Polyhedron model: Difference between revisions

Revision as of 05:37, 19 August 2005

Revision as of 03:56, 20 August 2005

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[[Image:Universiteit_Twente_Mesa_Plus_Escher_Object.jpg|thumb|A sculpture of a [[stellated dodecahedron]] (inspired by [[M. C. Escher]]) 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 polyhedra, including the five [[Platonic solid|regular convex polyhedra]], five [[polyhedral compound|polyhedral compounds]]s, four [[Kepler-Poinsot solid|Kepler-Poinsot solids]]s, 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 ==

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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 on a single sheet of cardboard. Choosing colours requires geometric understanding of the polyhedron. One way is to colour each [[Face_%28mathematics%29|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]].

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

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

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.

Templates are then made. One way is to copy templates from a polyhedron-making book, such as Magnus Wenninger's ''[[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 sectionsnets of the faces are then traced or printed on template material. A third way is using a software named ''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 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.

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 nets.

=== External links ===

*[http://www.software3d.com/Stella.html Stella: Polyhedron Navigator]

*[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 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.

=== External links ===

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

== See also ==

* [[Polyhedron]]

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