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231 How Rules Work When I Calculate The square gets smaller or the square gets bigger. And this rule moves a triangle to the right, while its inverse does so to the left. But inverses needn’t be distinct when rules are defined in the schema x fi tðxÞ. An identity A fi A and its inverse are evidently the same, only this is trivial. Of greater interest, the rule that rotates a square at a corner, and its inverse are the same. You can check this in general using the transformation t A t 1 , where t A is a transformation in the symmetry group of x in x fi tðxÞ, and t Aðt 1 ðxÞÞ ¼ tðxÞ.
232 II Seeing How It Works It’s also easy to see that t A t 1 is in the symmetry group of x þ tðxÞ. For the two squares in the rule t A t 1 defines the axis of reflection Still, in general, there’s more to check to ensure that there are right answers. More generally, rules with symmetric sums have inverses that aren’t the same—for example, this one When the squares are reflected, the left one is the left one and not the right one. There’s no flip. The precise details of applying rules and saying how they work are straightforward. It’s mostly using your eyes and saying what they see. A rule A fi B applies to a shape C in two easy stages. (1) Find a transformation t that makes the shape A part of C. This picks out some part of C that looks like A. In symbols it looks like this tðAÞ a C (2) Subtract the transformation of A from C, and then add the same transformation of the shape B. This replaces the part of C that’s like A with another part that looks like B. And once again, it’s nice to show it in symbols ðC tðAÞÞ þ tðBÞ
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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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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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