Uniform Polyhedra and other interesting solids

Making models of polyhedra is an interesting and educational hands-on project. The resulting polyhedra have a satisfying elegance and symmetry.

 A solid is a three-dimensional geometric object; a polyhedron (plural polyhedra) is a solid made up of polygons (faces) joined at their edges.  The edges of the polygons meet at vertices (singular vertex); if faces intersect, the edges and vertices formed are called false edges and vertices. A convex polyhedron is made up of convex polygons,  has no false edges or vertices, and the interior angles between adjacent faces are all less than a straight angle (180 degrees). A compound polyhedron is one that can be separated into two or more polyhedra, with faces and edges assigned to component polyhedra without reusing or omitting any.



A uniform polyhedron is made up of regular polygons arranged identically around each (true) vertex.  A regular polyhedron is made up of congruent (identical) polygons arranged identically around the vertices.  There are five convex regular (or Platonic) polyhedra: Tetrahedron (made up of four triangles), Hexahedron or Cube (made up of six squares), Octahedron (made up of eight triangles), Dodecahedron (made up of twelve pentagons), and Icosahedron (made up of twenty triangles); there are also four non-convex (Kepler-Poinsot) polyhedra: Great Dodecahedron (made up of twelve intersecting pentagons), Small Stellated Dodecahedron (made up of twelve pentagrams -- five-pointed stars -- around twelve vertices), Great Stellated Dodecahedron made up of twelve pentagrams around twenty vertices), and Great Icosahedron (made up of twenty intersecting triangles).

A semi-regular polyhedron is a uniform polyhedron made up of different types of regular polygons.  There are two infinite classes of convex semi-regular polyhedra: prisms have two identical parallel polygons connected by squares, and anti-prisms have two identical parallel polygons connected by triangles.  The convex semi-regular polyhedra that are not part of either of these classes are called Archimedean polyhedra.

The dual of a semi-regular polyhedron is made up of identical polygons arranged in different ways around the vertices (and is not a uniform polyhedron).  The duals of prisms (dipyramids) are made up of  triangles extending from both sides of a regular polygon (which is not a face) to a point on either side of the central polygon's plane, and the duals of anti-prisms (trapezohedra) are made of kites similarly arranged around a non-planar cycle of equal-length edges.

A deltahedron is made up of identical equilateral triangles. The tetrahedron, octahedron, icosahedron, and great icosahedron are deltahedra.

 These are models I built of the regular polyhedra, some uniform polyhedra, and other interesting polyhedra:

The regular (Platonic and Keplar-Poinsot) polyhedra

Tetrahedron	Dodecahedron	Great Dodecahedron
Hexahedron (Cube)	Icosahedron	Great Stellated Dodecahedron
Octahedron	Small Stellated Dodecahedron	Great Icosahedron

Some (convex and non-convex) uniform polyhedra and dual polyhedra

Truncated Icosahedron	Small Triambic Icosahedron	Small Icosihemidodecahedron
Rhombic Dodecahedron	Tetrahemihexahedron
Rhombic Triacontahedron	Octahemioctahedron

Some deltahedra

Triangular Dipyramid	Triaugmented Triangular Prism	Stella Octangula
Pentagonal Dipyramid	Gyroelongated Square Dipyramid
Snub Disphenoid	Excavated Dodecahedron

And finally, because of the recent interest in the Transformer movies, here is a transforming polyhedron:  A ring/necklace of non-uniform tetrahedra (triangular pyramids)

that transform into a rhombic dodecahedron.

Polyhedron Models by Magnus J. Wenninger (Cambridge University Press, 1974) has detailed instructions for building models of uniform polyhedra and stellations of the platonic polyhedra and the cuboctahedron.  Mathenatical Models by H. Martin Cundy and A. P. Rollett (Second Edition, Oxford University Press, 1977) has nets for the Platonic, Kepler-Poinsot, and Archimedean polyhedra and some other interesting polyhedra.  The Great Icosahedron and Small Triambic Icosahedron (First Stellation of the Icosahedron) were constructed from paper nets at www.korthalsaltes.com [having assembled them from paper, I would suggest using a light card stock such as used for index cards].  The instructions for assembling the tetrahedron ring / rhombic dodecahedron are at http://mathforum.org/pcmi/hstp/resources/dodeca/index.html [I used three colors of paper repeating in pairs in the same sequence (white, white, yellow, yellow, green, green); index card stock would not crinkle as much].