lambdaway :: sierpinsky4

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_h1 sierpinsky {sup ([[triangle|?view=sierpinsky]] | carpet {sup ([[2|?view=sierpinsky2]]|[[3|?view=sierpinsky3]]|4|[[5|?view=sierpinsky5]]|[[6|?view=sierpinsky6]]|[[7|?view=sierpinsky7]])}}

_p A version built essentially on text rewriting, using Scheme and Lambdatalk.

_h2 1) scheme/racket version

_p A code written by [[Jean-Paul Roy|http://jean.paul.roy.free.fr/]].
;; « {i La fractale sierpinsky(n) contient 8 fois la fractale sierpinsky(n-1), le code Python s'en déduit, ainsi que le code Scheme...} »

{pre
(define (repeat n f init)     ; appliquer la transformation f n fois sur init
  (if (= n 0) 
      init 
      (repeat (- n 1) f (f init))))
;; (repeat 30 cos 1)            ; une solution approchée de cos(x) = x
(define (replace old new L)   ; sans doute une primitive mais...
  (if (empty? L)
      L
      (cons (if (equal? (first L) old) new (first L)) 
            (replace old new (rest L)))))

(define str3 string-append)

(define (transf L)
    (append (map (lambda (x) (str3 x x x)) L)
            (map (lambda (x) (str3 x (string-replace x "#" " ") x)) L)
            (map (lambda (x) (str3 x x x)) L)))   ; recalculé, bof...

(define (sierpinsky n)  
  (repeat n transf (list "#")))

(sierpinsky 3)     ; ou bien :    (for [(x (sierpinsky 3))] (println x))
}

_p displays

{pre {@ style="line-height:0.8em;"}
'("###########################"
  "# ## ## ## ## ## ## ## ## #"
  "###########################"
  "###   ######   ######   ###"
  "# #   # ## #   # ## #   # #"
  "###   ######   ######   ###"
  "###########################"
  "# ## ## ## ## ## ## ## ## #"
  "###########################"
  "#########         #########"
  "# ## ## #         # ## ## #"
  "#########         #########"
  "###   ###         ###   ###"
  "# #   # #         # #   # #"
  "###   ###         ###   ###"
  "#########         #########"
  "# ## ## #         # ## ## #"
  "#########         #########"
  "###########################"
  "# ## ## ## ## ## ## ## ## #"
  "###########################"
  "###   ######   ######   ###"
  "# #   # ## #   # ## #   # #"
  "###   ######   ######   ###"
  "###########################"
  "# ## ## ## ## ## ## ## ## #"
  "###########################") 
}

_h2 2) lambdatalk version

_p See the version using lists in [[sierpinsky5]]. 
_p Here we forget lists and use two primitives {code [S.replace & S.map]} working on (sequences of) words and two HTML primitives, {code [h2 & div]}, for formatting.

{pre
'{S.replace x by _ in axbxcxdx exfxgx}
-> {S.replace x by _ in axbxcxdx exfxgx}

'{S.map {lambda {:x} :x:x:x} abcd efg}
-> {S.map {lambda {:x} :x:x:x} abcd efg}
}

_p Nothing more!

_h3 2.1) code

{pre
'{def sierpinsky

{def sierpinsky.r
  {lambda {:n :w}
   {if {= :n 0}
    then :w
    else {sierpinsky.r 
          {- :n 1}
          {S.map {lambda {:x} :x:x:x} :w}
          {S.map {lambda {:x} :x{S.replace ■ by o in :x}:x} :w}  
          {S.map {lambda {:x} :x:x:x} :w} }}}}

{lambda {:n}
  {h2 n=:n}{S.replace o by space in 
            {S.replace \s by {div} in 
             {sierpinsky.r :n ■}}}}}    
-> {def sierpinsky 
 {def sierpinsky.r
  {lambda {:n :w}
   {if {= :n 0}
    then :w
    else {sierpinsky.r {- :n 1}
                       {S.map {lambda {:x} :x:x:x} :w}
                    ;; {S.map {lambda {:x} :x{{lambda {:x ■} :x} :x o}:x} :w} 
                       {S.map {lambda {:x} :x{S.replace ■ by o in :x}:x} :w}
                       {S.map {lambda {:x} :x:x:x} :w} }}}}
 {lambda {:n}
  {h2 S:n}{S.replace o by space in {S.replace \s by {div} in {sierpinsky.r :n ■}}}}}
}

_p {b Notes}
_ul 1) the last argument of a lambda, here {code :w}, gets all the following values.
_ul 2) {b '{S.replace ■ by o in :x}} could be written {b '{{lambda {:x ■} :x} :x o}}

_h3 2.2) tests

{pre
'{pre
 {@ style="line-height:0.6em;   // to glue the lines
           color:#f80;"}        // and color the characters
 {S.map sierpinsky 0 1 2 3 4}
}
}

_p displays

{pre {@ style="line-height:0.6em; color:#a00;"}
 {S.map sierpinsky 0 1 2 3 4}   
}

_p These are not images but pure text blocks.

_p Computation time on a 2011 MacBook Air: ≈10ms for n=3, ≈20ms for n=4.

{center S4 is to be compared with the S3 [[SVG version|?view=sierpinsky3]], which is ≈10 times slower.}
_img data/sierpinsky_screen.jpg

{style 
pre { box-shadow:0 0 8px #000; padding: 10px; }
}