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315 Design Is Calculating gðxÞ fi gðyÞ according to the predicate. This can go from any rule—then the values for x and y are simply shapes —to rules with certain prescribed properties. Maybe y ¼ x to define identities for polygons Or perhaps y ¼ tðxÞ for a transformation t of x or y ¼ x þ tðxÞ so that once a rule is applied to a part like x it can be applied again to a part like tðxÞ. This kind of recursion is neat, but it’s not all that can happen in shapes, as the rule demonstrates with the surprise square in its right side. And single rules are also possible—then x and y are given shapes, say, this square and its diagonals
316 III Using It to Design that I draw or point to in the usual way to show what they are. But no matter what rule is defined, it applies exactly as my two formulas describe. Still, it may be worthwhile to say this for schemas directly. The schema x fi y applies to the shape C whenever an assignment g and a transformation t define a shape tðgðxÞÞ that’s part of C tðgðxÞÞ a C The result of this is another shape ðC tðgðxÞÞÞ þ tðgðyÞÞ produced by replacing tðgðxÞÞ with the shape tðgðyÞÞ. In most of the examples that follow, I’ll be using schemas. I’ll usually define them with one or two of the rules they determine under specific assignments with a short verbal description of the predicate involved, just as I’ve been doing. This avoids the need for a lot of technical details and lets me present everything visually. We’ll just agree that predicates can be filled in as necessary. And I promise not to do anything where the details are mysterious. A useful way to explain my formulas for schemas is to notice that the composition t g in the expressions tðgðxÞÞ and tðgðyÞÞ generalizes the transformations. What it means for shapes to look alike can range very broadly and be decided in all sorts of different ways. I tend to show examples that depend only on Euclidean transformations— for example, shapes are alike if they’re copies of the square either because they’re congruent or because they’re geometrically similar
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SHAPE
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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
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53 Background and count as I please
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55 Background and time. The way Eul
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57 Background them apart—a triang
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59 Background without ‘‘agreeme
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62 I What Makes It Visual? (2) What
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64 I What Makes It Visual? Whenever
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66 I What Makes It Visual? calculat
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68 I What Makes It Visual? And my r
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70 I What Makes It Visual? more tha
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72 I What Makes It Visual? Table 1
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74 I What Makes It Visual? but neve
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76 I What Makes It Visual? may be f
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78 I What Makes It Visual? formula
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80 I What Makes It Visual? But what
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82 I What Makes It Visual? they pro
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84 I What Makes It Visual? The one
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86 I What Makes It Visual? I see a
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88 I What Makes It Visual? But this
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90 I What Makes It Visual? contains
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92 I What Makes It Visual? and othe
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94 I What Makes It Visual? The lett
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96 I What Makes It Visual? and corr
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98 I What Makes It Visual? (Is hh i
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100 I What Makes It Visual? and the
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102 I What Makes It Visual? There
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104 I What Makes It Visual? to the
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106 I What Makes It Visual? Table 2
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108 I What Makes It Visual? is trie
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110 I What Makes It Visual? artifac
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112 I What Makes It Visual? and thi
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116 I What Makes It Visual? looks l
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120 I What Makes It Visual? when th
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122 I What Makes It Visual? until t
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124 I What Makes It Visual? The fir
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126 I What Makes It Visual? What Sc
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128 I What Makes It Visual? Here, g
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130 I What Makes It Visual? be desc
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132 I What Makes It Visual? An addi
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134 I What Makes It Visual? forget
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136 I What Makes It Visual? that bl
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138 I What Makes It Visual? Why is
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140 I What Makes It Visual? see wha
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142 I What Makes It Visual? to calc
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144 I What Makes It Visual? How cav
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146 I What Makes It Visual? togethe
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148 I What Makes It Visual? and six
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150 I What Makes It Visual? and its
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152 I What Makes It Visual? these h
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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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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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