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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|polyhedral compounds]]s, four [[Kepler-Poinsot solid|Kepler-Poinsot solidspolyhedra]], and thirteen [[Archimedean solid|Archimedean solids]]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 ==

== Construction ==

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''.

[[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 fromby folding a pattern, oncalled 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_%28mathematics%29face (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]].

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

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

Templates are then made. One way is to copy templates from a polyhedron-making book. 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 sections of the faces are then traced or printed on template material.

TemplatesNet 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 onwith [[computer-aided design]] software and then drawing on them the polyhedron's [[Edge (geometry)|edge]]s. The exposed sectionsnets 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.

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.

The templates are then replicated unto the material, matching carefully the chosen colours.

The net templates are then replicated onto 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 of polygons in their nets.

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.

== Interactive computer models ==

== See also ==

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 also ==

* [[Polyhedron]]

*[[List of Wenninger polyhedron models]]

==External links==

{{Geometry-stub}}

*[http://www.software3d.com/Stella.php 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://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_mathematicsRecreational mathematics]]

[[Category:Polyhedra|Model]]