Torus Knot Pattern using Python

 

import matplotlib.pyplot as plt

import numpy as np

from mpl_toolkits.mplot3d import Axes3D

R=3

r=1

p=2

q=3

theta=np.linspace(0,2*np.pi,1000)

x=(R+r*np.cos(q*theta))*np.cos(p*theta)

y=(R+r*np.cos(q*theta))*np.sin(p*theta)

z=r*np.sin(q*theta)

fig=plt.figure(figsize=(6,6))

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

ax.plot(x,y,z,color='purple',lw=2)

ax.set_xlabel('X Axis')

ax.set_ylabel('Y Axis')

ax.set_zlabel('Z Axis')

ax.set_title('Torus Knot',fontsize=16)

plt.show()

#source code --> clcoding.com

Code Explanation:

Imports:

import numpy as np

import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D

numpy: Used for numerical calculations and to generate the array of angles (theta) for the parametric equations.

 matplotlib.pyplot: Used to plot the 3D torus knot.

 mpl_toolkits.mplot3d: Provides the necessary tools to create 3D plots in matplotlib.

 Parameters:

R = 3

r = 1

p = 2 

q = 3

R: The radius of the central circle of the torus. This determines the distance from the center of the hole to the center of the tube (doughnut shape).

 r: The radius of the tube itself. This controls the thickness of the torus.

 p and q: These are the parameters that determine how many times the knot will wind around the tube and the hole of the torus.

 p: The number of times the knot winds around the tube of the torus.

 q: The number of times the knot winds around the central hole of the torus.

 In this case, the values are set as:

 p = 2: The knot will loop around the tube 2 times.

 q = 3: The knot will loop around the hole 3 times.

 Parameter Array (theta):

theta = np.linspace(0, 2 * np.pi, 1000)

theta is the angle parameter that will range from 0 to 2π, which represents one full rotation around the circle.

 The np.linspace(0, 2 * np.pi, 1000) generates 1000 evenly spaced values between 0 and 2π, which will be used to plot the curve.

 Parametric Equations for the Torus Knot:

x = (R + r * np.cos(q * theta)) * np.cos(p * theta)

y = (R + r * np.cos(q * theta)) * np.sin(p * theta)

z = r * np.sin(q * theta)

The 3D coordinates (x, y, z) are computed using parametric equations. These equations define the position of each point on the knot:

 X-coordinate (x):

x(θ)=(R+r⋅cos(qθ))⋅cos(pθ)

This equation describes the horizontal position of the point as it winds around the torus.

 Y-coordinate (y):

y(θ)=(R+r⋅cos(qθ))⋅sin(pθ)

This equation describes the vertical position of the point, complementing the X-coordinate to form the loop around the torus.

Z-coordinate (z):

z(θ)=r⋅sin(qθ)

This equation determines the height of the point relative to the horizontal plane. It ensures the knot's winding around the hole of the torus.

 Plotting the Torus Knot:

fig = plt.figure(figsize=(8, 8))

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

 ax.plot(x, y, z, color='purple', lw=2)

ax.set_title(f'Torus Knot (p={p}, q={q})', fontsize=16)

ax.set_xlabel('X axis')

ax.set_ylabel('Y axis')

ax.set_zlabel('Z axis')

plt.show()

 Creating the Figure:

fig = plt.figure(figsize=(8, 8))

Creates a figure of size 8x8 inches to hold the plot.

 Creating a 3D Axes Object:

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

Adds a 3D subplot to the figure. The projection='3d' argument specifies that the axes should be 3-dimensional.

 Plotting the Knot:

ax.plot(x, y, z, color='purple', lw=2)

Plots the points (x, y, z) on the 3D axes. The curve is colored purple, and the line width is set to 2.

 Setting Labels and Title:

ax.set_title(f'Torus Knot (p={p}, q={q})', fontsize=16)

ax.set_xlabel('X axis')

ax.set_ylabel('Y axis')

ax.set_zlabel('Z axis')

ax.set_title(): Sets the title of the plot, which includes the values of p and q to indicate which specific torus knot is being displayed.

 ax.set_xlabel(), ax.set_ylabel(), ax.set_zlabel(): Labels the X, Y, and Z axes respectively.

 Displaying the Plot:

plt.show()

Displays the plot in a new window.