Making new undip words

Any undip word can be inserted anywhere in any other undip word. The resulting basket will be only 2-edge-connected at the splice. Shuffling can fix this. Shuffling is when adjacent letters that are left/right (e.g., up) or right/left (e.g., pd) switch places. Shuffling is always permissible.

Conversely, anywhere an undip word appears in another undip word (that would most commonly be ud or np) it can be deleted.

The pairs du and pn can also be deleted anywhere they appear. I call this a rewire mutation since it redirects a photon. The converse, inserting these pairs, cannot be relied upon to be viable. There is a special context where absorb/emit events can be inserted: between emit/absorb events on the same side.

The most fun way to make new undip words is to doodle them. Using graph paper make a doodle that stays on the graph paper lines. The rules of the game are that your doodle must start and end at the same place (the origin) and never go below or to the left of that point. In math terms it must not cross the x or y axis.

The path is encoded to an undip word in this way: each step up is a u, each step down is a d, each step out (i.e., to the right) is an n, each step back (i.e.,to the left) is a p.

Make a Pony-O Basket from a Word

I'll just show you how to do it here. Details on the tiny bit of particle physics involved and on the undip language itself are covered in my 15-minute video, Make a Basket from a Word.

THE BASIC MOVE: QUOIT-THROUGH-THREAD

Three Pony-O's can be interlooped in what may properly be called a Mrs. Bright's knot (a three-way version of knot #2425 in The Ashley Book of Knots,) but to make things easier for us to remember, we'll call it Quoit-through-Thread, or sometimes just QTT. Try this right now:

Step 1: Hold NEEDLE upright.

Toss QUOIT on NEEDLE.

Insert THREAD through NEEDLE.

Pull QUOIT through THREAD.

Voila!

Once you feel comfortable making Quoit-through-Thread as shown, try inserting THREAD from the left so that QUOIT ends up on the right. Another way to do this variation is to feed THREAD halfway through every time and just pull QUOIT up through the side you want it on.

In making a Pony-O basket, QUOIT always represents a photon, while NEEDLE and THREAD always represent the electron in differing energy states. In the pictures above, the photon color is green, and the two energy colors of the electron are purple and pink.



UNDIP

Undip is a language that tells stories about an electron. We'll choose this undip word,

uunddupd,

and find out what shape it makes. Along the way you'll hear the story of the electron and learn to read undip.

You need three colors of pony-O's, 4 pony-O's of each, plus one extra pony-O of any color you like. (You'll always need three colors, and of each color you will always need half the number of letters in the undip word; you will also need one extra of any color to finish off the work.)

Hopefully one of your colors will really stand out. Make that color your photon color. In the photos I'm using bright green as my photon color.

The other two colors are energy colors. The energy of an electron always changes when it emits or absorbs a photon, so our electron's energy state alternates between these two colors at each photon event.

Print or write down the undip word above in lowercase letters. Then turn the page 90 degrees so that you are reading the word from the bottom up.

To begin, choose a pony-O in one of your energy colors. It will become NEEDLE in your first Quoit-through-Thread.

The first letter in this undip word, 'u,' says that our electron emits a photon to the left. Choose a pony-O in your photon color and use it as QUOIT. Emitting a photon changes (decreases) the energy level of an electron, so use the other energy color for THREAD. Make sure QUOIT ends up on the left at this event since we are emitting a photon to the left.

If that QTT went well, you've completed your first photon event.

The next letter is also 'u.' The undip word says that our electron emits another photon to the left. Use the free end of the OLD THREAD (it has the current energy color) as NEEDLE in the next QTT. Repeat the steps above to emit another photon on the left. Remember to change your energy color at each event.

The next letter is 'n.' The undip word says that the electron emits a photon to the right. That's going to be easy: its just like the first two events except that QUOIT ends up on the right. Remember to change your energy color; and by the way, never let go of the current energy state! A good electron never loses track of its energy. For that matter, don't let the work start getting twisted up or your electron will start confusing left and right.

If you set your work down,failing to mark where you were, note that there are only two energy-colored Pony-O's, those are the first and the last. Comparing the first few events with the begining of the undip word should sort out which end is which.

The next letter 'd' is a little different. The undip word says that at this point in the story our electron absorbs a photo on the left. As you might expect, we'll be bringing a photon-colored QUOIT in from the left, but this time it is one of the photons already emitted. We find that photon by running a hand back along the left side of the work. The first loose end we come to is our QUOIT. This time, put QUOIT through THREAD as the first move; then QUOIT on NEEDLE; and then THREAD through NEEDLE. You may choose to always make QTT in this order, but I tend to fumble with it. Absorbing a photon always changes (increases) the energy level of a photon, so THREAD is a different color from NEEDLE this time as well.

The next letter is another 'd'. The undip word says that our electron absorbs another photon on the left, so repeat the previous event, again tracing your hand back along the left side of the work to find the photon to use as the quoit, and bring it in from the left.

The next letter is 'u': again we emit a photon to the left.

The next letter is 'p'. That says that our electron absorbs a photon on its right. This time we trace a hand back along the right side of the work to find the photon we will bring in from the right side. You'll find this photon is a long way back, but as always, its the first one we come to.

The last letter is 'd.' An undip word tells a story that implicitly repeats, so the story is not over just because we have come to the last letter of the word. We are headed back to where we started (and our basket will be finished) but our electron will be headed out for another lap. The 'd' says that the electron absorbs a photon on the left, so trace back along the left to find the photon and bring it in from the left as QUOIT.

At this point we want to bring the only remaining free end (notice that it is none other than our first energy state) as THREAD. This presents a problem because this THREAD is already attached to the work. So this time we toss an extra Pony-O named OUTIE over THREAD before beginning the knot. Things now proceed normally until we would normally pull THREAD through NEEDLE: we pull OUTIE through instead. OUTIE, can be tied off in an a snug overhand knot to keep things from unravelling and to mark where it all began.

Bravo! You've done it!

undip dictionaries published

I have just published unabridged, illustrated dictionaries for the undip words of 2, 4, 6, 8 and 10 letters. The illustrations, unfortunately, are just the lattice walks corresponding to the words—no basket shapes yet.

Viable Adjacent Letter Mutations

There are some two-adjacent-letter edits (substitutions, insertions, and deletions) that can always be made in an undip word and the result will be another undip word. These special edits, viable adjacent letter mutations, or val mutations, intrinsically satisfy the undip grammar rules, the larger context of the two original adjacent letters need not be consulted. The term "viable," borrowed from biology, here means that the character string that results from a val mutation will indeed be a word in the undip language (and it will therefore weave unambiguously to a specific basket shape phenotype.)

In the tables below, all of the val mutations are summarized in truth table format. The first of the original adjacent letters labels the row, the second the column. The letters that will replace the two original letters are entries in the table, with e symbolizing deletion of both letters, and a blank entry indicating that no val mutation of that type (substitution, insertion, or deletion) can be made for those two letters.

Some other two-adjacent-letter edits may be viable in certain special contexts, for example insertion of the reversed spikes du and pn , but in the worst case, full context information would be needed to determine if the insertions are viable. In this sense, what are termed here val mutations are context-free.

Gene-Splicing Baskets

Starting with undp, an undip word for tetrahedrane, 

concatenate ud, an undip word for acetylene... 

...and get undpud, or benzvalene.

Insertion of undip words works just as well as concatenation. At any place in an undip word, insert another undip word and the result is a new undip word describing a hybrid shape. The new shape is 2-edge connected where the two original shapes are tenuously joined.

The structural weakness of this 2-edge join can later be healed by viable adjacent-letter (val) mutations. Val mutations are substitutions, insertions, or deletions of two adjacent letters that can be made without reference to context because they intrinsically satisfy the undip grammar rules.

Twongs Thronged at NY Maker Faire!

Visitors thronging the Twongs Table at the New York Maker Faire, September 25-26, 2010.

That's me in orange. Some tried making the carbon chemistry models (annulenes) displayed in steel, but most went freeform.

All ages advanced to level 101: having fun.

The steel models wore their undip codes attached.

Tetrahedrane in foreground, acetylene in back.

Catalan Numbers and Baskets

How many undip words are there?

The famous Catalan numbers turn up in many places ( see "Exercises on Catalan and Related Numbers" and "Catalan Addendum" by Richard P. Stanley.) One of the classic examples is counting the number of ways 2n people seated around a table can simultaneously form n handshakes without crossings. The answer to this problem is the nth Catalan number. For example, the Catalan numbers for n = {0, 1, 2, 3, 4, 5} are {1, 1, 2, 5, 14, 42}.

Another set counted by the Catalan numbers are the number of words in the Dyck language that contain n pairs of parentheses. The Dyck language comprises all strings of open and close parentheses that follow the usual rules of arithmetic expressions. For example:

()

(()) ()()

((())) ()(())) (())() (()()) ()()()

etc.

The undip language is a shuffle of two Dyck languages: a {u,d} language and an {n,p} language. We can think of {u,d} as {(,)} and {n,p} as {[,]}, but unlike a Dyck language on two types of parentheses, in a shuffle of two Dyck languages the two types of parentheses ignore each other grammatically speaking. For example, a word such as:

( [ ) ]

is permitted.

R. Cori, S. Dulucq, and G. Viennot explain how the Catalan numbers are related to the planar hamiltonian cubic maps (the structures described by undip codes) in "Shuffle of Parenthesis Systems and Baxter Permutations," Journal of Combinatorial Theory, Series A 43, 1-22 (1986):

"...hamiltonian cubic maps are planar maps with a hamiltonian circuit in which all vertices have degree three. In such a map any vertex is incident with only one edge not in the hamiltonian polygon, this edge may be inside the polygon or outside. Thus to build a 'Hamiltonian rooted cubic map' one has to choose 2k vertices among the 2n (those incident with inside edges) then draw a planar map inside the polygon (it is easy to see that this can be done in Ck ways*) and a planar map outside."

*This is easy to see because it is essentially the handshaking problem.

A map is rooted by distinguishing a vertex and direction. In a hamiltonian circuit this is combinatorially equivalent to labeling all the vertices, since, given a direction and a place to start, we can reliably label the vertices with sequential integers. Working with a rooted map, every rotation of a pattern counts as a distinct pattern because the root occupies a different place in each rotation. Counting rooted maps exhaustively counts all of the possible undip codes even when they trivially encode the same shape by virtue of merely starting at a different location on the hamiltonian cycle.

Cori, Dulucq, and Viennot go on to show that the number of rooted hamiltonian planar cubic maps are counted by the product of two sequential Catalan numbers. For a hamiltonian circuit with 2n vertices the number of maps is the product of the nth Catalan number and the (n+1)th Catalan number.

Tthe number of undip words of length 2n for n= {1, 2, 3, 4, 5} are {2, 10, 70, 588, 5544}. This is sequence A005568 in the On-Line Encyclopedia of Integer Sequences (OEIS).

For better or worse, there are a great many more undip words than there are shapes to be described by them. Synonyms will abound.