Lesson 2: Loops and remainders

"""Times table on a circle: N dots, each dot n joined to dot (k * n) mod N.
Run: python times-table.py   (needs: pip install numpy matplotlib)
"""
import numpy as np
import matplotlib.pyplot as plt

plt.style.use("dark_background")

N, k = 200, 2  # the number of dots and the multiplier

start = 2 * np.pi * np.arange(N) / N  # dot n at the angle 2*pi*n/N
end = start * k  # dot k*n: angles wrap around the circle by themselves (that is the "mod N")

plt.figure(figsize=(6, 6))
for a, b in zip(start, end):
    plt.plot([np.cos(a), np.cos(b)], [np.sin(a), np.sin(b)], color="#f5c76b", linewidth=0.6)
plt.axis("equal")
plt.axis("off")
plt.show()

# Try: k = 3 for the nephroid, k = 101 to see it collapse into diameters

No Python on this computer? The notebook runs the same script, the exercises and the answers in your browser, on Google Colab.

Line by line

1. N, k = 200, 2

   Two names at once: 200 dots and the multiplier 2. These are the two numbers to change.

2. start = 2 * np.pi * np.arange(N) / N

   The angle of each dot: dot n sits at 2π·n / N, so the 200 dots share the circle evenly.

3. end = start * k

   The angle of dot k·n. An angle past 2π simply goes round again, and that does the wrapping (k·n mod N) for us.

4. for a, b in zip(start, end):

   zip pairs the two arrays: each time round, the loop gets one start angle with its end angle.

5. plt.plot([np.cos(a), np.cos(b)], [np.sin(a), np.sin(b)], color="#f5c76b", linewidth=0.6)

   plt.plot takes the x values, then the y values, of the line’s two ends: one chord of the circle each time round.

Try it yourself

1. Draw times 3 and times 4. How many cusps (sharp points) does each curve have?

   Hint

   Change k.

   Answer

   Times 3 gives the nephroid with 2 cusps, times 4 a curve with 3: always k − 1.

2. With 200 dots, which dot does 150 × 2 land on? Work it out with the remainder operator.

   Hint

   In Python, % gives the remainder after dividing.

   Answer

   (150 * 2) % 200 is 100.

3. Use 20 dots instead of 200. Why is the heart harder to see?

   Hint

   Change N.

   Answer

   With 20 dots there are only 20 lines, too few for the curve they all touch to show.

Quiz