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
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.
How many wagons, for example, can a model locomotive pull?
Attach 10 wagons and see if the locomotive can pull them.
And if you should sit beneath a model railway bridge when the tiny engine rolls across it, pulling all those wagons, what will the sound be like?
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.
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.
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.)
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
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.
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.
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.
Remember that a turtle - walk scenario describes in words and sketches how you want the turtle to walk through a design.
Let yourself go, but be specific.
Addressing your instructions to the turtle and talking out loud may be helpful.
Let's use the sketches on the next page as the focal point of this scenario.
I have divided my turtle walk into small scenes and have given each a letter designa tion.
The previous shapes in this section were drawn with Logo procedures, but I have intentionally left the following figures in freehand form ; they are taken from my own Logo notebook.
I wanted to remind you that sketches come before Logo procedures that draw rounder circles.
The following sketches record my visual doodling about this particular problem.
But to appreciate the usefulness of sketches, you must do some yourself.
Don 't just look at my examples.
Because sketches can be effective visual aids for careful thinking, they need to be drawn carefully.
I occasionally use rulers and a compass, but not always.
Of course, the small diagrams on the next page are final sketches, not beginning ones.
Final drawings, like final Logo procedures, are the results of many preliminary studies, many of which did not "work out properly."
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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?
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
In my turtle-walk scenario I drew the pipe circle before doing anything else.
The procedure PIPEGONdraws it last.
Why did I change the order of things?
Well, I wanted to use recursion and to be able to specify the number of times recursion would happen.
I used the argument: N to take care of this.
PIPEGON's first line looks at the current value of: N; when: N"becomes zero, PIPEGON should be stopped.
It is easier to know when a procedure should be stopped than when it has just begun, and this seemed a nice place to draw the pipe, after all the rollers had been drawn.
Could you reorganize the procedure to draw the pipe before drawing any of the rollers?
Have you noticed that even the most complex designs we have done so far are drawn quite quickly?
Each is complete by the time the roller has made a single 360-degree trip around the pipe.
If the roller makes a second trip around the pipe, the design repeats exactly.
We can describe this kind of design as one that has "closed upon itself" or, more briefly, that has "closed" after one trip.
Not all designs produced by our PIPEGONmachine will close after only one trip ; some will take several trips to close, and others will require a great number of trips.
Experiments will show that altering the sizes of the roller and pipe leads to different closure patterns.
What determines the number of trips before closure occurs?
Can you calculate the trips until closure if you know the sizes of the roller and pipe?
Words that are visually descriptive, like closure, should call up a variety of images in your mind.
This elicitation of mind -images can be enormously useful in visual modeling.
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
chanting of the word.
Put it all in your notebook.
Suppose you needed to find a synonym for closure.
What would you suggest?
Any suggestion must be descriptive of all the image work we have completed.
By the way, you probably won 't find closure in a standard dictionary.
Why is this?
At the start of this chapter I mentioned that sometimes we would model machines from the real world - pipes and rollers are very real world - and other times we would model machines that aren 't so real.
Perhaps we can make one model do both real and imaginary things.
For example, can we make our PIPEGON machine draw some fantastic designs? (By the way, look at that word imaginary.
Why does it have image in it?
Can you imagine why?)
Let 's imagine a striped roller inside a pipe.
The procedure PIPEGON will generate a composite picture of this roller as it travels around the inside of a pipe.
So far, this is just like the situations viewed above.
Let's summarize what has happened in the last few pages.
We have built a Logo model that can produce a large variety of images, some of them very surprising.
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.
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.
I will show you, in the chapters to come, dozens of other examples that illustrate the ways in which visual modeling encourages the modeler to look at the world differently.
Each person 's scientist aspect encourages him to " improve his constructs by increasing his repertory, by altering them to provide better fits, and by subsuming them with subordinate constructs or systems." For Kelly, human behavior is the application of scientific methodin making sense of a particular environment.
Rather than merely responding to surroundings, people use an experimental approach to test and extend their system of personal constructs.
Each person 's goal, in Kelly 's view, is to build explanatory models that effectively explain and predict personal environments.
Kelly suggested shortcuts for improving construct systems.
Kelly 's shortcut was to encourage individuals to make their own constructs verbally explicit.
His most famous method for eliciting and verbalizing personal constructs is known as the repertory grid technique.
Using slightly different words, Kelly 's techniques encouraged individuals to build verbal models of their own constructs.
Once built, these verbal models could be analyzed in much the same way as we have analyzed our pipe -and-roller model.
Tinkering with constructs would occur naturally, and this would encourage further tinkering.
And as a result of this play, construct models might become more general and more powerful.
Kelly worked with verbal rather than visual models, but many of his ideas can be extended to the latter.
My interest, as was Kelly 's, is to suggest how to describe our inner models.
While Kelly was interested in the verbal description of models, I am interested in more graphical descriptions.
My goal is to encourage you to look at your own visual baggage.
Obviously, I need words, too, to help in my form of elicitation.
Sometimes, you may think that I rely on words too much.
Too much chat, you might say...
If you are intrigued by this very brief account of George Kelly 's work, find his book A Theory of Personality: the psychology of personal constructs (W.
W. Norton, New York, 1963). All the Kelly quotes were taken from it.
