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How to Calculate with Shapes by George Stiny MIT ‘An interesting question for a theory of semantic information is whether there is any equivalent for the engineer’s concept of noise. For example, if a statement can have more than one interpretation and if one meaning is understood by the hearer and another is intended by the speaker, then there is a kind of semantic noise in the communication even though the physical signals might have been transmitted perfectly.’ — George A. Miller What to do about ambiguity It’s really easy to misunderstand when there’s so much noise. Shapes are simply filled with ambiguities. They don’t have definite parts. I can divide them anyhow I like anytime I want. The shape includes two squares four triangles and indefinitely many K’s like the ones I have on my computer screen right now. There are upper case K’s ����� ����� ����� lower case k’s 1����� ���������� and their rotations and reflections. These are fixed in the symmetry group of the shape. All of its parts can be flipped and turned in eight ways And I can divide the shape in myriad myriads of other ways. I can always see something new. Nothing stops me from changing my mind and doing whatever seems expedient. These additional parts scarcely begin to show how easy it is to switch from division to division without rhyme or reason. (The fantastic number ‘myriad myriads’ is one of Coleridge’s poetic inventions. It turns a noun into an adjective. It’s another example of the uncanny knack of artists and madmen to see triangles and K’s after they draw squares. I do the trick every time I apply a rule to a shape in a computation. There’s no magic in this. I can show exactly how it works. But maybe I am looking too far ahead. I still have a question to answer.) What can I do about ambiguity? It causes misunderstanding, confusion, incoherence, and scandal. Adversaries rarely settle their disputes before they define their terms. And scientific progress depends on accurate and coherent definitions. But it’s futile trying to remove ambiguity completely with necessary facts, authoritative standards, or common sense. Ambiguity isn’t something to remove. Ambiguity is something to use. The novelty it brings makes creative design possible. The opportunities go on and on. There’s no noise, only the steady hum of invention. In this chapter, I use rules to calculate with shapes that are ambiguous. This is important for design. I don’t want to postpone computation until I have figured out what parts there are and what primitives to use — this happens in 2heuristic search, evolutionary algorithms, and optimization. Design is more than sorting through combinations of parts that come from prior analysis (how is this done?), or evaluating schemas in which divisions are already in place. I don’t have to know what shapes are, or to describe them in terms of definite units — atoms, components, constituents, primitives, simples, and so on — for them to work for me. In fact, units mostly get in the way. How I calculate tells me what parts there are. They’re evanescent. They change as rules are tried. How to make shapes without definite parts At the very least, shapes are made up of basic elements of a single kind: either points, lines, planes, or solids. I assume that these can be described with the linear relationships of coordinate geometry. My repertoire of basic elements is easily extended to include curves and exotic surfaces, especially when these are described analytically. But the results are pretty much the same whether or not I allow more kinds of basic elements. So I’ll stay with the few that show what I need to. A little later on, I say why I think straight lines alone are enough to see how shapes work. Some of the key properties of basic elements are summarized in Table 1. Points have no divisions. But lines, planes, and solids can be cut into discrete pieces — smaller line segments, triangles, and tetrahedrons — so that any two are connected in a series of pieces in which successive ones share a point, an edge, or a face. More generally, every basic element of dimension greater than zero has a distinct basic element of the same dimension embedded in it, and a boundary of other basic elements that have dimension one less. ____________________________________________________________ Table 1 Properties of Basic Elements Basic element Dimension Boundary Content Embedding point 0 none none identity line 1 two points length partial order plane 2 three or more area partial order lines solid 3 four or more volume partial order planes ____________________________________________________________ 3Shapes are formed when basic elements of a given kind are combined, and their properties follow once the embedding relation is extended to define their parts. There are conspicuous differences between shapes made up of points and shapes containing lines, planes, or solids. First, shapes with points can be made in just one way if the order in which points are located doesn’t matter, and in only finitely many ways even if it does. Distinct shapes result when different points are combined. But there are indefinitely many ways to make shapes of other kinds. Distinct shapes need not result from different combinations of lines, planes, or solids when these basic elements fuse. The shape can be made with eight lines as a pair of squares with four sides apiece, with twelve lines as four triangles with three sides apiece, or with 16 lines as two squares and four triangles. But there’s something very odd about this. If the 16 lines exist independently as units that neither fuse nor divide in combination — if they’re like points — then the outside square is visually intact when the squares or the triangles are erased. I thought the shape would disappear. And that’s not all. The shape looks the same whether the outside square is there or not. There are simply too many lines. Yet they make less than I see. I can find only four upper case K’s, and no lower case k’s. Lines are not units, and neither are other basic elements that are not points. Of course, shapes are ambiguous when they contain points and when they don’t. They can be seen in different ways depending on the parts I actually resolve (or as I show below, on how I apply rules


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MIT 4 273 - How to Calculate with Shapes

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