(Please note that this doesn’t mean the resulting image is scientifically meaningful. If anything, it’s a science-inspired code/art experiment.)
import numpy as np
from PIL import Image
from hilbertcurve.hilbertcurve import HilbertCurve
# 1) Grid size
W = H = 4096
N = W * H # 16,777,216
# 2) Camera parameters (pan and zoom) to frame the amplituhedron
# center_u, center_v specify the focal point in the unit square
# zoom >1 zooms in, <1 zooms out
center_u, center_v = 0.5, 0.3
zoom = 1.5
half_range = 0.5 / zoom
u_min, u_max = center_u - half_range, center_u + half_range
v_min, v_max = center_v - half_range, center_v + half_range
# 3) Build cyclic polygon data for A(n,1,2) on the moment curve
def build_amplituhedron_data(n=8):
"""
Build cyclic polygon data on the planar moment curve for A(n,1,2).
Returns Z (n×3 array) and precomputed triangle volumes and cross-products.
"""
t = np.linspace(0.1, 0.9, n)
Z = np.vstack([np.ones(n), t, t**2]).T
Z0 = Z.mean(axis=0)
tri_volume, cross_12, cross_20, cross_01 = [], [], [], []
for i in range(n - 1):
A, B = Z[i], Z[i+1]
tri_volume.append(np.linalg.det(np.stack([Z0, A, B], axis=1)))
cross_12.append(np.cross(A, B))
cross_20.append(np.cross(B, Z0))
cross_01.append(np.cross(Z0, A))
return Z, Z0, np.array(tri_volume, float), cross_12, cross_20, cross_01
Z, Z0, tri_volume, cross_12, cross_20, cross_01 = build_amplituhedron_data(n=8)
# 4) Define the true amplituhedron canonical-form evaluator
def amplituhedron_value(u, v, epsilon=1e-12):
"""
Evaluate the canonical-form density F(u,v) of A(n,1,2) at Y=(u,v,1).
Regularizes exact boundary poles by epsilon.
"""
Uy = u.ravel(); Vy = v.ravel()
pts = np.column_stack([Uy, Vy, np.ones_like(Uy)])
F_flat = np.zeros_like(Uy, float)
for i in range(len(tri_volume)):
d1 = pts.dot(cross_12[i])
d2 = pts.dot(cross_20[i])
d3 = pts.dot(cross_01[i])
F_flat += tri_volume[i] / (d1 * d2 * d3 + epsilon)
return F_flat.reshape(u.shape)
# 5) Sample the field with camera framing and rank pixels
us = np.linspace(u_min, u_max, W)
vs = np.linspace(v_min, v_max, H)
U, V = np.meshgrid(us, vs)
F = amplituhedron_value(U, V).ravel()
pixel_order = np.argsort(-F, kind='mergesort')
# 6) Generate full 24-bit palette and sort by a 3D Hilbert curve for smooth color continuity
colors = np.empty((256**3, 3), dtype=np.uint8)
idxs = np.arange(256**3, dtype=np.uint32)
colors[:,0] = (idxs >> 16) & 0xFF
colors[:,1] = (idxs >> 8) & 0xFF
colors[:,2] = idxs & 0xFF
# Build 3D Hilbert indexer on 8-bit coordinates
hc3 = HilbertCurve(p=8, n=3)
# Coordinates for each color
coords_rgb = np.column_stack([colors[:,0], colors[:,1], colors[:,2]])
# Compute Hilbert distances (slow but one-time)
distances = np.array(hc3.distances_from_points(coords_rgb.tolist()), dtype=np.uint32)
color_order = np.argsort(distances, kind='mergesort')
# 7) Assemble image and save
img_flat = np.empty((N,3), np.uint8)
img_flat[pixel_order] = colors[color_order]
img = img_flat.reshape((H, W, 3))
Image.fromarray(img).save('amplituhedron_true_unique_palette_hilbert.png')
print("Saved: amplituhedron_true_unique_palette_hilbert.png")
