Paulo Freire's Polyhedra

A few years ago, I became obsessed by polyhedra, in great part because of the art of M. C. Escher. I made a few models of uniform polyhedra: the Paulyhedra!
You can see some of the paulyhedra in the images below.


NEW! Polyhedra in my office!


I am building all the regular and quasi-regular polyhedra, including the star polyhedra. I might eventually also build the duals of the quasi-regular polyhedra.



Below: stained glass polyhedra in my apartment. As far as I know, these are unique!


Below: A great ditrigonal icosidodecahedron has just landed on my coffee table!!!


This is a recent, mild relapse, which is nothing compared with the severe polyhedron infatuation I went through in the 1990's (see below).
I don't have as much time now, so I try to be a bit more selective about which polyhedra I choose to build.


Fly through the paulyhedra! This is a 144 MB movie, so it takes time to download.

More pictures of the paulyhedra.

• "Regular Polytopes", by H. S. M. Coxeter. This is the book for people who want to understand regular polyhedra and regular polytopes.


• "Polyhedron Models", by Magnus Wenninger. This is the book for people that want to build polyhedra (see list of models on Wikipedia).

• Roman Maeder's uniform polyhedra page. The rotating polyhedron on top of this page was taken from this web site. See also the stellated icosahedra. Apart from a neat demonstration of the power of Mathematica, this is a beautiful way to start the day.


• George Hart's page. From his beautiful site and its links you may explore the whole polyhedral universe.
  Below I focus mostly on uniform polyhedra.

• Regular, quasi-regular and uniform polyhedra (each of these classes includes non-convex star polyhedra). See list of all uniform polyhedra by vertex figure.


• Polyhedral compounds (including a list of uniform compounds).


• A (3-D) uniform polyhedron can be related to a uniform tiling of a 2-D spherical surface, a spherical polyhedron. The football is a well-known example of a spherical polyhedron. The concept is a bit more general than that of polyhedra: as an example, the beach ball is a spherical polyhedron without an analog among the classical polyhedra.


• Obvious extensions to this concept are uniform tilings of the flat Euclidean plane (there are three regular tilings, with the plane divided in squares, triangles and hexagons) and also tilings of the hyperbolic plane (there is an infinite number of such regular tilings).


• The concept of a 2-D uniform tiling (of the sphere, plane, or hyperbolic surface) can readily be extended to a higher numer of dimensions through the concept of uniform tesselation.
  
  First, in three dimensions, it is easy to imagine a uniform tesselation of a flat 3-D space by convex polyhedra (a uniform honeycomb). Only one of these is regular, the cubic honeycomb.
  
  There are convex uniform hyperbolic honeycombs. Unlike the infinity of such configurations found in the 2-D surface, there are only 4 regular hyperbolic tesselations.
  
  The finite uniform honeycombs that divide a 3-D spherical space can be thought as the "surfaces" of 4-dimensional analogues of the uniform polyhedra, the uniform polychora. Of these there are 6 regular (convex) forms.
  
  Just as in the case of 3-D polyhedra (but interestingly, not for any higher dimensions), the concept or regular polychoron can include the regular Schäfli-Hess star polychora. These are the 4-D equivalents of the regular Kepler-Poinsot star polyhedra.


• The concept extended to an arbitrary number of dimensions is called a uniform polytope, of which the regular polytopes are a well-studied and well-understood case (see list of regular polytopes and polytope families).
  
  Apart from the uniform polyhedra and polychora (see above), we also know all the uniform polytera (5-D polytopes). Three of these are regular, none of them is a regular star polyteron. There are 3 regular polyteric tesselations of the Euclidean space and 9 regular polyteric tesselations of hyperbolic space, 4 of which are "star" (nonconvex) tesselations.
  
  We also know all the uniform polypeta (6-D polytopes). For this and all higher dimensions, only 3 regular convex polytopes are known and 1 tesselation of the Euclidean plane similar to the cubic honeycomb and no regular hyperbolic tesselations.


• Some important concepts:
  
  The Schwarz triangles, are special symmetric tilings of the spherical (or plane, or hyperbolic) surface. If their edges are (1-D) mirrors, they can be thought of as a Kaleidoscope. With one exception, uniform polyhedra can be generated by reflections in this Kaleidoscope by the Wythoff construction. Therefore, their Wythoff symbol provides a good way of listing them.
  
  Alternative notations for polytopes are the Schäfli symbol and the more general Coxeter-Dynkin diagram, which can describe many other (more general) situations where there is geometrical symmetry.