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329 What the Thinking Eye Sees But be careful. Inverses may not do what you expect. For example, the inverse x þ xA fi x of the schema x fi x þ xA doesn’t work for quadrilaterals and pentagons. When I use the rules it defines, I’m left with an ugly gap Yet beauty is in the eye of the beholder—perhaps this outcome is good for something else. And I can always go on from anything I have with shapes and rules. Working through Klee’s drawing in this way isn’t meant to recapitulate what he did—no one knows for sure, and it’s no good relying on what he says. Rather, I’m trying to show that schemas are up to Klee’s kind of designs. The rules I need are included in a handful of schemas that are easy to use. And the rules themselves are easy to apply to shapes in terms of embedding and transformations. Embedding ensures there’s no letter of the law—I can calculate with triangles and see quadrilaterals and pentagons—and general transformations allow for as much irregularity as I want—parametric variation of any kind is possible. Rules and freedom go hand in hand—the one implies the other. I’m staking the plausibility of the metaphor ‘‘design is calculating’’ as a heuristic and equivalence squarely on my success with schemas, especially as I’m able to use them more and more to calculate with shapes and to make designs in different styles from different places and times. This is another kind of freedom, and it provides a very practical kind of proof that calculating works. There’s seeing and doing, and that’s all it takes when rules apply to shapes. I guess ‘‘design is calculating’’ really is more of a thesis than a theorem. The evidence adds up, but not to logical certainty, just to empirical belief.
330 III Using It to Design Figure 3 Chinese Lattice Designs—Seeing What You Do There are myriad ways schemas can be used to produce designs. I especially like the Chinese lattice designs that fill window frames. Most of these form regular patterns of the kind shown in figure 3. They’re taken from Daniel Sheets Dye’s A Grammar of Chinese Lattice. It’s a wonderful and extensive catalogue that’s hard to put down. It’s a joy to see. And in fact, like The Thinking Eye, it holds the material for endless experiments with your eyes.
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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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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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272 II Seeing How It Works descript
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276 II Seeing How It Works points.
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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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