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103 A Second Look at Calculating The five-pointed star contains the translation of the original triangle exactly where it should be, right over the original chevron But lines fuse and divide. Look at the star again. In addition to the triangle from the initial shape I can see four new ones that are each over different chevrons. It’s hard to see more than one triangle at a time. Still, all five are parts of the star. And when any one is moved, the other four disappear along with the corresponding chevrons—just like magic. If I apply the rule
104 I What Makes It Visual? to the triangle using a reflection, then I get the result I want. My eyes show me how to do something I never had in mind. My representation of the initial shape as a pair of independent polygons—a triangle that moves around and a chevron that stays put—doesn’t tell me what to see, and in fact it keeps me from seeing what I need. It’s hard to believe, but there’s actual proof. Calculating is creative. It conjures up a triangle and a chevron out of thin air. But the alchemy is real—it’s all in the pentacle. The rule produces the star from a triangle and chevron, and I can see both in the star. But when I apply the rule to the star, it changes what I see in a surprising way. Lines fuse whenever they combine, so that all prior divisions disappear. Then new divisions are possible—anywhere. I can find triangles, and chevrons along with them, to reconfigure the star. I don’t have to foresee anything before I begin or have any idea of what’s going to happen. And I
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( 2006 Massachusetts Institute of T
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Contents Acknowledgments ix INTRODU
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INTRODUCTION: TELL ME ALL ABOUT IT
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3 Answer Number One—What Do You S
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9 Answer Number Two—Three More Wa
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11 Answer Number Two—Three More W
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33 Trying to Be Clear Points aren
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35 Trying to Be Clear and twenty-fo
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37 Trying to Be Clear things look.
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39 Trying to Be Clear on correspond
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41 Tables, Teacher’s Desks, and R
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43 Tables, Teacher’s Desks, and R
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45 It Always Pays to Look Again In
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47 Background always more when you
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49 Background Getting it first and
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51 Background and not These are the
Page 64 and 65: 53 Background and count as I please
Page 66 and 67: 55 Background and time. The way Eul
Page 68 and 69: 57 Background them apart—a triang
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Page 73 and 74: 62 I What Makes It Visual? (2) What
Page 75 and 76: 64 I What Makes It Visual? Whenever
Page 77 and 78: 66 I What Makes It Visual? calculat
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Page 81 and 82: 70 I What Makes It Visual? more tha
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Page 89 and 90: 78 I What Makes It Visual? formula
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Page 97 and 98: 86 I What Makes It Visual? I see a
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Page 149 and 150: 138 I What Makes It Visual? Why is
Page 151 and 152: 140 I What Makes It Visual? see wha
Page 153 and 154: 142 I What Makes It Visual? to calc
Page 155 and 156: 144 I What Makes It Visual? How cav
Page 157 and 158: 146 I What Makes It Visual? togethe
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154 I What Makes It Visual? So how
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156 I What Makes It Visual? (TRI(XY
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158 I What Makes It Visual? The fir
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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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168 II Seeing How It Works There ar
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170 II Seeing How It Works while th
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172 II Seeing How It Works there ar
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174 II Seeing How It Works This has
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176 II Seeing How It Works And noti
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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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190 II Seeing How It Works It’s e
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192 II Seeing How It Works of one i
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194 II Seeing How It Works how it w
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196 II Seeing How It Works Table 7
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198 II Seeing How It Works of these
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200 II Seeing How It Works maximal
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202 II Seeing How It Works They’r
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204 II Seeing How It Works So, bðA
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206 II Seeing How It Works made up
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208 II Seeing How It Works boundary
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210 II Seeing How It Works First, t
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212 II Seeing How It Works topology
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214 II Seeing How It Works This has
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216 II Seeing How It Works where po
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218 II Seeing How It Works Operatio
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220 II Seeing How It Works is part
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222 II Seeing How It Works In parti
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224 II Seeing How It Works suggest
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226 II Seeing How It Works intricat
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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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232 II Seeing How It Works It’s a
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234 II Seeing How It Works that can
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236 II Seeing How It Works But supp
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238 II Seeing How It Works And toge
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240 II Seeing How It Works with poi
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242 II Seeing How It Works It’s f
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244 II Seeing How It Works to limit
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246 II Seeing How It Works and two
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248 II Seeing How It Works And the
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250 II Seeing How It Works (inducti
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252 II Seeing How It Works Two seco
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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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262 II Seeing How It Works up of se
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264 II Seeing How It Works These co
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266 II Seeing How It Works Why? The
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268 II Seeing How It Works and its
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270 II Seeing How It Works square f
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272 II Seeing How It Works descript
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274 II Seeing How It Works produce
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276 II Seeing How It Works points.
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278 II Seeing How It Works x fi y t
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280 II Seeing How It Works Indeterm
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282 II Seeing How It Works Or I can
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284 II Seeing How It Works applies
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286 II Seeing How It Works the same
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288 II Seeing How It Works in terms
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290 II Seeing How It Works It’s e
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292 II Seeing How It Works If a rul
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294 II Seeing How It Works But then
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296 II Seeing How It Works And, cer
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298 II Seeing How It Works are all
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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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324 III Using It to Design —divid
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326 III Using It to Design Perhaps
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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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358 III Using It to Design that wor
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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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364 III Using It to Design and in f
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366 III Using It to Design to the p
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368 III Using It to Design rules th
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370 III Using It to Design x fi bð
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372 III Using It to Design These we
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374 III Using It to Design easy to
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376 III Using It to Design and spir
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378 III Using It to Design I said I
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380 III Using It to Design include
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382 III Using It to Design x fi x i
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384 III Using It to Design It’s g
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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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