do some typesetting, esp. of equations.

ch_2_visual_modelling.md

@@ -1,12 +1,11 @@
-# Chapter 2: Visual Modeling
+## Chapter 2
+# Visual Modeling
 
-> In the case of miniatures, in contrast to what happens when we try to
+> In the case of miniatures, in contrast to what happens when we try to understand an object or living creature of real dimensions, knowledge of the whole precedes knowledge of the parts.
-understand an object or living creature of real dimensions, knowledge of the
-whole precedes knowledge of the parts.
 And even if this is an illusion, the point of the procedure is to create or sustain the illusion, which gratifies the intelligence and gives rise to a sense of pleasure which can already be called a es the tic.
 > Claude Levi-Strauss, _The Savage Mind_ 
 
-## Models
+### Models
 Most physical models are miniatures, smaller versions of something else.
 Model railroad engines are good examples, and many of us have had pleasant experiences with them.
 They can be picked up and looked at from any angle, and they can be experimented with, too.
@@ -16,19 +15,9 @@ And if you should sit beneath a model railway bridge when the tiny engine rolls
 Will it be like the real thing?
 Listen carefully and you will experience a double thrill: an excitement that comes from using a model to hear how a full-sized locomotive might sound; an excitement that comesfrom simply playing, on your own terms, with a miniaturized piece of the world.
 
-  Logo
+I believe that this kind of play, because it encourages us to look more closely at our world, is very useful enjoyment.
-An
+In addition, I am convinced that the clarity of vision developed by such play is best pursued by involving ourselves, not in just the manipulation of models but in their design and construction as well.
-. But
+
-important
-first
-...
-pause in the narrative
-Visual Modeling
- I believe that this kind of play, because it encourages us to look more
-closely at our world, is very useful enjoyment.
-In addition, I am convinced that
-the clarity of vision developed by such play is best pursued by involving
-ourselves, not in just the manipulation of models but in their design and construction as well.
 This book is about a special kind of modeling that explores patterns and visual images.
 Sometimes we will create designs using visual models of machines that are very much from the real world.
 Other times, images will be produced by more abstract or imaginary machinery.
@@ -37,142 +26,139 @@ Logo is the raw material, but this book is not about Logo.
 In fact, you must try hard not to let Logo get in the way of your model building craft.
 Models first.
 OK?
+
 OK.
 No more introduction.
-Let 's build an image - producing machine using
+Let 's build an image-producing machine using Logo.
-  What
+But first...
-language.
+
+### An important pause in the narrative
+What comes next requires that you have a fair understanding of the Logo language.
 But what, you ask, does fair mean?
 If you haven't yet looked at the material in Chapter 1, now is the time.
 Chapter 1 will let you compare your current knowledge of Logo mechanics with what you will need to build the visual model described in the next section.
-Don 't just scan Chapter 1; go through
+Don't just scan Chapter 1; go through the examples in the text, and try out the exercises at the chapter's end.
-the examples in the text, and try out the exercises at the chapter 's end.
+
 Even if you already know some Logo, a review probably would be helpful.
 Chapter 1 describes much of the geometry and trigonometry used throughout the book, so now is a good time to review that material.
-Try to apply some geometry
+Try to apply some geometry to a few of my exercises.
-to a few of my exercises.
 In addition, Chapter 1 will introduce you to my Logo-talking style; maybe you should get used to that style right away.
 Be sure to spend some time working on the centered polygon assignment.
-This exercise com - bines a review of Logo, the introduction of turtle walks, and simple geometry.
+This exercise combines a review of Logo, the introduction of turtle walks, and simple geometry.
-comes
-next requires that you have a fair understanding of the Logo
- 35
 
-  Chapter2
 Go back and take at peek at Chapter 1 right now.
 Have a good read and take your time with it.
-If you are a real Logo high -flier, test your flair by doing the exercises at the end of Chapter I, before you glance through the hints given there.
+If you are a real Logo high-flier, test your flair by doing the exercises at the end of Chapter 1, before you glance through the hints given there.
 After having a go at these problems, compare your Logo style with mine.
-On to modeling
+
-I have decided to begin this work with something concrete, something selected
+### On to modeling
-from the world of things.
+I have decided to begin this work with something concrete, something selected from the world of things.
-We can do some abstract modeling of ideas or emotions
+We can do some abstract modeling of ideas or emotions later on.
-later on.
+The example that follows is a description of a device that I vaguely remember seeing described in an old Scientific American magazine.
-The example that follows is a description of a device that I vaguely
+I've called it a pipe- and-roller machine.
-remember seeing described in an old Scientific American magazine.
-I've called it
-a pipe - and -roller machine.
 I never built the thing, but I always wanted to see how it worked.
-Pipes and rollers
+
-Imagine that we are wandering about a construction site where there are useful
+### Pipes and rollers
-bits and pieces lying about, free for the taking.
+Imagine that we are wandering about a construction site where there are useful bits and pieces lying about, free for the taking.
-While this wandering will be
+While this wandering will be done only in our minds, the images seen there are based on impressions from past, real excursions.
-done only in our minds, the images seen there are based on impressions from past, real excursions.
+
 We will need some short sections of plastic pipe, the kind used for plumbing.
 Try to find as many different-diametered pipes as you can.
 Here is a sketch of what you might have picked up so far.
-     36
 
-  Visual Modeling
+FIGURE 1: Pipes
+
 Next, let's assemble a collection of wooden dowels of various diameters, from very small diameters to very large ones.
 Dowels, or rounded wooden pegs, are used to join together adjacent parts by fitting tightly into two corresponding holes.
 Dowels are used by cabinetmakers to assemble fine pieces of furniture when nails or screws would be unsightly; and large dowel-pegs are still used to fit together wooden beams when aesthetics are more important than cost.
 Call the dowels that you have assembled "rollers"; you will see why in just a minute.
-    Here is a sketch of my dowel collection. (Sketches are models, too.)
+Here is a sketch of my dowel collection. 
-Now, imagine that one section of pipe is floating in the air at eye level ; one end of the pipe is clearly visible to us, and the pipe 's length is parallel to the
+(Sketches are models, too.)
-      Assemblin
-19thepipe-and-roller machine
-  37
 
-  Chapter2
+FIGURE 2: Dowels
- ground.
+
+### Assembling the pipe-and-roller machine
+Now, imagine that one section of pipe is floating in the air at eye level; one end of the pipe is clearly visible to us, and the pipe's length is parallel to the ground.
 Now, hold one of the dowel-rollers parallel to the length of the pipe and place it on top of the pipe.
-   Next, imagine rolling the dowel around the circumference of the pipe until
+
-it arrives back at its starting position at the top of the pipe.
+FIGURE 3: Dowel-on-pipe
-You will have to
+
-hold the roller very carefully so that it doesn't slip on the pipe but rolls nicely
+Next, imagine rolling the dowel around the circumference of the pipe until it arrives back at its starting position at the top of the pipe.
-in contact with the pipe.
+You will have to hold the roller very carefully so that it doesn't slip on the pipe but rolls nicely in contact with the pipe.
 If you roll the dowel around the pipe in a counterclockwise direction, the dowel will also turn counterclockwise.
--'-
+
-"',.."
+FIGURES 4 & 5: Dowel rotations around the pipe
+
 The dowel would turn in a clockwise direction if the rolling-about-the-pipe was also in a clockwise direction.
 Note the two motions of the roller: the roller goes around the pipe as it turns around it own center.
 The two centers, the center of the pipe and the center of the roller, will feature in all our calculations.
- 38
 
-  VisualModeling
 Now that you have an image of the physical machine in your mind's eye, I can ask you to begin manipulating the parts of the image.
 Imagine playing with this model in your mind.
 What will the roller "look like" as it rolls around the pipe?
 Can you draw a picture of it?
-Or better still, can you create a Logo model of this roller / pipe machine that can illustrate the motions for us?
+Or better still, can you create a Logo model of this roller/pipe machine that can _illustrate_ the motions for us?
+
 Imagine, for example, that we glue an arrow onto the end of the roller.
+
+FIGURE 6: Arrow on roller
+
 What pattern will the tip of this arrow trace out as the roller moves around the pipe?
 Imagine a series of photographs taken at regular intervals as the roller moves around the pipe.
 Let's construct a Logo machine that will work in this photographic way.
 In other words, let's build a Logo machine to model the physical pipe-and-roller events visually.
-Roller talk
+
-At first glance, this exercise looks pretty difficult.! certainly more complicated than the centered polygon problem discussed in Chapter 1. But, if we could just break this problem down into smaller, more manageable parts, as we broke the polygon problem down, some of the complexity might vanish.
+### Roller talk
+At first glance, this exercise looks pretty difficult, certainly more complicated than the centered polygon problem discussed in Chapter 1. 
+But, if we could just break this problem down into smaller, more manageable parts, as we broke the polygon problem down, some of the complexity might vanish.
 So let 's set about doing just that.
+
 First, we can talk about the geometry of rolling cylinders to see what we already know.
 Then, we can do a turtle-walk scenario- with sketches and words.
 Finally, Logo will act as the glue to stick all the individual parts together.
-     39
 
-  Chapter 2
 Look at the figure on the left below.
-We see the roller at the top of the pipe, position (a), and then rolled counterclockwise around the pipe to position (b). How much has the roller turned between points (a) and (b)? The dark bands in the figure indicate the contact surfaces between the roller and the pipe.
+We see the roller at the top of the pipe, position (a), and then rolled counterclockwise around the pipe to position (b). 
+How much has the roller turned between points (a) and (b)? 
+The dark bands in the figure indicate the contact surfaces between the roller and the pipe.
 If there is no slipping, the length of the band on the roller must equal the length of the band of the pipe.
 Why?
-a-
+
-(b)(.
+FIGURES 7 & 8: Rotation diagrams.
+
 We could think of this rolling in another way.
 See the figure on the right above.
-Imagine that the roller stays fixed at position (a). The roller now rotates at (a) while the pipe is rotated- clockwise- underneath it from (b) to (a). In this alternative view, the roller also turns in a counterclockwise manner, by the distance indicated by the dark band.
+Imagine that the roller stays fixed at position (a). 
+The roller now rotates at (a) while the pipe is rotated- clockwise- underneath it from (b) to (a). 
+In this alternative view, the roller also turns in a counterclockwise manner, by the distance indicated by the dark band.
 Here, as in the figure on the left, the length of the band on the pipe must equal the length of the band on the roller.
+
 How can we calculate the lengths of these bands?
 First, note that the bands can be described as segments of a circle, that is, some fraction of the total circumference of a circle.
 Well, what do we know about circumferences?
-Any circle 's circumference, C, equals 2nR, where R is the radius of the circle.
+Any circle's circumference, C, equals 2πR, where R is the radius of the circle.
 Now look at the figure on the next page.
-    40
 
-  -'--
+FIGURE 9: Rotation diagram with labelled angles.
+
 The band on the pipe is some fraction of the circumference of the pipe.
-This
+This fraction is the angle θ (theta), measured in degrees, divided by 360, the total number of degrees in a circle.
-fraction is the angle 8 ( theta ), measured in degrees, divided by 360, the total
+The length of the band on the pipe is, therefore, 2πR<sub>p</sub>θ/360, where R<sub>p</sub> is the radius of the pipe.
-number of degrees in a circle.
+The same thinking provides the length of the band on the roller: 2πR<sub>r</sub>φ/360. 
-The length of the band on the pipe is, therefore,
+Here, R<sub>r</sub> is the radius of the roller.
-27tRp8 / 360, where Rp is the radius of the pipe.
+
-The same thinking provides the
-length of the band on the roller: 27tRr4 >/ 360. Here, Rr is the radius of the roller.
 What next?
-We can set these two expressions equal to each other, since the
+We can set these two expressions equal to each other, since the physical dimensions they represent are equal to each other in length.
-physical dimensions they represent are equal to each other in length.
+Then we can rearrange terms to express φ (phi), the degree rotation of the roller, in terms of θ, the degree distance between (a) and (b). 
-Then we
+Here it is:
-can rearrange terms to express <I> ( phi ), the degree rotation of the roller, in terms
+
-of 8, the degree distance between (a ) and (b ). Here it is:
+> φ = θR<sub>p/R<sub>r</sub>.
-CP=8Rp /Rr
+
 This rotation expression is very convenient.
-If we know how many times we want
+If we know how many times we want to photograph the roller on its way around the pipe, we can calculate 8, the degree distance between stoppings, by dividing 360 by the number of stoppings.
-to photograph the roller on its way around the pipe, we can calculate 8, the
+And knowing the radius of the pipe and of the roller, we can use the tidy expression above to calculate <P, the relative rotation of the roller from one stopping point to the next.
-degree distance between stoppings, by dividing 360 by the number of stoppings.
-And knowing the radius of the pipe and of the roller, we can use the tidy expression above to calculate <P, the relative rotation of the roller from one
-stopping point to the next.
 So much for the roller talk.
-But before we go on to the turtle walk, will you
+But before we go on to the turtle walk, will you admit that you know more about this problem than you thought you knew at the outset?
-admit that you know more about this problem than you thought you knew at the
-outset?
 Listen: breaking big problems down into smaller ones makes getting
 Visual Modeling
 ( a. )
@@ -226,27 +212,22 @@ I""
 ~ MOU ld2uoI O JeldE4~
 Aue al pue
 S o
-: a1l1 E sau
+: a1l1 E sau a1.I~ a ~t?l ~snln ~unOUle UO !~e~Ol dq ~d~elrole.UMElp !a aqa4 P~lnoJMD aD u~oa4~ A~loldiD liaD1l1!1!qSJrDqaM ~Sallpu!{Jn~S~~OU!lEl!iM!elJplJa1nUD3l lO:J a]a14l1UO~ ~!~Uele!aMaElalapal {SU:g!!UIalrWa!rJllEol2aEle.I allna!Ol}OA{}~O~uaUlo!~{!UM}a!s!oUlaO !~1!:us:o)d!arauoelS!~l{'sJ1e}0x!1aGu ~ lanOl
-a1.I~ a ~t?l ~snln ~unOUle UO !~e~Ol dq ~d~elrole.UMElp !a aqa4 P~lnoJMD aD u~oa4~ A~loldiD liaD1l1!1!qSJrDqaM ~Sallpu!{Jn~S~~OU!lEl!iM!elJplJa1nUD3l lO:J a]a14l1UO~ ~!~Uele!aMaElalapal {SU:g!!UIalrWa!rJllEol2aEle.I allna!Ol}OA{}~O~uaUlo!~{!UM}a!s!oUlaO !~1!:us:o)d!arauoelS!~l{'sJ1e}0x!1aGu ~ lanOl
 G
 MElp
-Jo
+Jo s1uawn
-s1uawn
 PlnoM
 Sn !pEl
 ~ lE
 09 a1l1
 OM }
-Sa ){E1 NO ~ N:) 1U !od.IEl:JuaJ E punolE suo ~ Alod
+Sa ){E1 NO ~ N:) 1U !od.IEl:JuaJ E punolE suo ~ Alod sell ldIIOl dq ~ dSne; ) dH.
-sell ldIIOl dq ~ dSne ; ) dH.
+MO paAour 1aA 10U sell laIIol al snUl noA ()~u!od~V.()UO !a all } Jo.Ia }uaJ all } O
-MO paAour 1aA 10U sell laIIol al
-snUl noA ()~u!od~V.()UO !a all } Jo.Ia }uaJ all } O
 !JapUO!1!sod 1UallnJ
 ~ ~UnOUle dUlOS pdUln ~ dq II !M s.aNlm~a(lZl1)-uo.09!~!sSond!pEuloJpoala-u~OUa~;A)lol(dZ~)SUnOpap!!p1t!s!?osld-JoOz:uraOEplJl-!a;I2J)ult?!!1MJlet?~lpSMOMlloEN a,ll~{1M+elOd~S'!2ua;!)JuetJ?~sale!PnOAl{J!l{MU!UO aa
 OU II ! M MOlle dq ~ ' ( Z ) OZ NO~N:):aldwExa !1.I.L 'lallol a1.I~ Jo la ~Ua ;) a1.I~ I(Z) uo !~!sod o ~ aAOW put? dn uad a1.I~ ~ ;)!d '~xa N MOlle lad.uO~Alod a1l1{1saqS!!l1JeSlWn{J1lpue!J'1dEn1l11laI{J2l!e!lJ1S 2U!1(U1)!oudo !U}M!esolpd 'adaq!duaetJ{}JMoOll.eIa}uaJal{a1t{'}aJOua}H){Jeq.
 l
-'av ~ PUE 'UMElP aq 01 UO ~Alod a1l1 Jo sapuop!s Jo~Iuslealqwnu{ 1! 'UOa1!l11!so'dN 2u !1le1S al {1 urOlJ
+'av ~ PUE 'UMElP aq 01 UO ~Alod a1l1 Jo sapuop!s Jo~Iuslealqwnu{ 1! 'UOa1!l11!so'dN 2u !1le1S al {1 urOlJ aa
-aa
 '
 lEln ~ al SMElp
 1E1l1
@@ -258,21 +239,17 @@ lEln ~ al SMElp
 The roller has moved from position (2) to position (3) by rotating about its own center.
 We used the symbol <I>to indicate this rotation.
 The angle <I>is measured relative to the dotted line linking the centers of the pipe and roller: (1) to (3).
-You have arrived at position (3), pointing along the axis (1) to (3). If you
+You have arrived at position (3), pointing along the axis (1) to (3). If you now turn left by angle <I>= 8Rp/ Rr, you will be facing in the correct direction to draw the arrow.
-now turn left by angle <I>= 8Rp/ Rr, you will be facing in the correct direction to draw the arrow.
 Draw the roller circle, too.
 DiagramG: Gettingbacktothecenteorfthe12i12e
 Turn right by 4>, pick up the pen, and move back down to (1).
 DiagramH: Pre~aringforthenextrollersto~~ing~osition
-Get ready to draw the next roller image: turn left by angle 8 and move out to position ( 4 ). The roller rotation angle at the point ( 4 ) is again measured relative
+Get ready to draw the next roller image: turn left by angle 8 and move out to position ( 4 ). The roller rotation angle at the point ( 4 ) is again measured relative to the dotted line linking points (1) to (4). Why?
-to the dotted line linking points (1) to (4). Why?
 Angle cj> at position ( 4 ) equals 28Rp / Rr Why 28? Because cj> must be cal- culated relative to the starting position, and the roller has moved 28 degrees
 from the starting position ( 2 ). Turtle: you may now turn left arrow, turn right by cj>, and go back down to the center of the pipe.
 Angle cj> at the next stopping position ( 5 ) is not shown in the will be 38Rp / Rr Why?
 Ora w a few diagrams to convince yourself back and look at the figures on page 40 for some help.
-A turtle walk transfonned into Logo procedures
+A turtle walk transfonned into Logo procedures by cj>, draw the diagrams.
-by cj>, draw the
-diagrams.
 But it of all this.
 Go
 To start, recall that we have to glue an arrow onto the face of the roller.
@@ -288,8 +265,7 @@ LT 140 FD.2*:L BK.2*:L
 RT 280 FD.2*:L BK.2*:L LT 140 BK:L
 PU
 END
-PIPEGONs
+PIPEGONs length
-length
 :L.
 The length of
 Let's call the procedure that will carry out this turtle walk PIPEGON.
@@ -299,8 +275,7 @@ The length of
 Here is the list of arguments so far:
 : RP, the radius of the pipe: RR, the radius of the roller
 : THETA, the angle distance between stopping places: N, the number of stopping places
-Let's add one more,: CUM, that will keep track of the total of the angle
+Let's add one more,: CUM, that will keep track of the total of the angle turned from the starting roller position.
-turned from the starting roller position.
 We can now write the first line of PIPEGON:
 TO PIPEGON:RP:RR:THETA:CUM:N 46
 
@@ -324,16 +299,11 @@ Visual Modeling
 Supporting procedures
 TO CNGON.
 : N
-: RAD
+: RAD an N- position
-an N - position
 ,
 ,
-To draw current
+To draw current sided polygon
-sided
+RAD is the centered turtle circle
-polygon
-RAD is the
-centered
-turtle circle
 ' s that
 .
 .
@@ -342,46 +312,32 @@ radius
 , would pass through all of the polygon 's vertices.
 .
 , See Chapter 1 for a full PU FD: RAD
-RT180 -(90*(:N-2)/:N)PD
+RT180-(90*(:N-2)/:N)PD description of
-description
-of
 CNGON
 .
 NGON: N ( 2 *: RAD * SIN ( 180 LT 180- (90*(:N-2)/:N)
 /: N
 TO NGON:N
 END
-Some
+Some pip
-pip
 ; To
 ; drawn
 ; is
-REPEAT
+REPEAT draw given
-draw
+. egon from
-given
+: N by
-. egon
+[ FD productions
-from
-: N
-by
-[ FD
-productions
 PU BK END
 : RAD
 PD
-: EDGE
+: EDGE an
-an
 I typed
 PIPEGON
-N - sided polygon
+N- sided polygon the turtle 's current position, and its length
-the turtle 's current position, and its length
 EDGE.
 :EDGE RT360/:N]
 60 30 60 0 6. This models, in a visual way, the rolling of a
-. The
+. The first edge will be
-first
-edge
-will
-be
 47
 
 Chapter2
@@ -406,8 +362,7 @@ One of the pleasures of modeling is playing with the little model you have 48
 Visual Modeling
 built.
 Let's fiddle with PIPEGON's parts to seewhat happens.
-I will show you
+I will show you only a few things to give you the idea.
-only a few things to give you the idea.
 Let's start with some different argument values.
 Here is the portrait of PIP EGaN 60 30 2 0 180.
 But I don 't like all those circles.
@@ -422,10 +377,7 @@ LT:CUM*:RP/:RR ARROW: RR* 1. 5
 (***) <--- RT:CUM*:RP/:RR
 PU BK:RP +:RR
 LT: THETA
-A.PIPEGON:RP:RR:THETA(:CUM+:THETA) (:N-l)<--- END
+A.PIPEGON:RP:RR:THETA(:CUM+:THETA) (:N-l)<--- END new name removed newname
-new name
-removed
-newname
 line
 49
 
@@ -455,8 +407,7 @@ END
 new name
 <--- 2- sided
 (: COM +: THETA
-CNGON
+CNGON installed here
-installed here
 50
 
 And here is a portrait of s.PIPEGON 60 30 2 0 180.
@@ -489,8 +440,7 @@ FLASH (***)
 RT:CUM *:RP /:RR PU BK:RP +:RR LT: THETA
 L.
 PIPEGON: RP
-  <- - - new
+<--- new name and arg
-name and arg
 PD
 :RR:L:THETA (:CUM+:THETA) <---
 (:N-l) new
@@ -498,8 +448,7 @@ name and arg
 END
 51
 
-  Chapter 2 Here
+Chapter 2 Here is the flash portrait of:
-is the flash portrait of:
 L.PIPEGON 60 30 40 2 0 180
 ~.~*"'*.~-
 *--.
@@ -565,8 +514,7 @@ In each of the following exercises, I will stress the importance of words.
 We must talk a lot in conjunction with sketching a lot.
 Take a few minutes here to think visually about the word closure.
 Say " closure ": what images does it bring to mind?
-Tell the turtle to " hurry up and bring a design to a close." Jot down, or sketch, the image ideas elicited in your own mind by the
+Tell the turtle to " hurry up and bring a design to a close." Jot down, or sketch, the image ideas elicited in your own mind by the chanting of the word.
-chanting of the word.
 Put it all in your notebook.
 Suppose you needed to find a synonym for closure.
 What would you suggest?
@@ -605,16 +553,12 @@ We have built a Logo model that can produce a large variety of images, some of t
 But more important, we have seen how the act of modeling can facilitate the visual exploration of some of the characteristics of a real-world machine.
 Once we began to tinker with our model, we wanted to tinker further.
 Some of our designs posed difficult questions whose answers were not at all obvious.Closurewassuchaquestion.Weneededtodomoretinkering andmore experimenting to come to grips with what was going on.
-Could we have predicted the directions this tinkering and experimentation
+Could we have predicted the directions this tinkering and experimentation would take before we started?
-would take before we started?
 I don't think so.
-Once we begin to model parts of
+Once we begin to model parts of our world, the act of modeling takes on a life of its own.
-our world, the act of modeling takes on a life of its own.
+I think we have touched what Levi-Strauss said happens when one plays with miniatures.
-I think we have touched
-what Levi -Strauss said happens when one plays with miniatures.
 Model play
-"gratifies the intelligence and gives rise to a sense of pleasure which can
+"gratifies the intelligence and gives rise to a sense of pleasure which can already be called aesthetic." I hope you would also describe visual modeling as fun.
-already be called aesthetic." I hope you would also describe visual modeling as fun.
 Why pipes and rollers?
 I started with this particular machine because I was interested in it.
 I could have used any number of alternative illustrations, but this was my own direction.
@@ -675,13 +619,10 @@ Do the generalizations hold up after more experimenting?
 Whether you feel successful in this activity or not, find the following book in your local library: E.
 H. Lockwood, A Book of Curves (Cambridge University Press, Cambridge, 1963).
 This book may help you think about closure.
-It may also suggest other
+It may also suggest other image ideas to think about visually.
-image ideas to think about visually.
+Don 't worry too much about the book 's math.
-Don 't worry too much about the book 's
-math.
 Look at the diagrams, and read the chapter names.
-Listen to these
+Listen to these chapters: cardioids, limac;ons, astroids, right strophoids, tractrices, roulettes,
-chapters: cardioids, limac;ons, astroids, right strophoids, tractrices, roulettes,
 and glissettes.
 What images do these names dredge up?
 Sketch them before you find the book.
@@ -709,15 +650,13 @@ Visual Modeling
 Exercis2.e4
 Do you know the term " kinetic sculpture "? If not, you can guess what they are, or rather what they do?
 Kinetic sculptures are mechanical or electronic sculpture- machines that move, clank, or flash.
-Some even squirt water (for example, the
+Some even squirt water (for example, the wonderful kinetic fountain designed by Nikki de Saint-Phalle and Jean Tinguely opposite the Pompidou Center in Paris).
-wonderful kinetic fountain designed by Nikki de Saint-Phalle and Jean Tinguely opposite the Pompidou Center in Paris).
 61
 
 Chapter2
 Carnival rides are a special class of kinetic sculptures.
 They may not seem suitable for art gallery installation, but I have seen films of amusement park rides included in exhibitions.
-The French sculptor Jean Tinguely does kinetic
+The French sculptor Jean Tinguely does kinetic sculpture on a more modest scale.
-sculpture on a more modest scale.
 Below is a reproduction of his " Homage a Marcel Duchamp," done in 1960. It is human scale, about 5 feet high.
 Design and build a kinetic sculpture using Logo.
 You might start by trying to model the Tinguely machine.