# Polyhedron Operations Package

A Platonic solid is a convex polyhedron whose faces and vertices are all of the same type. There are five such solids. There are also a few nonconvex polyhedra known that have faces and vertices all of the same type. This package contains functionality for modifying some of the properties of the polyhedra available in PolyhedronData.

Tetrahedron | Cube |

Octahedron | Dodecahedron |

Icosahedron | Hexahedron |

GreatDodecahedron | SmallStellatedDodecahedron |

GreatStellatedDodecahedron | GreatIcosahedron |

Some of the available polyhedra.

In[1]:= |

Stellate[PolyhedronData[polyname]] | display a stellated polyhedron |

Stellate[PolyhedronData[polyname],ratio] | |

display a stellated polyhedron with stellation ratio ratio | |

Geodesate[PolyhedronData[polyname],n] | |

display the projection of the order- regular tessellation of each face of the polyhedron onto the circumscribed sphere | |

Geodesate[PolyhedronData[polyname],n,{x,y,z},radius] | |

display the projection of the order- regular tessellation of each face of the polyhedron onto the circumscribed sphere having center and size radius | |

Truncate[PolyhedronData[polyname]] | display a truncated polyhedron |

Truncate[PolyhedronData[polyname],ratio] | |

display a truncated polyhedron with the polygon edges truncated by the ratio ratio | |

OpenTruncate[PolyhedronData[polyname]] | |

display a truncated polyhedron without the polygons that close the truncated vertices | |

OpenTruncate[PolyhedronData[polyname],ratio] | |

display a truncated polyhedron with edges truncated by the ratio ratio without the polygons that close the truncated vertices |

Transformation functions for polyhedra.

The polyhedra are by default centered at the origin with a unit distance from the origin to the midpoint of the edges. Any of the convex solids can be stellated using Stellate. This replaces each of the polygon faces by a pyramid with the polygon as its base. The user can also adjust the stellation ratio. Note that ratios less than give concave figures and that the default value of this ratio is . Geodesate triangulates five‐sided or greater polygons before projecting onto the circumscribed sphere. If the order- of the regular tessellation of each face is not given, a default value of is assumed. The default position of the sphere is with radius . Truncate and OpenTruncate act on every polygon and truncate at each vertex of the polygon. The default value of the truncation ratio is .

You can extract the polygon list in the form of a GraphicsComplex from the object produced by PolyhedronData using First. In addition, the "VertexCoordinates", "Faces", and "Edges" properties available in PolyhedronData give you the vertex coordinates, a list of graphics primitives for each face of a polyhedron, and graphics primitives for the edges of a polyhedron, respectively.

First[PolyhedronData[polyname]] | give the GraphicsComplex representing the graphic of the named polyhedron |

PolyhedronData[polyname,"VertexCoordinates"] | |

give a list of the coordinates of the vertices of the polyhedron | |

PolyhedronData[polyname,"Faces"] | give a list of the faces of the polyhedron as Polygon primitives |

PolyhedronData[polyname,"Edges"] | give a list of the edges of the polyhedron as Line primitives |

Getting vertex, face, and edge data.