These pages will list nearly all the models that B.M. Stewart described in his book "Adventures Among the Toroids: A Study of Orientable Polyhedra with Regular Faces" (second edition). This site is not intended to replace that book, but to supplement it. I will not be describing all the details of how and why he generated each of the models, or their properties, etc. That information is all contained in his book. This site is here to allow readers of the book to see 3D VRML models of the figures, to get a better understanding of them. If you already have the book, go on in, otherwise, please go and buy the book.
I will start with a chapter listing of models, and plan to add other methods of lookup in the future (sorted by convex hull, by genus, by symmetry group etc.)
This column lists the name of the model, using Stewart's nomenclature. The name of the model is linked to a VRML file that will allow you to explore the model in 3 dimensions.
If you have a copy of Robert Webb's Stella application, the appropriate .STEL files can be downloaded by clicking the ".Stel" next to the name of the model. Many models are best understood by opening them up and looking inside. Stella lets you hide faces, to get a better view of the interior.
This column displays the genus of the model. Since we are dealing with non-self-intersecting models, often the genus is easily determined by visual inspection. But for multi-storied models, often the interior of the model is harder to explore, so I have listed this property of the model for easy reference.
The abbreviations of properties used throughout the book deserve some description in a convenient location for reference. It is a bit long, but here goes:
- (R) - Regular-Faced. All faces of the polyhedron must be regular polygons, and he excluded star-polygons from this category. (R) seems to be the most important criteria for his exploration. There is very little in the entire book about polyhedra with irregular faces. Once you break this criterion, your search area is too big to be manageable.
- (A) - Aplanar. Adjacent faces of the polyhedron cannot be coplanar. While this might seem to be an arbitrary condition (if you allow convex and concave edges, why rule out coplanar?), but it is actually a strengthening of the (R) condition. Two coplanar adjacent triangles could also be replaced by a rhombus, which is not a regular-faced figure. There are few examples of models that are not (A). He often went out of his way to fix models that were not (A).
- (D) - Disjoint Interiors. The interiors of all the faces must be disjoint. In other words, no two faces may intersect, except at the edges. This feature is so important to him that it is generally assumed, and not explicitly mentioned for each model. If you allow faces to intersect, you allow non-orientable figures in, as well as other toroids with no visible holes. This would also greatly expand the field of exploration. Note: Although Stewart did not generally list (D) in his property descriptions, I like to list it explicitly. Mostly because I have some associates that are quite fond of self-intersecting polyhedra, and I want to remind them that they are being explicitly excluded from these explorations.
- (T) - Tunneled. All excavations of the polyhedra must go towards increasing the genus of the figure. There should be no unnecessary excavations, unless required to achieve property (A). This is one of his weaker restrictions. There are numerous examples that are not (T), particularly the ringed models. Technically speaking, a polyhedron is (T) if there is a set of polyhedra that can be removed from a starting polyhedron, and each removed polyhedron is either a tunnel or a rod. He specifically excludes models that include "worm-holes" (a tunnel that loops back on itself, and therefore only has one opening instead of two or more).
- (C) - Convex. Obviously toroidal polyhedra cannot be convex, so this property is not used to describe them. Often it is used to describe some component of the model, such as the tunnel for a particular model may be (C).
- (Q) - Quasi-Convex. The nearest you can get to convex in a toroidal model is to have all of the edges of the convex-hull of the toroid also be edges of the toroid. In appearance, this makes it look like you started off with a convex polyhedron, and dug holes through it, starting and ending with a single face. Although this is a very important criterion for Stewart, there were numerous times when he departed from this objective, particularly the ringed models.
- (Q') - Weak Quasi-Convexity. All edges in the Convex Hull of the polyhedron is in the Union of edges of that polyhedron. Same as (Q), except the edge-length of the polygons in the hull can be a multiple of the edge lengths of the original polygons. e.g. a 3x3x1 box can be seen as a conglomeration of 9 1x1x1 cubes. So if we remove the central smaller cubes, the hull is still a Quasi-convex polyhedron, but the edges of length 3 are the union of three edges of the smaller cubes. Note that this figure is (Q') but not (Q) or (Q") or even (A)
- (Q") - Regular-Faced Quasi-Convexity. All the faces of the Convex Hull are regular faces. Most of the (Q) models that Stewart explored are also (Q"), and on page 79 he actually restricts his search to (Q") models, thinking that there were few models that were (Q) and not (Q"), and those few were rather uninteresting. We have since found a few quite interesting models in this class, that I think would have interested him. I will try to put together a page on this soon.
To be as encompassing as possible, I would say that a "Stewart Toroid" would be (R)(A) and (D), with special interest on the models with (Q) and (T).
While the vast majority of his book was concerned with (Q) models, there was quite a bit of interest in what I refer to as "ringed" models. These are conglomerations of regular faced polyhedra that form a ring (or multiple rings), and therefore have a higher genus, e.g. a ring of 8 octahedra. They are generally not (Q), and most of them are found in chapter 7. The more interesting ones are those that the rings aren't coplanar with other rings, but loop back around and form figures with the appearance of a cube, or a rhombic dodecahedron, or rhombic triacontahedron. A lot of this most interesting area, was due in large part to a student of Stewart's, Kurt Schmucker. I would propose calling these ringed models Stewart-Schmucker Toroids. I have some interesting extensions of his research in this area, which again will be appearing on a page as soon as I can get it organized.
Pages with Diagram:
When a visual depiction of the model was included in the book, I tried to include the page number, at least for the first occurrence. I have probably not included all references to all drawings, so feel free to let me know if I missed one.
Pages with Description:
Whenever there was a description of the model, how it was constructed, or why it was investigated, I have tried to include the page numbers, at least for the first occurrence. Again, please let me know of any particularly good descriptive material that refers to a model, if I have omitted it.
I have tried to keep my notes column brief, since this is not intended to replace the great descriptions in the book. If you want to find out all about the model, see the pages listed. This is often just a helpful note about some salient feature of the model, to help as a reminder.
These pages are dedicated to B.M. Stewart and his very thorough and extensive book. They are intended to be used by readers of his book to supplement the illustrations in his book, using three dimensional models. His book has tons of information that I do not intend to represent here, and I highly encourage anyone interested in this subject to buy the book. All page numbers will refer to the second edition. If you like my illustrations, please let me know. And if you are interested in a particular figure being modeled, please let me know and I will do what I can.
When I said that I modeled nearly all the figures in his book, obviously I did not model all the permutations that he described, but rather a representative sample of those figures. It would be an interesting project to come up with interactive VRML models of some of the exercises, that would allow the user to add and rotate the different components that had some degree of flexibility, and therefore allow the user to create any particular permutation of that model they desired (e.g. p. 194 - K5 / eZ4(E5) with Quintillions of distinct possibilities)
Generation of VRML models was expedited tremendously by the use of Robert Webb's Stella application. The appropriate .STEL files will be referenced along with the .wrl files.
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Link to this page as http://Polyhedra.Doskey.com/Stewart00.html