GPT-4 did this in 1 shot.
The prompt is at https://chat.openai.com/share/b2af7f22-2953-4b25-af69-bd16077a6770
Code ran first time, always a rush when that happens.
For a periodic minimal hypersurface, one would typically expect the surface to repeat itself in a regular pattern across the space. This periodicity could be along one or more axes, depending on the dimension and the specific geometry of the space.
\[\text{div}\left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = 0\]To define such a surface mathematically, you would typically use a combination of differential geometry and partial differential equations. The key equation here is the minimal surface equation, which in its simplest form (for graphs) is a nonlinear partial differential equation. For a surface \(S\) given as a graph of a function \(u(x_1, x_2, \ldots, x_{n-1})\), the minimal surface equation in \(\mathbb{R}^n\) can be written as:
I had it do the Schwarz P Surface, defined by the equation
\[\cos(x) + \cos(y) + \cos(z) = 0\]import torch
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from skimage import measure
# Define the range and resolution for the grid
grid_size = 50
x = torch.linspace(-np.pi, np.pi, grid_size)
y = torch.linspace(-np.pi, np.pi, grid_size)
z = torch.linspace(-np.pi, np.pi, grid_size)
# Create a 3D meshgrid
X, Y, Z = torch.meshgrid(x, y, z, indexing='ij')
# Evaluate the Schwarz P surface equation
F = torch.cos(X) + torch.cos(Y) + torch.cos(Z)
# Convert the grid and function values to numpy for processing
F_np = F.numpy()
# Using the marching cubes algorithm to get the surface mesh
verts, faces, _, _ = measure.marching_cubes(F_np, level=0)
# Plotting the surface
fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111, projection='3d')
ax.plot_trisurf(verts[:, 0], verts[:,1], faces, verts[:, 2], cmap='Spectral', lw=1)
# Setting labels and title
ax.set_xlabel('X axis')
ax.set_ylabel('Y axis')
ax.set_zlabel('Z axis')
plt.title('Schwarz P Minimal Surface Visualization')
plt.show()