Polyhedra: Polyhedra Interactive
A website devoted to the enumeration of polyhedra and their properties. The Polyhedra Interactive website seeks to collect and catalog the properties of all interesting three-dimensional solids, and makes use of the symbolic computational power of Mathematica to enumerate their mathematical characteristics such as surface area, volume, edge lengths, and so on.
[http://www.wolfram.com/]
Polyhedra: Virtual Reality Polyhedra
Welcome to this collection of thousands of virtual reality polyhedra for you to explore.
Your web browser must be set up for viewing virtual reality models.
[http://www.georgehart.com/]
Polyhedra: All 80 Polyhedra
Uniform polyhedra consist of regular faces and congruent vertices. Allowing for non-convex faces and vertex figures, there are 75 such polyhedra, as well as 2 infinite families of prisms and antiprisms. A recently discovered uniform way of computing their vertex coordinates [Harel93] is the basis for a program to display all of these solids, among which are many beautiful and stunning shapes.
[http://www.mathconsult.ch/]
Polyhedra: Free Paper Polyhedra Models
Polyhedra are beautiful 3-D geometrical figures that have fascinated philosophers, mathematicians and artists for millennia. On this site are more than eighty paper models available for free.
[http://www.korthalsaltes.com/]
Polyhedra: Downloadable Polyhedra Templates
Here are templates for making paper models for each of the 5 Platonic solids and the 13 Archimedean semi-regular polyhedra. You are free to use them for any non-commercial purpose, as long as the copyright notice on each page is retained.
[http://isotropic.org/]
Polyhedra: Archimedian Solids
If we continue to require that all vertices be indentical and that the solid be convex, but we remove the requirement that only one kind of regular polygon be used, the family of solids that results is called the Archimedean Solids (also called the semiregular solids), of which there are 13.
[http://home.teleport.com/]
Polyhedra: Johnson Solids
Here is a list of all 92 Johnson solids. The numbering follows the sequence in his paper, listed in the references.
[http://www.georgehart.com/]
Polyhedra: Johnson Solids
The Johnson solids are the convex polyhedra having regular faces and equal edge lengths (with the exception of the completely regular Platonic solids, the "semiregular" Archimedean solids, and the two infinite families of prisms and antiprisms). There are 28 simple (i.e., cannot be dissected into two other regular-faced polyhedra by a plane) regular-faced polyhedra in addition to the prisms and antiprisms (Zalgaller 1969), and Johnson (1966) proposed and Zalgaller (1969) proved that there exist exactly 92 Johnson solids in all.
[http://mathworld.wolfram.com/]
Polyhedra: Kepler-Poinsot Solids
If we do not require polyhedra to be convex, we can find four more regular solids. As in the Platonic solids, these solids have identical regular polygons for all their faces, and the same number of faces meet at each vertex. What is new is that we allow for a notion of "going around twice" which results in faces which intersect each other.
[http://www.georgehart.com/]
Polyhedra: Kepler-Poinsot Solids
One of the first people in modern times to study polygons, polyhedra, and crystals was the astronomer Johannes Kepler, who discovered the two solids above. In each case, faces of a dodecahedron are extended outward into a star. Both are stellated dodecahedra; they are called "small" and "great" to distinguish them.
[http://www.uwgb.edu/]
Polyhedra: Platonic Solids
The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids (Steinhaus 1983, pp. 252-256): the cube, dodecahedron, icosahedron, octahedron, and tetrahedron, as was proved by Euclid in the last proposition of the Elements. The Platonic solids are sometimes also called "cosmic figures" (Cromwell 1997), although this term is sometimes used to refer collectively to both the Platonic solids and Kepler-Poinsot solids (Coxeter 1973).
[http://mathworld.wolfram.com/]
Polyhedra: Platonic Solids
The so-called Platonic Solids are regular polyhedra. “Polyhedra” is a Greek word meaning “many faces.” There are five of these, and they are characterized by the fact that each face is a regular polygon, that is, a straight-sided figure with equal sides and equal angles.
[http://www.mathacademy.com/]
Polyhedra: Platonic Solids
This is where it all starts... What solids are convex (ball-shaped), have vertices that are all alike, and each face is a single kind of regular polygon? It turns out that there are only five solids that satisfy these constraints, collectively known as the Platonic Solids.
[http://home.teleport.com/]
Polyhedra: Prism and Antiprism
Triangular Antiprism, Square Antiprism, Pentagonal Antiprism,Hexagonal Antiprism,Octagonal Antiprism, Nonagonal Antiprism, Decagonal Antiprism and more.
[http://library.wolfram.com/]
Polyhedra: Pyramids, Dipyramids, & Trapezohedra
A particularly popular polyhedron is the pyramid. If we restrict ourselves to regular polygons for faces, there are three possible pyramids: the triangle-based tetrahedron, the square pyramid, and the pentagonal pyramid. Being bounded by regular polygons, these last two fall within the class of Johnson solids. One interesting property of pyramids is that like the tetrahedron, their duals are also pyramids. (Incidentally, the Egyptian pyramids have square bases but the triangular side faces are not quite equilateral; they are very close to half a golden rhombus.)
[http://www.georgehart.com/]
Polyhedra: Pyramids
A polyhedron with one face (known as the "base") a polygon and all the other faces triangles meeting at a common vertex (known as the "apex"). A right pyramid is a pyramid for which the line joining the centroid of the base and the apex is perpendicular to the base. A regular pyramid is a pyramid whose bases is a regular polygon. An n-gonal regular pyramid (denoted ) having equilateral triangles as sides is possible only for n = 3, 4, 5. These correspond to the tetrahedron, square pyramid, and pentagonal pyramid, respectively.
[http://mathworld.wolfram.com/]
Polyhedra: Uniform Polyhedra
This list follows the order in the paper "Uniform Compounds of Uniform Polyhedra" by J. Skilling, cited in the references. Many of these 75 compounds (plus duals) have never been made in paper, and their pictures appear nowhere else but here. The notations T, O, I, or iP indicate that the compound has tetrahedral, octahedral, icosahedral, or i-sided prism symmetry, respectively. I haven't finished the prisms yet.
[http://www.georgehart.com/]
Polyhedra: Zonohedron
A convex polyhedron whose faces all possess a central symmetry (Coxeter 1973, pp. 27-30). Equivalently, a convex polyhedron whose faces are parallel-sided -gons.
[http://mathworld.wolfram.com/]
Polyhedra: Zonohedron
A zonohedron (by one restrictive definition) is a convex polyhedron all of whose faces are parallelograms.
[http://www.georgehart.com/]
Polyhedra: Other Polyhedra
In addition to the above, there are innumerable other polyhedra. There are 92 additional convex polyhedra with regular faces for which all of the vertices are not of the same type. There are also concave polyhedra with regular faces, and of course polyhedra with irregular faces. R. Buckminster Fuller used triangles to approximate spheres to create his geodesic polyhedra.
[http://www.mathartfun.com/]
Polyhedra: Other Polyhedra
There are five and only five of these figures, also called the Platonic Solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
[http://mathforum.org/]
