Undip as a sculptural language

I'm going to talk about a sculptural construction toy I've designed.

This image shows a close-up of some of the pieces used in the toy. As you see, the pieces are all the same shape in three different colors. They are made out of polypropylene plastic about a half millimeter thick. (Actually, I die-cut them out of Office Depot report covers using a homebrew steel-rule die.)



The pieces of the toy interweave, three-at-a-time, to form a Y-shaped structure. This is an unconventional way to put a construction toy together, and learning to make this three-way join is a big part of the learning curve with this toy.



My wife and I had the fun of showing this toy at World Maker Faire New York a couple of weeks ago. Maker Faire is a show of things that people make themselves sponsored by the publisher of Make magazine. At Maker Faire, there are usually a number of tech-art projects, but also lots of crafts and homebrew electronics.

Our booth was in the "Young Makers" tent. The kids in this photo learned how to make the three-way join. From there the same skill can be used repeatedly to add on to what you've already made, and to make baskets in free-play.

Our booth was actually entitled "Make a Basket from a Word," and the ambition was to move on to a more sophisticated kind of play with the older kids and adults.



The "Word" mentioned in our title is written in a made-up language called undip. Undip uses a four-letter alphabet {u, n, d, p}. I'm going to skip over what we were actually teaching at our booth—how a weaver can interpret an undip word to make a basket—here I want to talk about the language itself, which is a language describing sculptural shapes. You'll have to trust me that a weaver could read the undip words in these captions and weave the baskets shown above them.

English is a natural language, so there is no rule you can rely on to tell whether a given sequence of letters really spells a word in English—you just have to know the language.

Undip is an algebraic language. That means the question of whether a given sequence of letters spells a word in undip is settled by whether or not the sequence can be generated by repeatedly applying a set of rewriting rules.

The two rewriting rules for undip are really simple.

The first rule is that we can insert ud or np anywhere we like in an undip word and the result will be another undip word.

In generating the words of an algebraic language, the only acceptable starting point is the empty word. The empty word is really just a blank space, but it is represented, when necessary, by the Greek letter epsilon. The empty word is considered to be a word in undip (and every other algebraic language.) Confronted with the empty word, the weaver weaves nothing.

Part of the appeal of an algebraic language is this Genesis-like origin. This is a glimpse of the modern, bottom-up, combinatorial esthetic of today's mathematics. Sad to say, it is antithetical to the esthetic of the old mathematics that kids are still being taught in school.

In this sequence, we start with the empty word—itself a word in undip—and choose to invoke the first rewriting rule to insert ud as a suffix, thereby generating the undip word ud. Weaving ud makes the little basket shown.



We can now apply the same rewriting rule again, this time we'll choose to insert np as a suffix. The resulting undip word is udnp. Weaving udnp makes the slightly larger basket shown.

The only other rewriting rule in undip is this one: wherever a left-letter (u or d) is next to a right-letter (n or p) they can switch places.

Having generated udnp, we've got an opportunity to invoke the second rewriting rule because d, a left letter, is next to n, a right letter. We choose to switch their places, generating undp. Weaving undp makes a little basket shaped like a tetrahedron. The tetrahedron is the first of the famed platonic solids.

It is interesting that we can describe platonic solids in undip (a few more letters will suffice to describe the cube or the dodecahedron,) but it is even more interesting that less familiar, less symmetrical—but nonetheless sculpturally interesting shapes—are actually more fundamental.



Inserting two more letters, we can make the four shapes above in variety of ways. Exhausting all possibilities, we can generate 70 different undip words that are 6-letters in length. Weaving all 70, we are surprised to discover that between them they only describe these four shapes. Unfamiliar are they not? Yet, from one perspective they are all more fundamental than the cube.

I'll end with photos of some of the young weaving champions at Maker Faire.



This gentleman wove uunddp from the word and dubbed it barrel.



This gentleman wove uundpd from the word and dubbed it light bulb.



After struggling initially, this young lady persevered and mastered undip better than anyone. She wove two 10-letter baskets (the longest words we had brought along.)