[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81683-en":3,"doc-seo-81683-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81683,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Height Functions on the m × n Miura ori Flip Graph: Degree Sequence and Diameter","The origami crease pattern state space forms an origami flip graph, where vertices are flatfoldable mountain–valley assignments and edges correspond to single face flips. For the m × n Miura-ori, degree sequences and graph diameter are established beyond the previously known two-row case. Each vertex degree equals the number of strict local extrema of an associated integer height function on the grid. The paper gives explicit polynomial counts for degrees up to five and proves closed-form lower diameter bounds, with an upper bound tied to extremal inequalities for 1-Lipschitz grid functions.","arXiv :2606 .22614v2 [math .CO] 24 Jun 2026  \nHeight functions on the m × n Miura-ori flip graph: degree sequence and diameter  \nChakshu Gupta  \nCollege of Computing, Georgia Institute of Technology  \n[cgupta65@gatech.edu](cgupta65@gatech.edu)  \nAbstract  \nThe state space of an origami crease pattern forms a flip graph, whose vertices are the flatfoldable mountain–valley assignments and whose edges join assignments differing by a single face flip. For the m × n Miura-ori, the degree sequence and diameter of this graph are known only for two rows. Each assignment maps to an integer height function on the grid, under which a vertex’s degree equals its number of local extrema. In this model the vertices of each degree up to five are counted by an explicit polynomial in m and n, valid once both exceed a bound that grows with the degree, and the height functions realising those degrees are described explicitly. A closed-form lower bound for the diameter holds for all m and n, and the matching upper bound reduces to an extremal inequality for 1-Lipschitz functions on the grid, recovering the two-row distance at m = 2 . Since each invariant is read from the extrema or height differences of a grid function, the same reduction applies to any flip-graph quantity expressible in those terms.  \n1 Introduction  \nOrigami is the art of folding a flat sheet of paper into a three-dimensional figure through a sequence of folds, each bending the paper along a straight line [LD06] . When the figure is unfolded, the sheet retains a pattern of crease lines, each recording whether the paper was folded convexly (a mountain fold) or concavely (a valley fold) . A valid assignment of mountain and valley labels to these creases, one that allows the sheet to fold flat at every vertex, is called a flat-foldable mountain– valley assignment; the collection of crease lines together with such an assignment is an origami crease pattern. Such patterns have found broad application beyond the art form itself, spanning biomedical devices, architectural facades, robotics, and deployable space structures [MCZ+21] . A particularly prominent example is the Miura-ori [Miu94], a rigid origami tessellation that allows a flat surface to pack tightly and expand along a single degree of freedom.  \nIn combinatorics, a flip graph is a graph whose vertices represent the valid states of a discrete structure and whose edges connect pairs that differ by a single elementary operation called a flip [STT88, WW22], such as replacing one diagonal of a quadrilateral with the other in a polygon triangulation. For origami, the states are the flat-foldable mountain–valley assignments of a crease pattern C, and two assignments are adjacent if they differ by switching every crease on a single face between mountain and valley, provided the result is again flat-foldable. The resulting origami flip graph OFG(C) [HMNTS22] can fail to be connected for some crease patterns but is connected for the Miura-ori [ADE+20] . For the m × n Miura-ori Mm,n (a grid of m rows and n columns of parallelograms), a bijection [GH14] identifies OFG(Mm,n) with the 3-colouring reconfiguration graph of the m × n grid graph. Every 3-colouring of this grid lifts to an integer-valued height  \nfunction [CvdHJ09], so the flip-graph distance between two assignments is half the ℓ 1 distance between their lifts, minimised over a global integer offset [JKK+16] . For m = 2, both natural invariants of OFG(Mm,n) are known [CHO+25], with the degree sequence (the number of vertices of each degree) determined by a recurrence on the small per-column state space of the two-row pattern, and the diameter (the longest shortest path between two vertices) by a median argument on the relative height of two such lifts.  \nThis paper drives the height-function reduction to exact counts of degree-d vertices for each d ≤ 5 and to a closed-form lower bound on the diameter. The reduction identifies the degree of a vertex with the number of strict l","cbCaitiRLgYUU9aK","https://ap.wps.com/l/cbCaitiRLgYUU9aK","pdf",370125,2,1,18,"English","en",105,"# Abstract\n# Introduction\n# The height-function reduction\n## Degree as number of local extrema\n## Distance via height functions","[{\"question\":\"What is the origami flip graph for the Miura-ori crease pattern?\",\"answer\":\"Its vertices are flatfoldable mountain–valley assignments and edges join assignments that differ by flipping every crease on a single face, provided the result remains flatfoldable.\"},{\"question\":\"How does the paper relate vertex degree to height functions?\",\"answer\":\"For the grid-based height-function model, a vertex’s degree equals the number of strict local extrema of its corresponding integer height function.\"},{\"question\":\"What diameter results are proved for the m × n Miura-ori flip graph?\",\"answer\":\"A closed-form lower bound D(m,n) on the diameter is given for all m and n, and the matching upper bound is reduced to an extremal inequality for integer 1-Lipschitz functions on the grid.\"}]",1784175395,45,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"height-functions-on-the-m-n-miura-ori-flip-graph-degree-sequence-and-diameter","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/height-functions-on-the-m-n-miura-ori-flip-graph-degree-sequence-and-diameter/81683/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-21","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the origami flip graph for the Miura-ori crease pattern?","Question",{"text":75,"@type":76},"Its vertices are flatfoldable mountain–valley assignments and edges join assignments that differ by flipping every crease on a single face, provided the result remains flatfoldable.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper relate vertex degree to height functions?",{"text":80,"@type":76},"For the grid-based height-function model, a vertex’s degree equals the number of strict local extrema of its corresponding integer height function.",{"name":82,"@type":73,"acceptedAnswer":83},"What diameter results are proved for the m × n Miura-ori flip graph?",{"text":84,"@type":76},"A closed-form lower bound D(m,n) on the diameter is given for all m and n, and the matching upper bound is reduced to an extremal inequality for integer 1-Lipschitz functions on the 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