Structure and Interpretation of Computer Programs

            

  

            

  

Figure 2.10:  Images produced by the 

wave

 painter, with respect to four different frames. The

frames, shown with dotted lines, are not part of the images.

Figure 2.10:  Images produced by the 

wave

 painter, with respect to four different frames. The frames,

shown with dotted lines, are not part of the images.

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Figure 2.11:  Images of William Barton Rogers, founder and first president of MIT, painted with

respect to the same four frames as in figure 2.10 (original image reprinted with the permission of

the MIT Museum).

Figure 2.11:  Images of William Barton Rogers, founder and first president of MIT, painted with

respect to the same four frames as in figure 2.10 (original image reprinted with the permission of the

MIT Museum).

Figure 2.12 shows the drawing of a painter called 

wave4

 that is built up in two stages starting from 

wave

: 

(define wave2 (beside wave (flip-vert wave)))

(define wave4 (below wave2 wave2))

---

            

  

(define wave2                         (define wave4

  (beside wave (flip-vert wave)))       (below wave2 wave2))

Figure 2.12:  Creating a complex figure, starting from the 

wave

 painter of figure 2.10.

Figure 2.12:  Creating a complex figure, starting from the 

wave

 painter of figure 2.10.

In building up a complex image in this manner we are exploiting the fact that painters are closed under

the language’s means of combination. The 

beside

 or 

below

 of two painters is itself a painter;

therefore, we can use it as an element in making more complex painters. As with building up list

structure using 

cons

, the closure of our data under the means of combination is crucial to the ability

to create complex structures while using only a few operations.

Once we can combine painters, we would like to be able to abstract typical patterns of combining

painters. We will implement the painter operations as Scheme procedures. This means that we don’t

need a special abstraction mechanism in the picture language: Since the means of combination are

ordinary Scheme procedures, we automatically have the capability to do anything with painter

operations that we can do with procedures. For example, we can abstract the pattern in 

wave4

 as

(define (flipped-pairs painter)

  (let ((painter2 (beside painter (flip-vert painter))))

    (below painter2 painter2)))

and define 

wave4

 as an instance of this pattern:

(define wave4 (flipped-pairs wave))

We can also define recursive operations. Here’s one that makes painters split and branch towards the

right as shown in figures 2.13 and  2.14: 

(define (right-split painter n)

  (if (= n 0)

      painter

      (let ((smaller (right-split painter (- n 1))))

        (beside painter (below smaller smaller)))))

---

           

  

     right-split n                   corner-split n

Figure 2.13:  Recursive plans for 

right-split

 and 

corner-split

.

Figure 2.13:  Recursive plans for 

right-split

 and 

corner-split

.

We can produce balanced patterns by branching upwards as well as towards the right (see 

exercise 2.44 and figures 2.13 and  2.14):

(define (corner-split painter n)

  (if (= n 0)

      painter

      (let ((up (up-split painter (- n 1)))

            (right (right-split painter (- n 1))))

        (let ((top-left (beside up up))

              (bottom-right (below right right))

              (corner (corner-split painter (- n 1))))

          (beside (below painter top-left)

                  (below bottom-right corner))))))

---

            

  

     (right-split wave 4)         (right-split rogers 4)

            

  

    (corner-split wave 4)         (corner-split rogers 4)

Figure 2.14:  The recursive operations 

right-split

 and 

corner-split

 applied to the

painters 

wave

 and 

rogers

. Combining four 

corner-split

 figures produces symmetric 

square-limit

 designs as shown in figure 2.9.

Figure 2.14:  The recursive operations 

right-split

 and 

corner-split

 applied to the painters 

wave

 and 

rogers

. Combining four 

corner-split

 figures produces symmetric 

square-limit

designs as shown in figure 2.9.

By placing four copies of a 

corner-split

 appropriately, we obtain a pattern called 

square-limit

, whose application to 

wave

 and 

rogers

 is shown in figure 2.9: 

(define (square-limit painter n)

  (let ((quarter (corner-split painter n)))

    (let ((half (beside (flip-horiz quarter) quarter)))

      (below (flip-vert half) half))))

Exercise 2.44.  Define the procedure 

up-split

 used by 

corner-split

. It is similar to 

right-split

, except that it switches the roles of 

below

 and 

beside

. 

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