|Cubic graph (the dual)|
|Triangulation (the primal)|
|Cubic graph (the dual)|
|Triangulation (the primal)|
Twongs and twogs can make more shapes than the four-letter undip code can code.
A graph must have a Hamilton circuit in order to be undip encoded. (Note that Hamiltonicity is a property of the graph, not the embedding.) That leaves out cubic graphs with self-loops (cubic pseudographs proper) since they can never be Hamiltonian. Only cubic multigraphs can be undip coded.
Furthermore, the embedding must be in the plane (or equivalently the sphere) in order for the coded side bonds to link up predictably. On a higher genus surface, the whole surface area might be encoded in the predictable planar way, but there will remain a closed curve (i.e., the polygon of the plane model, see A Combinatorial Introduction to Topology by Michael Henle) that cannot be seamed across without encountering a surprise.
The photo above shows a digonal prism. Even though a digonal prism is a graph embedded in the sphere, it is not considered a polyhedron by most definitions. For one thing, it contains two-sided faces (digons); for another, it is only 2-edge connected. (Cutting just the two edges shown vertical in the photo would separate the graph into two components.) Note that these are properties of the graph itself, not the embedding. Graphs that are only 1-edge or 2-edge connected make weak structures but they certainly are of sculptural interest. Polyhedron or not, 'udud' codes the digonal prism nicely.
Taking the adjective "plane" to mean "embedded in the plane or sphere," the undip code can encode Hamiltonian plane cubic multigraphs.
Twongs and twogs can make 3D embeddings of cubic pseudographs.
A graph is called cubic if three edges meet at each vertex. Simple graphs permit only a single edge to link two distinct vertices, multigraphs allow more than one edge to link two distinct vertices, pseudographs also allow a vertex to link to itself (i.e., form a self-loop.) A graph is an abstraction: a set of symmetrical relations (edges) between pairs of members of a set (the set of vertices). For example, the friendships (edges) between persons (vertices) listed in a phonebook. Such an abstraction has no geometry until we make some decisions not specified in the graph itself in order to place (embed) its vertices and edges in 3D space, or on the Euclidean plane, or on some other surface or in some other space. Sometimes a given graph can be embedded in a space in fundamentally different ways, e.g., a left-handed and a right-handed version. The embedding, not the graph itself, is our guide to these important practical details. See Topics in Trivalent Graphs by Marijke van Gans for a clear mathematical exposition.
If twongs and twogs are made long enough, they can be used to realize any 3D embedding of a cubic pseudograph. The construction above is an embedding of the smallest cubic pseudograph having loops. I call it loop-loop. An embedding of the smallest cubic pseudograph without loops (thus also a multigraph) is shown below. I call this one bang-bang.
These are what I call twongs.
A twong is a helical length of wire that has been bent in four places. Twongs are made by twisting up a pair of wires (these particular ones have been twisted to a helical wavelength of about 5 wire diameters), unravelling them, cutting them to length (these have been cut to a length of 12 helical wavelengths), and then bending. I have been working by hand, so I have been limited to 9 gauge (0.14 inch diameter) steel wire and smaller. Wire is made over a vast range of diameters, so twongs can be almost any size.
The main thing about twongs is that they twine together, three at a time, to form vertices. I've made a video about twining them together. With enough twongs you can make any shape having exclusively 3-valent vertices (three edges meeting at a vertex.) The tetrahedron, cube and dodecahedron are famous 3-valent (i.e., cubic) polyhedra, but there are many more. For example, there are 14,501 isomers of the dodecahedron, that is, different shapes but all with the same number of faces, vertices, and edges as the dodecahedron, and they are likewise all 3-valent. The C-60 buckyball is also all trivalent. Its isomers are effectively uncountable, being in excess of 10 to the 22nd. Yikes!
The are so many cubic polyhedra that, given enough vertices, we can use one to approximate any simple closed (i.e. genus zero) surface. The approximation is never smooth because all lengths are the same, nonetheless, there are many cases where a crinkly surface is a good enough, or where a crinkly surface can be made smooth by some mechanical process. In any case, the alternative--custom cutting every piece (I've tried it)--is a royal headache, and no custom-cut part is ever re-useable.
An astonishing thing about genus zero cubic graphs (let's just toss out the very small number that are non-Hamiltonian) is that they can be identified with words in a language having a very simple grammar. Formally the language is known as the shuffled Dyke language on two types of parentheses. Instead of parentheses we will want to use the following letters:
u, n, d, p
corresponding to (look at them tilting your head to the right) the following twiner's actions
"open left", "open right", "close left", and "close right"
The moves are as follows:
To start, pick up a very first twong and mark it with a twist-tie. This is insurance against getting lost--you can always retrace from the beginning if you know where that is. It also symbolizes that the very first twong is the work in progress. At any later vertex, at least one twong will be already part of the work in progress.
The first letter is always an "open" action, i.e., 'u' or 'n'. In an "open" action you just build a vertex--that's the same for "open left" or "open right." The difference comes when you exit the newly built vertex. To leave a twong "open (on the) left", we must build onto the right twong, Likewise to leave a twong "open (on the) right" we must build onto the left twong. That's all there is to 'u' and 'n,' they just tell us which way to go next.
Close actions require us to incorporate a previously placed twong into the current vertex. According to the letter, we are to look either to the left or the right for this previously placed twong. If there is more than one available on that side we simply choose the nearest one on that side (i.e., most recently placed). This is the same way nested parentheses close, hence the connection to parenthesis languages. Departing a close action is simple since only one of the three twongs will still have a free end.
The last vertex is always a close action. Implicitly, the first twong (the one marked by the twist tie) must be incorporated in this vertex along with the twong indicated by the final letter.
I have more about these "undip" codes in this year's Proceedings of the ISAMA.