Wythoff
displays uniform polyhedra, using Wythoff’s kaleidoscopic construction to
compute the locations of the vertices. A **uniform polyhedron** is one
that looks the same at every vertex;
it has the same kinds of faces meeting at every vertex in the same sequence,
and all of its faces are regular polygons.*
More details.

By clicking on the applet, you can view 74 of the 80 possible uniform polyhedra (including single examples from each of the five infinite classes of prisms and antiprisms); six of the polyhedra are omitted because they would require an exorbitant amount of time and memory to be rendered.

* Here, we allow the definition of
a polygon to include **star polygons**, where the sides intersect
each other (e.g. a pentagram is considered to be a polygon with five equal sides and
five vertices; the points where the sides intersect are not counted as
vertices). Similarly, we allow the definition of a polyhedron to include cases
where the faces intersect. The fragments into which the faces divide each
other, if they intersect, are known as **facets**; they’re important to
the way the polyhedron looks, but the underlying definition always refers to the whole,
undivided faces. The applet colours the facets according to the shape of the face
they come from, and draws them face by face, after removing all facets that will not
be visible when the whole polyhedron is drawn.

**Reference:** Zvi Har’El,
“Uniform Solution for Uniform Polyhedra”,
*Geometriae Dedicata* **47**: 57–110, 1993.
Zvi Har’El has a terrific
web site,
which includes a PDF copy of this paper, and a Java applet that draws wireframe
versions of *all* the uniform polyhedra and their duals.
Another great web site is Roman Mäder’s
Uniform Polyhedra.

Greg Egan
science fiction writer

Applets Gallery / Wythoff / created Saturday, 19 January 2002 / revised Sunday, 20 January 2002