# Stellations of the Icosahedron

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

## Coxeter's Classification and Notation

If we extend one face of an icosahedron outward, the remaining faces intersect it as shown below. Only the opposite parallel face does not intersect it. Each of the polygons enclosed by face intersections is a possible face of a stellation. We can identify 13 distinct faces as shown.

### Cells and Shells

Certain combinations of faces naturally occur together to enclose solids, which Coxeter termed cells. The different cells and their constituent faces are shown below. The cells in turn form layers or shells which are denoted by letter. Upper case letters denote shells and lower case letters the component cells. Some cells exist in both left- and right-handed form.

Coxeter used primes to denote the interior faces of cells and italics to denote chiral (left- or right-handed) faces. These pages use a slightly different notation: r or l for right- or left-handed, respectively. The conventions are purely arbitrary; we could define left or right in the opposite sense and get a mirror image of the stellation.

### Shells

The innermost first few layers of faces completely enclose space. therefore, there are no cells for shells A-D (they could be defined, but their interior faces would be completely enclosed and hidden). In levels E-G, it is possible to have stellations that expose the interior faces of the cells. Shell H, the outermost stellation, again encloses space completely. Coxeter classified the stellations according to the shells and cells that make up each.

Layers E, F and G each cosist of two types of cell. Coxeter used subscripts to differentiate them. Type 1 cells meet at or enclose a three-fold symmetry axis and type 2 cells meet at or enclose a five-fold symmetry axis.

Many of the g-cell stellations are all but indistinguishable from the outside and differ only in the form of their interior tunnels. Coxeter provided interior views of many stellations. Some other g-cell stellations can be viewed as an outer shell of g-cells enclosing and touching some simpler stellation, and new views of some of these stellations are presented here.

## Images of the Stellations

Created 16 February 1998, Last Update 3 March 1999