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{{for|the compiler optimization|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.

Since there are 75 [[uniform polyhedron|uniform polyhedra]], including the five [[Platonic solid|regular convex polyhedra]], five [[polyhedral compound]]s, four [[Kepler-Poinsot solidpolyhedra]]s, and thirteen [[Archimedean solid]]s, constructing or collecting polyhedron models has become a common mathematical recreation. Polyhedron models are found in [[mathematics]] classrooms much as [[globe]]s in [[geography]] classrooms.

Polyhedron models are notable as three-dimensional [[proof-of-concept]]s of geometric theories. Some polyhedra also make great centerpieces, [[tree topper]]s, Holiday decorations, or symbols. The [[Merkaba]] religious symbol, for example, is a [[stellated octahedron]]. Constructing large models offer challenges in engineering [[structural design]].

== 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 '''[[Netnet (polyhedron)|net]]''', onfrom a single sheet of cardboard. Choosing colours requires geometric understanding of the polyhedron. One way is to colour each [[Faceface (geometry)|face]] differently. A second way is to colour all square faces the same, all pentagonpentagonal faces the same, and so forth. A third way is to colour opposite faces the same. AMany fourthpolyhedra wayare isalso tocoloured asuch differentthat colourno same-coloured faces touch each faceother clockwisealong an edge or at a certain [[vertex]].

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''.

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

The second decision involves colours. A single-colour cardboard model is easiest to construct -- and some models can be made by folding a pattern, a '''[[Net (polyhedron)|net]]''', on 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 pentagon faces the same, and so forth. A third way is to colour opposite faces the same. A fourth way is to a different colour each face clockwise a certain [[vertex]].

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

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

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 0521098599|0-521-09859-9}}). A second way is drawing faces on paper or onwith [[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 athe software named ''[[Stella (software)|Stella]]'' to print nets.

An alternate way for [[polyhedral compound]] models is to colour each polyhedron component the same.

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.

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 0521098599). A second way is drawing faces on paper or on [[computer-aided design]] software and then drawing on them the polyhedron's [[edge]]s. The exposed nets of the faces are then traced or printed on template material. A third way is using a software named ''[[Stella (software)|Stella]]'' to print nets.

The net templates are then replicated untoonto the material, matching carefully the chosen colours. Cardboard nets are usually cut with tabs on each edge, so the next step for cardboard nets is to score each fold with a knife. Panelboard nets, on the other hand, require molds and cement adhesives.

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.

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 or over a thousandof polygons in their nets.

The net templates are then replicated unto the material, matching carefully the chosen colours. Cardboard nets are usually cut with tabs on each edge, so the next step for cardboard nets is to score each fold with a knife. Panelboard nets, on the other hand, require molds and cement adhesives.

== Interactive computer models ==

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 or over a thousand polygons in their nets.

Recent [[computer graphics]] technologies allowedallow 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 External links =also==

* [[Polyhedron]]

*[[List of Wenninger polyhedron models]]

=== External links ===

*[http://webwww.aanetsoftware3d.com.au/robertw/Stella.htmlphp Stella: Polyhedron Navigator] -: Software to generateexplore virtual polyhedra and printingprint their nets forto polyhedraenable physical construction

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

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

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

*[http://wwwbulatov.polyedergarten.deorg/polyhedra/wooden/ PaperWooden Polyhedra Models] of Uniform (and other) Polyhedra]

*[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]

== Interactive computer models ==

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

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

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

== See also ==

* [[Polyhedron]]

=== External links ===

*[http://web.aanet.com.au/robertw/Stella.html Stella: Polyhedron Navigator] - Software to create and explore polyhedra

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

[[Category:Recreational mathematics]]

[[Category:Polyhedra|Model]]