lambdaway :: hilbert

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{center
{svg {@ width="580px" height="580px" style="background:#666; color:#fff;"}
 {path 
 {@ id="spline"
     d="M {turtle 10 10 0 {left 18 5}}"  
      fill="transparent" stroke="#fff" stroke-width="3"
 }}
}}

{center {b [[chatgpt]] &} [[lambdatalk|?view=chatgpt_lambdatalk]] | [[roots|?view=chatgpt_20250220]] | [[langage|?view=chatgpt_langue]] | [[évaluation|?view=chatgpt_evaluation]] | [[hilbert|?view=hilbert_chatgpt]] | [[magali|?view=chatmag]] | [[relativité|?view=relativite_complexe]] |  [[costa|?view=chatgpt_costa]] | [[feynman|?view=chatgpt_feynman]] | incertitude ([[1|?view=chatgpt_incertitude]],[[2|?view=chatgpt_incertitude_2]]) | hélices ([[1|?view=chatgpt_helices]],[[2|?view=chatgpt_helices2]]) | [[synthèse|?view=chatgpt_synthese]] |[[casteljau|?view=decasteljau4]] | [[soliton]] | [[dirac|?view=chatgpt_dirac]] | [[young|?view=chatgpt_young]] | [[relatif|?view=relativite_complexe_3]] | [[charon]] | [[compton|?view=chatgpt_compton]]}

_h1 chatGPT & hilbert's curve

_p Voir [[hilbert]].

_h2 1) λtalk

{prewrap
'{def left
 {lambda {:d :n}
  {if {< :n 1}
    then
    else T90      {right :d {- :n 1}}
         M:d T-90 {left  :d {- :n 1}}
         M:d      {left  :d {- :n 1}}
         T-90 M:d {right :d {- :n 1}}
         T90
}}}
-> {def left {lambda {:d :n}
 {if {< :n 1}
   then
   else T90      {right :d {- :n 1}}
        M:d T-90 {left  :d {- :n 1}}
        M:d      {left  :d {- :n 1}}
        T-90 M:d {right :d {- :n 1}}
        T90}}}
'{def right
 {lambda {:d :n}
  {if {< :n 1}
    then
    else T-90    {left  :d {- :n 1}}
         M:d T90 {right :d {- :n 1}}
         M:d     {right :d {- :n 1}}
         T90 M:d {left  :d {- :n 1}}
         T-90
}}}
-> {def right {lambda {:d :n}
 {if {< :n 1}
   then
   else T-90    {left  :d {- :n 1}}
        M:d T90 {right :d {- :n 1}}
        M:d     {right :d {- :n 1}}
        T90 M:d {left  :d {- :n 1}}
        T-90}}}
}

_p The word {code Tθ} rotates the drawing direction of a pen from {b θ} degrees and the word {code Md} moves it on {b d} pixels. Writing {b '{left 18 5}} produces 2387 words begining with [T90 T-90 T90 T-90 T90 M10 T-90 M10 T-90 M10 T90 M10 T90 T-90 M10 T90 M10 T90 M10 T-90 M10 T-90 M10 T90 M10 T90 M10 T-90 T90 M10 T90 M10 T-90 M10 T-90 ...] which are sent to a graphic primitive, {b turtle}, which draws the curve in a SVG context.

_p So writing:
{pre
'{center
 {svg {@ width="580px" height="580px" 
         style="background:#666; color:#fff;"}
  {path 
   {@ id="spline"
      d="M {turtle 10 10 0 {left 18 5}}"  
      fill="transparent" stroke="#fff" stroke-width="3"
}}}} 
}

_p displays in real-time the picture on top of this wiki page.

_h2 2) python

_p Le code python suivant a été produit par ChatGPT le 19/02/2025, à partir du code lambdatalk que je lui ai donné en exemple.

{pre
import matplotlib.pyplot as plt
import numpy as np

# Initialisation des paramètres
angle = 90  # Angle de rotation (90°)
length = 10  # Longueur d'un segment
iterations = 5  # Nombre de niveaux de récursion

# Directions possibles (tortue LOGO)
directions = '{
    "T90":  np.pi / 2,   # Tourne de 90° à gauche
    "T-90": -np.pi / 2   # Tourne de 90° à droite
}

# Fonction récursive générant la courbe de Hilbert
def hilbert_curve(order, length, angle, left=True):
    if order == 0:
        return []
    
    if left:
        return [
            "T90", *hilbert_curve(order-1, length, angle, not left),
            "M", length, "T-90", *hilbert_curve(order-1, length, angle, left),
            "M", length, *hilbert_curve(order-1, length, angle, left),
            "T-90", "M", length, *hilbert_curve(order-1, length, angle, not left),
            "T90"
        ]
    else:
        return [
            "T-90", *hilbert_curve(order-1, length, angle, not left),
            "M", length, "T90", *hilbert_curve(order-1, length, angle, left),
            "M", length, *hilbert_curve(order-1, length, angle, left),
            "T90", "M", length, *hilbert_curve(order-1, length, angle, not left),
            "T-90"
        ]

# Génération des instructions
instructions = hilbert_curve(iterations, length, angle, left=True)

# Dessin de la courbe avec une tortue virtuelle
x, y = [0], [0]  # Coordonnées initiales
direction = 0  # Direction initiale (vers la droite)

for instr in instructions:
    if instr == "T90" or instr == "T-90":
        direction += directions[instr]  # Rotation de la direction
    elif instr == "M":
        # Calcul du déplacement
        new_x = x[-1] + np.cos(direction) * length
        new_y = y[-1] + np.sin(direction) * length
        x.append(new_x)
        y.append(new_y)

# Affichage du tracé
plt.figure(figsize=(6, 6))
plt.plot(x, y, 'b-', linewidth=2)
plt.axis("equal")
plt.axis("off")
plt.title("Courbe de Hilbert - Niveau '{}".format(iterations))
plt.show()
}

_p Il est sensé produire le même résultat dans un environnement Python.
_p {i alain marty | 2025/02/19}