lambdaway :: sierpinsky3

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_h1 sierpinsky {sup ([[triangle|?view=sierpinsky]] | carpet {sup ([[2|?view=sierpinsky2]]|3|[[4|?view=sierpinsky4]]|[[5|?view=sierpinsky5]]|[[6|?view=sierpinsky6]]|[[7|?view=sierpinsky7]])}} 
{center Refresh the screen to see other random colors.}
{svg {@ width="580" height="580" style="box-shadow:0 0 8px #000;"}

{sierp 580 3}

}

{pre

'{def sierp
 {def sierp.r
  {lambda {:x :y :w :c :n}
   {square :x :y :w :c}  
   {if {< :n 0}
    then   
    else {let { {:lx {- :x :w}} {:x :x} {:rx {+ :x :w}}
                {:ly {- :y :w}} {:y :y} {:ry {+ :y :w}}
                {:w {/ :w 3}}  {:rc {rc}} {:n {- :n 1}}
              } {sierp.r :lx :ly :w :rc :n}
                {sierp.r :x  :ly :w :rc :n}
                {sierp.r :rx :ly :w :rc :n}
                {sierp.r :rx :y  :w :rc :n}
                {sierp.r :rx :ry :w :rc :n}
                {sierp.r :x  :ry :w :rc :n}
                {sierp.r :lx :ry :w :rc :n}
                {sierp.r :lx :y  :w :rc :n} }}}}
 {lambda {:w :n}
  {square {/ :w 2} {/ :w 2} :w {rc}} 
  {sierp.r {/ :w 2} {/ :w 2} {/ :w 3} {rc} :n} }} 
-> {def sierp
 {def sierp.r
  {lambda {:x :y :w :c :n}
    {square :x :y :w :c}  
   ;; {text {@ x="{- :x {/ :w 4.0}}" y="{+ :y {/ :w 3.2}}" style="font-family:times; font-size::w; stroke:#000; fill::c;"}λ}
   {if {< :n 0}
    then   
    else {let { {:lx {- :x :w}} {:x :x} {:rx {+ :x :w}}
                {:ly {- :y :w}} {:y :y} {:ry {+ :y :w}}
                {:w {/ :w 3}}  {:rc {rc}} {:n {- :n 1}}
              } {sierp.r :lx :ly :w :rc :n}
                {sierp.r :x  :ly :w :rc :n}
                {sierp.r :rx :ly :w :rc :n}
                {sierp.r :rx :y  :w :rc :n}
                {sierp.r :rx :ry :w :rc :n}
                {sierp.r :x  :ry :w :rc :n}
                {sierp.r :lx :ry :w :rc :n}
                {sierp.r :lx :y  :w :rc :n}
 }}}}
 {lambda {:w :n}
  {square {/ :w 2} {/ :w 2} :w {rc}} 
  {sierp.r {/ :w 2} {/ :w 2} {/ :w 3} {rc} :n} }}

'{def rc
 {lambda {} 
  rgb({* 255 {random}},
      {* 255 {random}},
      {* 255 {random}})}}
-> {def rc {lambda {} rgb({* 255 {random}},{* 255 {random}},{* 255 {random}})}}

'{def square
 {lambda {:x :y :w :c}
  {rect {@ x="{- :x {/ :w 2}}" 
           y="{- :y {/ :w 2}}" 
           width=":w" height=":w"
           fill=":c"}}}}
-> {def square
 {lambda {:x :y :w :c}
  {rect {@ x="{- :x {/ :w 2}}" 
           y="{- :y {/ :w 2}}" 
           width=":w" height=":w" fill=":c"}}}}

'{sierp 580 3}
}

_h2 surface

_p The [[area of the carpet|https://en.wikipedia.org/wiki/Sierpiński_carpet]] is {b s = w{sup 2} * (8/9){sup n}} and tends to zero (in standard Lebesgue measure) when {b n -> ∞}.

{prewrap
'{def surf
 {lambda {:w :n} 
  {round {* {pow :w 2} {pow {/ 8 9} :np}}}}}
-> {def surf {lambda {:x :y} {round {* {pow :x 2} {pow {/ 8 9} :y}}}}}

'{S.map {surf 100} {S.serie 0 85}}  // a 10*10 square, 85 timesp
-> 
{S.map {surf 100} {S.serie 0 85}}
}

_p Mobile phone and Wi-Fi fractal antennas have been produced in the form of few iterations of the Sierpiński carpet. Due to their self-similarity and scale invariance, they easily accommodate multiple frequencies. They are also easy to fabricate and smaller than conventional antennas of similar performance, thus being optimal for pocket-sized mobile phones.

_img data/sierpinsky_screen.jpg