatorcontrib>Flatland, Robin</creatorcontrib><creatorcontrib>Ginepro, Jessica</creatorcontrib><creatorcontrib>Hull, Thomas</creatorcontrib><title>Minimum Forcing Sets for Miura Folding Patterns</title><description>ACM-SIAM Symposium on Discrete Algorithms (SODA15), (2015),
136-147 We introduce the study of forcing sets in mathematical origami. The origami
material folds flat along straight line segments called creases, each of which
is assigned a folding direction of mountain or valley. A subset $F$ of creases
is forcing if the global folding mountain/valley assignment can be deduced from
its restriction to $F$. In this paper we focus on one particular class of
foldable patterns called Miura-ori, which divide the plane into congruent
parallelograms using horizontal lines and zig-zag vertical lines. We develop
efficient algorithms for constructing a minimum forcing set of a Miura-ori map,
and for deciding whether a given set of creases is forcing or not. We also
provide tight bounds on the size of a forcing set, establishing that the
standard mountain-valley assignment for the Miura-ori is the one that requires
the most creases in its forcing sets. Additionally, given a partial
mountain/valley assignment to a subset of creases of a Miura-ori map, we
determine whether the assignment domain can be extended to a locally
flat-foldable pattern on all the creases. At the heart of our results is a
novel correspondence between flat-foldable Miura-ori maps and $3$-colorings of
grid graphs.</description><subject>Computer Science - Data Structures and Algorithms</subject><subject>Computer Science - Discrete Mathematics</subject><subject>Mathematics - Combinatorics</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotjssKwjAQRbNxIdW9K-kPVJukU7MV8QWKgu7LtJORgK2SVtG_t1VXFw6XwxFiJONJYgDiKfqXe05k0gKltOyL6d5VrnyU4ermC1ddwpNt6pBvPty7h8cWX6nDR2wa66t6IHqM19oO_xuI82p5Xmyi3WG9Xcx3EaYgI1SxygGlJZ2zzq0hFWtOC2NThBzYEIEpFEnihEGSNQVIbj-GNPOMdSDGP-03OLt7V6J_Z1141oXrD4P7PdU</recordid><startdate>20141008</startdate><enddate>20141008</enddate><creator>Ballinger, Brad</creator><creator>Damian, Mirela</creator><creator>Eppstein, David</creator><creator>Flatland, Robin</creator><creator>Ginepro, Jessica</creator><creator>Hull, Thomas</creator><scope>AKY</scope><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20141008</creationdate><title>Minimum Forcing Sets for Miura Folding Patterns</title><author>Ballinger, Brad ; Damian, Mirela ; Eppstein, David ; Flatland, Robin ; Ginepro, Jessica ; Hull, Thomas</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a651-a202b5a1ed3bf3be8d203f6c8e6a5b5f8dd58c2d1df4f51de8c51f2038d3ff7f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Computer Science - Data Structures and Algorithms</topic><topic>Computer Science - Discrete Mathematics</topic><topic>Mathematics - Combinatorics</topic><toplevel>online_resources</toplevel><creatorcontrib>Ballinger, Brad</creatorcontrib><creatorcontrib>Damian, Mirela</creatorcontrib><creatorcontrib>Eppstein, David</creatorcontrib><creatorcontrib>Flatland, Robin</creatorcontrib><creatorcontrib>Ginepro, Jessica</creatorcontrib><creatorcontrib>Hull, Thomas</creatorcontrib><collection>arXiv Computer Science</collection><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Ballinger, Brad</au><au>Damian, Mirela</au><au>Eppstein, David</au><au>Flatland, Robin</au><au>Ginepro, Jessica</au><au>Hull, Thomas</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Minimum Forcing Sets for Miura Folding Patterns</atitle><date>2014-10-08</date><risdate>2014</risdate><abstract>ACM-SIAM Symposium on Discrete Algorithms (SODA15), (2015),
136-147 We introduce the study of forcing sets in mathematical origami. The origami
material folds flat along straight line segments called creases, each of which
is assigned a folding direction of mountain or valley. A subset $F$ of creases
is forcing if the global folding mountain/valley assignment can be deduced from
its restriction to $F$. In this paper we focus on one particular class of
foldable patterns called Miura-ori, which divide the plane into congruent
parallelograms using horizontal lines and zig-zag vertical lines. We develop
efficient algorithms for constructing a minimum forcing set of a Miura-ori map,
and for deciding whether a given set of creases is forcing or not. We also
provide tight bounds on the size of a forcing set, establishing that the
standard mountain-valley assignment for the Miura-ori is the one that requires
the most creases in its forcing sets. Additionally, given a partial
mountain/valley assignment to a subset of creases of a Miura-ori map, we
determine whether the assignment domain can be extended to a locally
flat-foldable pattern on all the creases. At the heart of our results is a
novel correspondence between flat-foldable Miura-ori maps and $3$-colorings of
grid graphs.</abstract><doi>10.48550/arxiv.1410.2231</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Computer Science - Data Structures and Algorithms Computer Science - Discrete Mathematics Mathematics - Combinatorics |
| title | Minimum Forcing Sets for Miura Folding Patterns |
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