A Rose from the Realm of Mathematics

 

Mathematics isn't just about numbers; it's a profound way to explore and represent patterns that resonate with beauty. Recently, I stumbled upon a fascinating way to blend mathematical equations with graphical representation using Python. Let me introduce you to two gorgeous 3D visualizations, both rooted in a simple yet intricate equation.

import numpy as np

import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D

# Parameters

n = 800

A = 1.995653

B = 1.27689

C = 8

petalNum = 3.6

# Create grid

r = np.linspace(0, 1, n)

theta = np.linspace(-2, 20 * np.pi, n)

R, THETA = np.meshgrid(r, theta)

# Calculate coordinates

x = 1 - (1/2) * ((5/4) * (1 - np.mod(petalNum * THETA, 2 * np.pi) / np.pi)**2 - 1/4)**2

phi = (np.pi/2) * np.exp(-THETA / (C * np.pi))

y = A * (R**2) * (B * R - 1)**2 * np.sin(phi)

R2 = x * (R * np.sin(phi) + y * np.cos(phi))

X = R2 * np.sin(THETA)

Y = R2 * np.cos(THETA)

Z = x * (R * np.cos(phi) - y * np.sin(phi))

# Create colormap

red_map = np.linspace(1, 0.25, 10).reshape(-1, 1)

red_map = np.hstack([red_map, np.zeros_like(red_map), np.zeros_like(red_map)])

# Plot the surface

fig = plt.figure()

ax = fig.add_subplot(111, projection='3d')

ax.plot_surface(X, Y, Z, cmap=plt.cm.Reds, linewidth=0, antialiased=False)

ax.view_init(elev=42, azim=-40.5)

plt.show()

import numpy as np

import plotly.graph_objects as go

# Parameters

n = 800

A = 1.995653

B = 1.27689

C = 8

petalNum = 3.6

# Create grid

r = np.linspace(0, 1, n)

theta = np.linspace(-2, 20 * np.pi, n)

R, THETA = np.meshgrid(r, theta)

# Calculate coordinates

x = 1 - (1/2) * ((5/4) * (1 - np.mod(petalNum * THETA, 2 * np.pi) / np.pi)**2 - 1/4)**2

phi = (np.pi/2) * np.exp(-THETA / (C * np.pi))

y = A * (R**2) * (B * R - 1)**2 * np.sin(phi)

R2 = x * (R * np.sin(phi) + y * np.cos(phi))

X = R2 * np.sin(THETA)

Y = R2 * np.cos(THETA)

Z = x * (R * np.cos(phi) - y * np.sin(phi))

# Define the surface

fig = go.Figure(data=[go.Surface(x=X, y=Y, z=Z, colorscale=[[0, 'red'], [1, 'darkred']])])

# Update plot layout for better interactivity

fig.update_layout(scene=dict(

    xaxis_title='X',

    yaxis_title='Y',

    zaxis_title='Z',

))

# Show plot

fig.show()