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43 Tables, Teacher’s Desks, and Rooms ‘‘It’s chasing after a smaller fish’’ (draws eyes) ‘‘that swims up to the top.’’ And Alexandra’s younger sister, Catherine, has a nice way of seeing, and of saying what she sees. Only sometimes she has a little trouble counting. In her class at school, she learned how to use maps and plans. One had a key that showed a table her teacher’s desk and a flag, pencils, a globe, and a wastebasket When Catherine was asked how many tables were in the room
44 Introduction: Tell Me All About It she replied, ‘‘three.’’ I liked this—a desk is a table, isn’t it?—but Catherine’s teacher wasn’t pleased. I thought it might be the room—it was a bigger rectangle. Or was the small rectangle on her teacher’s desk a table? Maybe the room and the tables were desks—how many did her teacher have? Somehow, I missed the point. I was looking at it the wrong way. Flags, pencils, and globes weren’t wastebaskets. And pencils— long, thin things—couldn’t be globes—round, fat ones. This was right. Only Catherine didn’t care. She looked at the six (?) drawings in the key and saw shapes. No one told her not to. She used her eyes to count. Rectangles made tables and desks. I didn’t ask about things with circles. The room was magic—embedding was involved. It changed every time she looked. What good were standards now? But counting required identity. Why was this so important? Who said maps and plans worked that way? Were they hundred-squares? Why not the other way around, so that grids were to see? What happened to ambiguity and reasoning? (A math specialist at Catherine’s school tells me she’s amazed at how many children say her room has changed whenever they visit again. This may complicate counting or cause problems in metaphysics—it shows how artificial they are—but it doesn’t worry children. The rooms they see are the rooms they draw.) Alexandra and Catherine don’t make definite things. They draw lines to see what they get—Catherine actually says so—and blithely go on from there to try it again. It’s like an experiment, or more like painting. The outcome can’t be known beforehand because there isn’t time to look. Within the lines, Alexandra and Catherine are free to see as they please. It’s easy to find new things in new situations. They’re dealing with novelty all the time. Is this something to use? Seeing and counting aren’t the same, but there’s reasoning—yes, calculating—in both. Why not take advantage of it? To insist on counting may limit children, if they learn with their eyes and on their hands (digits). And it isn’t necessary when you can go on from what they see. Seeing is as useful as counting. It’s thinking, and there’s a lot to show. It Always Pays to Look Again This book is divided into three parts that correspond pretty closely to the three stages in my autobiography. But the parts are presented in reverse order. I need to show you more about what it means to calculate with your eyes—how seeing and counting differ when you use rules—before I can show how to calculate with shapes. Then there are the formal details to make sure that it works. And you need to know how to calculate with shapes before you try it out to design. So the three parts are these— Part I: Part II: Part III: What Makes It Visual? Seeing How It Works Using It to Design ‘‘It’’ is in the titles on purpose, and sometimes it and pronouns like it are elsewhere and have no set antecedent. It can’t be helped with shapes and rules—what you see is what you get, whatever was there before—and it can happen with words. Ambiguity is part of everything I do, so I try to use it.
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Page 8: Contents Acknowledgments ix INTRODU
Page 12 and 13: INTRODUCTION: TELL ME ALL ABOUT IT
Page 14 and 15: 3 Answer Number One—What Do You S
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Page 56 and 57: 45 It Always Pays to Look Again In
Page 58 and 59: 47 Background always more when you
Page 60 and 61: 49 Background Getting it first and
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Page 64 and 65: 53 Background and count as I please
Page 66 and 67: 55 Background and time. The way Eul
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154 I What Makes It Visual? So how
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160 II Seeing How It Works was thre
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162 II Seeing How It Works and lowe
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164 II Seeing How It Works Back to
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166 II Seeing How It Works Divide e
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178 II Seeing How It Works looks ex
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180 II Seeing How It Works meaning.
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182 II Seeing How It Works and the
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184 II Seeing How It Works Embeddin
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186 II Seeing How It Works both of
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188 II Seeing How It Works but in a
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196 II Seeing How It Works Table 7
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228 II Seeing How It Works see what
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230 II Seeing How It Works to show
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236 II Seeing How It Works But supp
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240 II Seeing How It Works with poi
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254 II Seeing How It Works Or I can
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256 II Seeing How It Works I can’
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258 II Seeing How It Works —and s
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260 II Seeing How It Works Now, no
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272 II Seeing How It Works descript
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288 II Seeing How It Works in terms
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300 II Seeing How It Works combine
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302 II Seeing How It Works In fact,
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304 II Seeing How It Works next pap
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306 II Seeing How It Works Table 12
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308 II Seeing How It Works initial
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310 II Seeing How It Works Kinemati
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312 III Using It to Design (3) desi
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314 III Using It to Design does mor
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316 III Using It to Design that I d
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318 III Using It to Design don’t
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320 III Using It to Design fix the
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322 III Using It to Design Figure 1
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328 III Using It to Design I seem t
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332 III Using It to Design Then the
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334 III Using It to Design and ‘
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338 III Using It to Design But this
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340 III Using It to Design but occa
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342 III Using It to Design numbers
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344 III Using It to Design to start
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346 III Using It to Design The rule
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348 III Using It to Design Everythi
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352 III Using It to Design Let’s
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354 III Using It to Design ever hav
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356 III Using It to Design (2) Furt
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360 III Using It to Design with the
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362 III Using It to Design are equi
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366 III Using It to Design to the p
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374 III Using It to Design easy to
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378 III Using It to Design I said I
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386 III Using It to Design take car
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388 III Using It to Design wealth o
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Notes Introduction 1. S. Pinker, Th
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393 Notes to pp. 48-50 Langer’s d
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395 Notes to pp. 51-55 19. H. L. Dr
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397 Notes to p. 156 9. C. Alexander
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399 Notes to p. 304 of lowest-level
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401 Notes to p. 305 24. The ‘‘n
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403 Notes to pp. 306-387 35. G. Sti
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405 Notes to pp. 388-389 M. A. Bode
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407 Notes to p. 389 The essence in
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410 Index Aristotle, 81, 167, 337 A
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412 Index creativity and (cont.) re
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414 Index greatest lower bound, 224
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416 Index maximal elements and basi
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418 Index Quine, W. V., 57, 58, 210
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420 Index shape grammars (cont.) an
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422 Index Van Gogh, V., 403n1 Vanto
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