A few days ago I wrote about a random process that creates a fractal known as the Twin Dragon. This post gives a deterministic approach to create the same figure.

As far as I can tell, the first reference to this fractal is in a paper by Davis and Knuth in the Journal of Recreational Mathematics from 1970. Unfortunately this journal is out of print and hard or impossible to find online [1]. Knuth presents the twindragon (one word, lowercase) fractal in TAOCP Vol 2, page 206.

Knuth defines the twindragon via numbers base b = 1 − i. Every complex number can be written in the form

z = sum_{k=-infty}^infty a_k (1 - i)^k

where the “digits” ak are either 0 or 1.

The twindragon fractal is the set of numbers that only have non-zero digits to the right of the decimal point, i.e. numbers of the form

z = sum_{k=1}^infty a_k (1 - i)^{-k}

I implemented this in Python as follows.

import matplotlib.pyplot as plt
from itertools import product

for bits in product([0, 1], repeat=15):
    z = sum(a*(1-1j)**(-k) for k, a in enumerate(bits))
    plt.plot(z.real, z.imag, 'bo', markersize=1)
plt.show()

This produced the image below.

Related posts

[1] If you can find an archive of Journal of Recreational Mathematics, please let me know.

The post Knuth’s Twindragon first appeared on John D. Cook.

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