The definition I’m using for a polyhedra is that they are:

1) composed of regular polygons

2) modeled on one of the Platonic or Archimedean polyhedra ie (regular and semiregular polyhedra)

3) are discontinuous (have positive Genus in other words have spaces/gaps between polygons and or bounded and unbounded regions)

4) not solids, but rather surfaces

Some are though not all are

1) not convex

2) some are composed (constructed from) subunits (dissections) of the Platonic or Archimedean polyhedra

3) others are built from structures such as cups, crowns, rings/stars and or belts.

In my mind, these are Toric Polyhedra

As Branko Grunbaum in his article, ‘Are Your Polyhedra the Same as My Polyhedra?’, has pointed out ““Polyhedron” means different things to different people” and I realize mine is not the standard definition, however, I do not think these things are written in stone. We can redefine what a polyhedron is as we see fit after all we (humans) invented (not discovered) the idea. There are many different possible definitions as has been pointed out by Grunbaum. This is one among them. Polyhedron literally means ‘many faces’ and by that criteria alone these qualify in the loosest sense of the word.

The concept of a polyhedron has undergone change over the centuries. According to Peter Cromwell, in his book ‘Polyhedra’, prior to the 19th century polyhedra were exclusively thought of as solids. The more modern conception is that they are hollow structures bounded by 2d polygons. I have simply taken that definition and explored some of the possibilities of such structures with a combination of bounded and unbounded regions that share symmetries with the Platonic and Archimedean polyhedra.

Here is a section that appears at Wikipedia talk section for Polyhedron:

## “Polyhedron – Surface or Solid?[edit]

Isn’t the word “polyhedron, -hedra” intended to describe a Surface (and its associated Area) rather than a Solid (and its Volume)71.116.178.98 09:30, 8 May 2007 (UTC)? Even the Greek root of the word seems to suggest this. Shouldn’t those famous figures be called “Platonic Surfaces” rather than “Platonic Solids”? Is there a word-ending which would convey this idea? “Polyhedroid” might work, except that the “-oid” ending as currently used seems to convey the feeling of “almost” or “-like”. Any comments or opinions? Ed Frank

- The oldest known polyhedra are definitely solid – solid stone! Modern thought is that the Greeks considered polyhedra as solids – I have no idea if this is correct. Leonardo da Vinci made skeletal (stick) models, and Kepler drew the seven regular polyhedra he knew of as thin-walled “surfaces”. Most recently, Grünbaum has advocated a theory based on partially-ordered pairs of points. There are also purely abstract definitions.
**So in this respect a “polyhedron” is pretty much what you want it to be at the time.**Some of the above stuff is written up on the page, but I think it needs the rest adding. Maybe I’ll find the time. Hopefully someone else will (grin). BTW the term “polyhedroid” has been used for various polyhedron-like things from time to time, but never seems to have stuck. HTH — Steelpillow 17:28, 8 May 2007 (UTC)

- Polyhedra which enclose a subset of space have a volume, or could also be said to divide space into an interior and exterior volume. The infinite skew polyhedrons can divide space into two disjoint volumes, both infinite. Self-intersectingstar polyhedrons also divide space if they are orientable, although definition of interior is ambiguous. Lastly nonorientable polyhedra have only one side and don’t enclose any volume. Lastly I guess you can consider – is an edge “two points” or “a line segment” – it has an interior length at least. And similarly are polyhedron faces “solid” or “empty”, but they have to be solid to have a surface area. As well, the Convex uniform honeycombs are made of polyhedron cells, which have interior volumes. I guess there’s no right answer – if you say a polyhedron is a surface, then you could say its faces are only perimeter, and then it also has no “surface area” and is just a wire frame, or if edges have no interior length, then it’s just a set of points in space! Tom Ruen 00:59, 9 May 2007 (UTC)”

From a Materials Science point of view my interest in these structures (polyhedra) would best be characterized in terms of porosity as it related to molecular (nano) polyhedra.

As representation of molecules/molecular polyhedra, the lines of the models represent chemical bonds and the vertices represent the location of would be atoms.

I have recently (as of 2/15/16) changed my view and understanding of these polyhedra. I now think that they belong to Topological Geometry and can be understood as follows:

- The ring-star polyhedra are a conceptual blend of three elements;

a. a polygonal polyhedra- or one that is constructed from or with two dimensional polygons.

b. the rings and stars are two dimensional toroids. I will catalog these structures in the web page I’m now working on entitled: Topological Geometry

c. these two dimensional toroids are being substituted for regular polygons that would otherwise be used to construct a polygonal polyhedron based on one of the Archimedean or Platonic ployhedra.

2. The Crown polyhedra represent a class of toroids in which the polygons of which they are composed are situated on the edges of one the Archimedean or Platonic polyhedra in a closed network. they represent a subtractive manipulation on ring polyhedra in the formation of crowns of which they are constructed.

3. The Exploded polyhedra are a class of polyhedra that result from the insertion of polygons between other polygons that make up one of the Archimedean or Platonic polyhedra. As such they, also are toric polyhedra.

4. The Foil polyhedra are generalizations of ring polyhedra in the sense that the rings are concave and convex not just convex. They are also a variety of toric polyhedra.

5. The Crinkled polyhedra are related to ring polyhedra as well in that they are an additive manipulation of the ring structures creating a crinkled ring or star that in turn is then the structural unit for building polyhedra based on the Archimedean and Platonics. Like the others they are toric.

From these five hybrids can be created. A systematic catalog of these hybrid toric polyhedra would involve a 5 x 5 table with each class running in rows and columns would be conceptually blended with the other in a binary blend. The result is that there are 25 kinds of hybrid toric plyhedra possible. For example;

- Exploded Foil polyhedra
- Foil Exploded polyhedra
- Crown Exploded Polyhedra
- Exploded Crown Polyhedra

Etc.