Editing Modular origami

{{Short description|Multi-stage paper folding technique}}
{{More footnotes|date=May 2009}}
[[File:Origami Triangle Edge Icosahedron.tif|thumb|''Triangle Edge [[Icosahedron]]'' designed by Bennett Arnstein. Diagrammed in the book [https://search.worldcat.org/title/32626399 3-D Geometric Origami: Modular Polyhedra (1995])]]
[[File:2014 Origami modułowe.jpg|thumb|An example of golden venture folding.<ref>{{Cite web |title=Golden Venture Folding {{!}} 3D Origami |url=https://origami-resource-center.com/golden-venture-folding/ |access-date=2024-07-15 |website=Origami Resource Center |language=en}}</ref>]]
[[File:Tomoko Fuse Hexagonal Box with Six-Petal Lid.png|thumb|Modular origami hexagonal box with six-petal lid. Designed by [[Tomoko Fuse]].]]
[[File:A page from Ranma zushiki 欄間図式 Volume 3 (1734).jpg|thumb|The page from Ranma zushiki 欄間図式 Volume 3 (1734) where modular origami models are depicted.<ref name=":0" />]]
[[Image:OrigamiStar-BlackPen.png|thumb|A [[stellated]] [[icosahedron]] made from custom papers]]
'''Modular origami''' or '''unit origami''' is a multi-stage [[origami|paper folding]] technique in which individual modules or units are created out of sheets of paper and assembled into a flat shape or three-dimensional structure.<ref name=":2">{{Cite book |last=Mukerji |first=Meenakshi |url=https://www.worldcat.org/title/232922105 |title=Ornamental origami: exploring 3D geomentric [sic] designs |date=2009 |publisher=AK Peters |isbn=978-1-56881-445-2 |location=Wellesley, Mass |oclc=232922105}}</ref> This is usually done by inserting flaps into pockets created by the folding process, which create tension or friction and hold the model together.<ref name=":2" /> Some assemblies can be somewhat unstable when adhesives or string are not used.<ref name=":1">{{Cite book |last=Fusè |first=Tomoko |title=Unit origami: multidimensional transformations |date=2009 |publisher=Japan Publications |isbn=978-0-87040-852-6 |edition=14. Pr |location=Tokyo}}</ref>

==Definition and restrictions==
[[Image:Modular Origami.jpg|thumb|right|Examples of modular origami made up of different variations of [[Sonobe]] units.]]

Modular origami can be classified as a subset of multi-piece origami, since the rule of restriction to one sheet of paper is abandoned. However, all the other rules of origami still apply, so the use of glue, thread, or any other fastening that is not a part of the sheet of paper is generally unacceptable in modular origami.

Not all multi-piece origami is modular, as modular origami must involve linking identical copies of a module in a symmetrical or repeating fashion. However, ''linking units'', which are hidden from sight to hold parts of the construction together are acceptable. Any other usage of non-identical modules are generally discouraged.

==History==
[[Image:Origami ball.jpg|thumb|right|A ''[[kusudama]]'', the traditional Japanese precursor to modular origami]]

The first historical evidence for a modular origami design comes from a Japanese book by Hayato Ohoka published in 1734 called ''Ranma Zushiki''. It contains a print that shows a group of traditional origami models, one of which is a modular cube.<ref name=":0">{{Cite web |title=illustrated book; print {{!}} British Museum |url=https://www.britishmuseum.org/collection/object/A_1979-0305-0-75-3 |access-date=2024-07-15 |website=www.britishmuseum.org |language=en}}</ref> The cube is pictured twice (from slightly different angles) and is identified in the accompanying text as a ''[[Tamatebako (origami)|tamatebako]]'' (magic treasure chest).  

Isao Honda's ''World of Origami'' (published in 1965) appears to have the same model, where it is called a "cubical box". The six modules required for this design were developed from the traditional Japanese paperfold commonly known as the ''menko''. Each module forms one face of the finished cube.  

There are several other traditional Japanese modular designs, including balls of folded paper flowers known as ''[[kusudama]]'', or medicine balls. These designs are not integrated and are commonly strung together with thread. The term ''kusudama'' is sometimes inaccurately used to describe any three-dimensional modular origami structure resembling a ball.

There are also a few modular designs in the [[Chinese paperfolding]] tradition, notably the pagoda (from Maying Soong) and the lotus made from [[Joss paper]].

Most traditional designs are however single-piece and the possibilities inherent in the modular origami idea were not explored further until the 1960s when the technique was re-invented by [[Robert Neale (paperfolder)|Robert Neale]] in the US and later by [[Mitsunobu Sonobe]] in Japan. The 1970s saw a sudden period of interest and development in modular origami as its own distinct field, leading to its present status in origami folding. One notable figure is Steve Krimball, who discovered the potential in [[Sonobe| Sonobe's cube unit]] and demonstrated that it could be used to make alternative polyhedral shapes, including a 30-piece ball.<ref>{{cite web |url=http://www.britishorigami.info/academic/lister/sonobe.php |title=David Lister on Origins of the Sonobe Module |website=www.britishorigami.info |url-status=dead |archive-url=https://web.archive.org/web/20090605230847/http://www.britishorigami.info/academic/lister/sonobe.php |archive-date=2009-06-05}} </ref>

Since then, the modular origami technique has been popularized and developed extensively, and now there have been thousands of designs developed in this repertoire.

Notable modular origami artists include [[Robert Neale (paperfolder)|Robert Neale]], Mitsunobu Sonobe, [[Tomoko Fuse]], [[Kunihiko Kasahara]], [[Tom Hull (mathematician)|Tom Hull]], [[Heinz Strobl]], [[Rona Gurkewitz]], Meenakshi Mukerji,<ref>{{Cite web |title=Origamee: Origami by Meenakshi |url=http://www.origamee.net/index.php |access-date=2025-04-22 |website=www.origamee.net}}</ref> and Ekaterina Lukasheva.<ref>{{Cite web |title=Kusudama Me! - site about modular origami |url=https://kusudama.me/ |access-date=2025-04-22 |website=kusudama.me}}</ref>

==Types==
[[File:Modules of modular origami.jpg|thumb|Modules of modular origami]]
Modular origami forms may be flat or three-dimensional. Flat forms are usually [[polygon]]s (sometimes known as coasters), stars, rotors, and rings. Three-dimensional forms tend to be [[polyhedron|regular polyhedra]] or tessellations of simple polyhedra.

Modular origami techniques can be used to create a wide range of lidded boxes in many shapes. Many examples of such boxes are shown in [[Tomoko Fuse]]'s books ''Origami Boxes'' (1989),<ref>{{Cite book |last=Fuse |first=Tomoko |title=Origami Boxes |publisher=Japan Publications |year=1989 |isbn=0-87040-821-6 |location=Tokyo, Japan |language=en |oclc=20372390}}</ref> ''Fabulous Origami Boxes'' (1998)<ref>{{Cite book |last=Fuse |first=Tomoko |title=Fabulous origami boxes |date=1998 |publisher=Japan Publications Trading Co |isbn=978-0-87040-978-3 |location=Tōkyō}}</ref>'','' and ''Tomoko Fuse's Origami Boxes'' (2018).<ref>{{Cite book |last=Fuse |first=Tomoko |title=Tomoko Fuse's origami boxes |publisher=Tuttle Publishing |year=2018 |isbn=978-0-8048-5006-3 |edition=First |location=Tokyo [Japan] ; Rutland, Vermont |language=en}}</ref> 

There are some modular origami that are approximations of [[fractals]], such as [[Menger's sponge]]. Macro-modular origami is a form of modular origami in which finished assemblies are themselves used as the building blocks to create larger integrated structures. Such structures are described in [[Tomoko Fuse]]'s 1990 book ''Unit Origami-Multidimensional Transformations''.<ref name=":1" />

==Modeling systems==
===Robert Neale's penultimate module===
Neale developed a system to model [[Equilateral|equilateral polyhedra]] based on a module with variable [[vertex (geometry)|vertex]] angles. Each module has two pockets and two tabs, on opposite sides. The angle of each tab can be changed independently of the other tab. Each pocket can receive tabs of any angle. The most common angles form polygonal faces:

{{div col}}
* 60 degrees ([[triangle]])
* 90 degrees ([[Square (geometry)|square]])
* 108 degrees ([[pentagon]])
* 120 degrees ([[hexagon]])
{{div col end}}

Each module joins others at the vertices of a polyhedron to form a polygonal face. The tabs form angles on opposite sides of an edge. For example, a subassembly of three triangle corners forms a triangle, the most stable configuration. As the internal angle increases for squares, pentagons and so forth, the stability decreases.

Many polyhedra call for unalike adjacent polygons. For example, a [[Pyramid (geometry)|pyramid]] has one square face and four triangular faces. This requires hybrid modules, or modules having different angles. A pyramid consists of eight modules, four modules as square-triangle, and four as triangle-triangle.

Further polygonal faces are possible by altering the angle at each corner. The Neale modules can form any equilateral polyhedron including those having [[rhombus|rhombic]] faces, like the [[rhombic dodecahedron]].

===Mukhopadhyay module===
The Mukhopadhyay module can form any equilateral polyhedron. Each unit has a middle crease that forms an edge, and triangular wings that form adjacent stellated faces. For example, a cuboctahedral assembly has 24 units, since the [[cuboctahedron]] has 24 edges.
Additionally, [[bipyramids]] are possible, by folding the central crease on each module outwards or convexly instead of inwards or concavely as for the [[icosahedron]] and other stellated polyhedra. The Mukhopadhyay module works best when glued together, especially for polyhedra having larger numbers of sides.

==Notes and references==
{{Reflist|2}}

==Bibliography==
*{{cite book|author=Tomoko Fuse|authorlink=Tomoko Fuse|title=Unit Origami: Multidimensional Transformations
|publisher=Japan Publications| year=1990| isbn=0-87040-852-6}}
*{{cite book|author=Tomoko Fuse|authorlink=Tomoko Fuse|title=Fabulous Origami Boxes
|publisher=Japan Publications Trading|year=1998|isbn=0870409786}}

==External links==
{{Commons category}}
* [http://www.3dorigamiart.com 3dOrigamiArt.com] Learn how to 3d Origami, tutorials and artist network.
* [https://www.youtube.com/channel/UCwNBvoJ2WZGUf2C9VmY4jyQ] 3D origami video tutorials by Arthur Vershigora.
* [http://www.ask.ne.jp/~kanzasi/en/e-kusu.html Kusudama Pictures] {{Webarchive|url=https://web.archive.org/web/20120609095302/http://www.ask.ne.jp/~kanzasi/en/e-kusu.html |date=2012-06-09 }}
* [http://www.origamee.net Photo Gallery and Folding Instructions For Many Polyhedra and Variations]
* [http://www.giladorigami.com/File.php?File=P_OUSA2005_EXN_Menger_2.JPG Image of Menger's Sponge in origami]
* [https://origami.kosmulski.org/ Modular origami page]
* [https://www.flickr.com/photos/pascalin/213394110 Origami Geosphere] Paper model of a Geodesic Sphere.
* [http://www.origamee.net/diagrams/misc/isos.html Mukhopadhyay's super simple isosceles triangle module]
* [http://www.cs.utk.edu/~plank/plank/pics/origami/penultimate/intro.html James S. Plank's Penultimate Modular Origami]
* [https://origami.kosmulski.org/models/oxi Oxi Module by Michał Kosmulski]
* [http://www.kusudama.me Kusudama Me! Kusudamas of Lukasheva Ekaterina, also diagrams and tutorials]
* [http://www.origami.edu.pl Paper Structures by Krystyna and Wojtek Burczyk]
* [https://web.archive.org/web/20130923164941/http://kusudam.in/ Kusudama by Mikhail Puzakov & Ludmila Puzakova: models, folding instruction, history, geometry]

{{Mathematics of paper folding}}

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[[Category:Modularity|Origami]]
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