Minimal Surfaces

GPT-4 did this in 1 shot.

The prompt is at

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.

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:

\[\text{div}\left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = 0\]

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')

Related Posts

Just because 2 things are dual, doesn't mean they're just opposites

Boolean Algebra, Arithmetic POV

discontinuous linear functions

Continuous vs Bounded

November 2, 2023

NTK reparametrization

Kate from Vancouver, please email me

ChatGPT Session: Emotions, Etymology, Hyperfiniteness

Some ChatGPT Sessions

2016 ML thoughts