Embedded graphs come in families of four

Cubic graph (the dual)

Triangulation (the primal)

Radial graph

Medial graph

Kagome weave

Anyam Gila

In unit weaving we are constantly dealing with families of four closely related graphs: a triangulation (the primal graph), a cubic graph (the dual graph), a medial graph, and a radial graph. Kagome (triaxial weaving) is closely related to the medial graph of a triangulation. Anyam Gila (or "mad weave"), a triple-layered version of kagome, has an outward appearance that suggests the radial graph of a triangulation.

Some facts from the Handbook of Graph Theory:

1. An embedded graph M is a medial graph of some graph G if and only if M is 4-regular and the faces can be properly 2-colored.

2. An embedded graph R is a radial graph of some graph G if and only if R is bipartite and every face is a quadrilateral.

3. The medial graph M of an embedded graph G is identical to the medial graph of the dual G*. The radial graph R of G is identical to the radial graph of the dual G*. The embedded graph and its dual are the only two graphs whose medial and radial graphs are M and R respectively.

Any surface-embedded graph can play the role of "primal," and it is guaranteed to have a dual, medial, and radial, all of which are discoverable through simple constructions called map operations. It is a convenience of speech that we identify one graph as the primal and the other as the dual: the roles could just as easily be reversed with no effect on the rest of the foursome. The same, however, is not true for the medial and radial (which are also mutually dual,) because radial graphs and medial graphs each have special characteristics as indicated in Facts 1 and 2 above. Thus, most graphs belong to only one foursome: the one formed when they are the primal (or equivalently the dual.) The few graphs that qualify as medial (or radial) graphs belong as well to a second foursome: the one where they are the medial (or radial.)

The familiar square grid makes for a special case. It is dizzyingly at once self-dual, self-medial, and self-radial. This deep symmetry of the grid pattern helps explain why 10,000 years of biaxial weaving has not gotten us very far in understanding weaving more generally.

When plain weaving is not in some artificially arranged conformation, just two threads cross at each crossing. A graph of a plain weaving (crossings being represented by vertices and threads being represented by edges) is therefore four-regular (i.e., exactly four edges meet at each vertex.) That satisfies the first requirement in Fact 1.

The relevance of the second requirement in Fact 1 can be seen from an observation by Snelson (U.S. Patent 6,739,937), that each thread in a plain weaving is a boundary between a left-handed opening and a right-handed opening. Such an arrangement is only possible if the fabric openings can be properly 2-colored (such a coloring is popularly known as a chess-coloring or checkerboard-coloring.)

The necessary and sufficient requirements for a graph to be a medial graph are thus the same requirements necessary and sufficient for a graph to specify a plain weave: the 4-regular graph specifies the two-dimensional arrangement of threads, and the 2-coloring of its faces tells us which openings are to be left-handed or right-handed. The handedness of the openings, in turn, tells us how to arrange the threads at each crossing, in the third dimension, to produce a true plain weave.

In this light, Fact 1 establishes that every medial graph describes two enantiomorphic plain weaves (one for each of the two alternate left/right color assignments.)

In the same light, the converse assertion in Fact 1 (i.e.: if M is 4-regular and the faces can be properly 2-colored, then M is a medial graph) establishes that no plane weave exists that is not described by a medial graph.

In short, weaving has found the light switch after thousands of years of fumbling in the dark:

1. We now know that every compact surface can be plain-woven-- no matter the topological complexity or the orientability of the surface. (The term "compact" merely rules out some mathematically defined surfaces that cannot be realized by any technological means. In layman's terms we can plain-weave any surface.)

2. We can plain-weave that surface in essentially any pattern (we are not limited to familiar biaxial or triaxial weaves.) Any graph, any drawing of edges and vertices we choose to put on the surface, can direct the weaving via its medial graph.

3. We know constructively how to proceed in every case.

4. We know that our construction process is universal: there exist no other plain weaves but those described by medial graphs.

We have 1-3 above from the 2009 topological proof by Akleman, Chen, Xing, and Gross (see my earlier post Proving Weaving) I believe the universality result is new.

Undip codewords as lattice walks

Any undip word can be identified with a 2D lattice walk where the allowed steps are N, S, E, W, the walk begins and ends at the origin (such a lattice walk is called a loop), and the walk is furthermore confined to the first quadrant (quarter plane).

Identify 'u' with an North step, and 'd' with an South step (mnemonically 'up' and 'down'). Similarly identify 'n' with an East step and 'p' with a West step.

The "balanced parentheses" rules for undip words (where |x| denotes the number of occurences of the letter 'x') are:

|u| = |d|

|n| = |p| ,

These rules guarantee that total South steps equal total North steps, and total West steps equal total East steps; thus, if a codeword-designated walk begins at the origin, it will end at the origin.

The "nested parentheses" rules guarantee that the walk stays in the first quadrant. The rules are:

|u| >= |d|

|n| >= |p|

for any predicate of an undip word.

In other words, at no point in the spelling of an undip word will there have been more d's than u's, or more p's than n's. On the lattice this means that the walk can never go south or west of the origin-- it is confined to the NorthEast quadrant.

Undip codes and Euler's Formula

Any graph embedded in the sphere must obey Euler's formula:

F - E + V = 2 ,

where F, E, and V are the number of faces, edges, and vertices. Since every hamiltonian cubic graph embedded in the sphere has a valid undip codeword, a reasonable question is how (and whether) having a valid undip codeword can guarantee compliance with Euler's Formula.

Euler's formula is a bit simpler for cubic graphs. Cutting a cubic graph at every mid-edge, leaves a disconnected set of vertices, each still holding on to three half-edges. Thus,

E/V = 3/2 .

That fixed E/V ratio reduces Euler's formula for a cubic graph to

F - V/2 = 2 ,

or

F = V/2 + 2 .

In a valid the undip codeword there is one letter for each vertex. For every open letter, 'u' and 'n,' there is a matching close letter, 'd' and 'p'. Therefore a count of the occurrences of d and p, |d, p|, will count one half of the vertices:

|d,p| = V/2 .

Euler's formula then becomes

F = |d,p| + 2 .

It is easy to see that in weaving a valid undip codeword this rule is always obeyed. Each close letter, 'd' or 'p', completes a face, until we come to the final close letter. The final close letter also completes the hamilton cycle and thereby closes an additional face on both the left and the right-- that provides the final "+ 2" faces.

What can undip codewords code?

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.

What can twongs make?

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.

twongs

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.