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.

This applet lets you 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.

Click on the applet to pause or redraw. Use the pop-up menu to choose the polyhedron, the and buttons to move through the list, or the Random button for a random choice.

* 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.

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