Polyhedron model: Difference between revisions

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:''For {{for|the compiler optimization, see [[|Polytope model]].''}}

[[Image:Universiteit Twente Mesa Plus Escher Object.jpg|thumb|A sculpture of the [[small stellated dodecahedron]] in [[M. C. Escher]]'s ''[[Gravitation (M. C. Escher)|Gravitation]]'', near the Mesa+ Institute of [[Universiteit Twente]]]]

A '''polyhedron model''' is a physical construction of a [[polyhedron]], constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material.

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==Construction==

[[File:Dodecahedron flat.svg|thumb|A net for the regular [[dodecahedron]]]]

Construction begins by choosing a ''size'' of the model, either the ''length'' of its edges or the ''height'' of the model. The size will dictate the ''material'', the ''adhesive'' for edges, the ''construction time'' and the ''method of construction''.

The second decision involves colours. A single-colour cardboard model is easiest to construct — and some models can be made by folding a pattern, called a '''[[net (polyhedron)|net]]''', from a single sheet of cardboard. Choosing colours requires geometric understanding of the polyhedron. One way is to colour each [[face (geometry)|face]] differently. A second way is to colour all square faces the same, all pentagonal faces the same, and so forth. A third way is to colour opposite faces the same. Many polyhedra are also coloured such that no same-coloured faces touch each other along an edge or at a vertex.

:For example, a 20-face [[icosahedron]] can use twenty colours, one colour, ten colours, or tenfive colours, respectively.

An alternative way for [[polyhedral compound]] models is to use a different colour for each polyhedron component.

Net templates are then made. One way is to copy templates from a polyhedron-making book, such as [[Magnus Wenninger]]'s ''[[List of Wenninger polyhedron models|Polyhedron Models]]'', [[1974]] ({{ISBN |0-521-09859-9}}). A second way is drawing faces on paper or with [[computer-aided design]] software and then drawing on them the polyhedron's [[Edge (geometry)|edge]]s. The exposed nets of the faces are then traced or printed on template material. A third way is using the software named ''[[Stella (software)|Stella]]'' to print nets.

A model, particularly a large one, may require another polyhedron as its inner structure or as a construction mold. A suitable inner structure prevents the model from collapsing from age or stress.

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Assembling multi-colour models is easier with a model of a simpler related polyhedron used as a colour guide. Complex models, such as [[stellation]]s, can have hundreds of polygons in their nets.

===External links===

*[http://web.aanet.com.au/robertw/Stella.html Stella: Polyhedron Navigator] - Software to generate and printing nets for polyhedra

*[http://web.aanet.com.au/robertw/MyModels.html Paper models of many polyhedra]

*[http://www.korthalsaltes.com/ Paper Models of Polyhedra] Many links

*[http://www.polyedergarten.de/ Paper Models of Uniform (and other) Polyhedra]

==Interactive computer models==

Recent [[computer graphics]] technologies allow people to rotate 3D polyhedron models on a computer video screen in all three dimensions. Recent technologies even provide shadows and [[texture (computer graphics)|textures]] for a more realistic effect.

==See also==

*[[Polyhedron]]

*[[List of Wenninger polyhedron models]]

===External links===

*[http://webwww.aanetsoftware3d.com.au/robertw/Stella.htmlphp Stella: Polyhedron Navigator]: Software to explore virtual polyhedra and print their nets to enable physical construction

*[https://web.archive.org/web/20050403235101/http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D polyhedra in Java]

*[http://www.flickr.com/photos/pascalin/sets/72157594234292561/ Origami Polyhedra]: Photos of models made with modular origami

*[http://bulatov.org/polyhedra/wooden/ Wooden Polyhedra Models]

*[http://www.georgehart.com/virtual-polyhedra/vp.html]: George Hart's extensive encyclopedia of polyhedra]

*[http://www.georgehart.com/pavilion.html]: George Hart's Pavilion of Polyhedreality]

*[http://polyhedra.org Online rotatable polyhedron models]

*[http://woodenpolyhedra.web.fc2.com/woodenpolyhedra30.html WOODEN POLYHEDRA 30]

[[Category:Recreational mathematics]]