This paper was published in the special polyhedra issue of the Symmetry: Culture and Science journal (see contents). The journal was released in 2003, but backdated to 2000 for some reason, so if you wish to reference this article, the reference might read: Webb, Robert "Stella: Polyhedron Navigator", Symmetry: Culture and Science, Vol. 11, Nos. 1-4, p231-268, 2000 (available online at https://software3d.com/PolyNav/PolyNavigator.php) Note: While this paper still serves as a great introduction to Stella's capabilities, it was written in relation to Great Stella 2.0 in 2003. Many new features have been added since then, along with the new product Stella4D.

Table of contents

• Abstract
• 1. Introduction
• 2. Duals
• 3. Stellation
  • 3.1. What is a stellation?
  • 3.2. So what is a stellation really?
  • 3.3. The stellation diagram
  • 3.4. The cell diagram
  • 3.5. Selecting cells
  • 3.6. Stellation criteria
  • 3.7. Extended stellations
  • 3.8. Stellation examples
• 4. Nets
  • 4.1. Identifying the parts
  • 4.2. Combining parts into nets
  • 4.3. Arrangement of nets on the page
  • 4.4. Data for alternative construction
  • 4.5. 3D folding nets
  • 4.6. Using supporting facets
• 5. Faceting
• 6. Augmentation
• 7. Symmetry
  • 7.1. Finding and showing symmetries
  • 7.2. Sub-symmetry groups
• 8. Dual morphing
• 9. Other features
• 10. Conclusion
• Acknowledgements
• References

Abstract

We introduce Stella, a computer program for navigating the world of polyhedra. The user starts by choosing from a long list of built-in models, then uses advanced functions such as stellation, faceting, augmentation and excavation to explore the trillions of other possibilities. The symmetry group of any model is established and symmetries can be displayed. Nets for the physical construction of any model discovered can be printed out. Models may also be morphed into their duals in real-time on the computer screen, using one of five different techniques.

In order to explain the concepts involved, this paper also represents a whirlwind tour of some of the major ideas in polyhedral theory today.

1. Introduction

Many polyhedra are beautiful to behold. The mind is kept busy trying to grasp the various relationships that may exist in any given model. Polyhedra are a great example of a connection between art, craft and mathematics. However, the level of maths and draftsmanship required to build many models makes it somewhat prohibitive as a hobby for the less mathematically minded, and adds a considerable amount of time to the construction of any given model. Some simple polyhedra, such as the Platonic and Archimedean solids, pose no great challenge, since their faces are regular polygons which do not intersect. But the other uniform polyhedra, their duals, and stellations are more challenging. Measurements for some of these models are available on the internet or in books, most notably Wenninger's wonderful "Polyhedron Models" [28] and "Dual Models" [29], but a lot of work is still left to the reader, especially in the latter book (and since the calculations were done by hand, there are even some errors in the more complicated polyhedra).

Using a computer can make life easier. In this paper we present a computer program called Great Stella [27], or Stella for short, which allows the user to explore a great many polyhedra and print out the nets required for their physical construction. This eliminates the need for the user to perform tricky calculations, which are time consuming and error prone. It also removes the need to draft the various nets, which is also time consuming and prone to accumulation of inaccuracies. All that remains is the craft of scoring the edges, cutting out the pieces, and gluing them together. This automation brings the craft of making polyhedron models to a much wider audience.

Whether or not the user is interested in building physical models of their own, Stella lets them create and visualise models on the computer screen. The program allows the user to browse through a huge set of polyhedra and rotate them on the screen in real-time. All the uniform polyhedra are available through the list of built-in models. This set consists of the familiar Platonic, Archimedean and Kepler-Poinsot solids, an infinite array of prisms and antiprisms, and fifty-three other nonconvex polyhedra. The set, first described in its entirety by Coxeter et al. in 1954 [2], is very popular for its attractiveness (see figure 2 for an example). Skilling proved the set to be complete in 1975 [22], and introduced one new model which is also uniform, but doesn't quite classify as a true polyhedron due to more than two faces meeting at some edges. Skilling's new model is also built into Stella. The Johnson solids [10] (all convex non-uniform regular-faced polyhedra), many Stewart Toroids [25] (regular-faced non-intersecting polyhedra with genus greater than zero), and other models are also available from the list of built-in polyhedra. The program can also generate duals of all these models (see section 2), and the user has an array of polyhedral tools at their disposal for creating new models.

Tools such as stellation, faceting, augmentation and drilling increase the set of available polyhedra tremendously, and open up an avenue for creativity in the discovery of new models. A number of research papers and books have been published on stellation theory ([1], [6], [7], [8], [14], [16], [17], [20], [21], [23]), and Wenninger's books ([28], [29]) also presented a collection of stellated polyhedra. Stella represents the culmination of all this theory, and can be used to produce most of the models presented in these publications, and many more. Faceting is another very powerful tool, but has not appeared much in the literature to date ([3], [9]).

Figure 1 shows what the program looks like. The big window on the left shows the compound of five tetrahedra, a stellation of the icosahedron. The smaller windows show one of the nets required top left, the icosahedron itself top right, the stellation pattern bottom left, and the cell diagram bottom right (these terms are discussed in more detail below). The window layout is configurable.

All the images with black backgrounds in this paper are photographs of models made by the author using nets printed by Stella. Nets were printed directly onto the coloured paper used for construction. All other images were also created using Stella. The program runs on Windows 95/98/ME/NT4/2000/XP/Vista. It has also been successfully used on a Mac via the Virtual PC Windows emulator.

In section 2 of this paper we explain the concept of duals. In section 3 we describe the process of stellation and some of the theory behind it. Section 4 explains the creation and printing of nets and how stellation theory can help. Section 5 discusses faceting, the dual process of stellation. Section 6 covers augmentation, excavation and drilling. Section 7 deals with symmetry, and how sub-symmetry can be used in conjunction with the previously described operations. Section 8 presents our techniques for morphing between dual models. And finally, in section 9 we will examine some additional capabilities not already covered.

2. Duals

For any polyhedron, there exists another polyhedron which is its dual. Taking the dual of this dual polyhedron returns us to the original polyhedron again. Roughly speaking, vertices are replaced with faces, faces with vertices, and edges with new edges orthogonal to the originals.

More precisely, the technique used to create a dual is spherical reciprocation. This is done with respect to some sphere, and the choice of sphere affects the resulting dual polyhedron. Typically the centre of the sphere is placed at the centre of symmetry, if one exists (where all the axes of rotational symmetry or planes of reflective symmetry intersect). The radius of the sphere is usually the midradius of the polyhedron, if one exists (the radius of a sphere touching all edges of the model tangentially). Stella will choose an appropriate sphere for the operation.

Let's suppose r is the radius of the sphere to be used, and C its centre. If the distance from a face plane to C is d, then the distance from C to the corresponding dual vertex is r²/d. Similarly, if the distance from a vertex to C is d, then the distance from C to the plane containing the corresponding dual face is again r²/d. Additionally, the direction from C to the dual face plane or dual vertex is the same as the direction from C to the corresponding original vertex or face plane.

We refer above to the face plane rather than just the face because the distance from C should be measured to the closest point in the face's plane, which may indeed be outside the face.

From these simple forumlae, the vertices and faces of the dual model may be obtained. Note that multiple faces in the same plane will all be mapped to the same dual vertex position. Note also that if a face passes through the centre of the sphere, its dual vertex will be at infinity, since the distance to that vertex will be based on division by d which is zero.

Stella allows the user to view a polyhedron and its dual at the same time in separate windows. It may also display a compound of the two. For example, the cube and octahedron are duals. Their compound is shown in figure 3. Nets for the physical construction of these compounds are also available within Stella.

3. Stellation

3.1. What is a stellation?

The act of stellation opens up an almost endless collection of fascinating models. The number of stellations of a single polyhedron will often be measured in the trillions. The most general definition of the term says that two polyhedra are stellations of each other if their faces lie in the same set of planes. The exact definition has been somewhat debated, but the author believes all other definitions are subsets of that which is given here. More justification for this definition will be given below, but first, a little more theory is required.

Sometimes several stellations may appear to be identical. For example, consider the great dodecahedron, which is a stellation of the dodecahedron, and consists of 12 intersecting pentagons (see figure 4). When observing this model, part of each pentagon is hidden from view, internal to the solid. As a result, only five triangular regions are visible from each pentagon, so the model could also be thought of as comprising only those visible triangles. The polyhedron would then have 60 triangular faces, and no hidden parts. There are also other polyhedra that appear identical to the great dodecahedron from outside. However, within any set of stellations that are identical in appearance, there is always exactly one that consists only of the parts that are visible, or rather accessible, from outside. The stellations created in Stella are of this form, but otherwise there is no restriction on what stellations can be made, (except that they must be finite). For example, if the user were to create the great dodecahedron as a stellation of the dodecahedron, the model they would get would really consist of 60 triangular faces, not 12 pentagons, but would otherwise be identical in appearance to the true great dodecahedron. It is important to recognise however that the two polyhedra are indeed different. If the user wishes to distinguish between multiple possibilities, faceting can be used (see section 5).

So how do we find the stellations of a polyhedron? Each face of the polyhedron lies in some plane. We can think of each of these planes as carving up space, partitioning it into a collection of three-dimensional convex cells. The first plane divides space into two parts, the second divides each of these parts in two giving us four parts (unless it is parallel to the first plane), and similarly, the third gives us seven or eight parts. All these parts are infinite though, that is, none of the parts are entirely bounded by planes yet. Once we add the fourth plane things get more interesting, as now there is one finite cell, bounded by planes on all sides. This is the situation when stellating the tetrahedron. The only cell that is finite is the tetrahedron itself, and so there are no other stellations.

The dodecahedron is a more interesting case. Its 12 faces lie in 12 planes, which carve up space into 63 finite cells. Due to the symmetry of the original polyhedron, the cells fall into symmetric sets, referred to as cell types. The 63 cells fall into four cell types. The dodecahedron itself is the central cell, and the only one of its type (see figure 5a). The next type consists of 12 pentagonal pyramid cells, which sit on each face of the dodecahedron, giving rise to the small stellated dodecahedron (see figure 5b). Next are 30 tetrahedral wedges, which fit between the spikes of the previous model and form the great dodecahedron (see figure 5c). And finally, 20 dipyramids fit into the dimples of the great dodecahedron to form the great stellated dodecahedron (see figure 5d). We construct different stellations by including different combinations of cell types. A stellation is usually required to have the same rotational symmetry as the original polyhedron, so we either include all cells of a type, or none. For this reason the two are synonymous for most purposes, and from here on we shall refer to cell types simply as cells.