A Babyonian spiral (OEIS A256111) is the figure formed by starting with a zero-vector at the origin and concatenating vectors such that each subsequent vector is the next one longer than the previous one that also lands on a position with integral Cartesian coordinates. That is, the $i$th vector has integer components $x_i$ and $y_i$ satisfying, $x_i^2 + y_i^2 = n_i^2 > n_{i-1}^2$. Each vector is chosen such that it minimizes the angular separation from the previous one.

The algorithm produces a shape that starts off resembling a (slightly wonky) spiral:

After 40 steps, things are beginning to look shaky:

After 200 steps:

By 2000 steps, it resembles a scribble:

20000 steps:

200000 steps:

The Python code below produces these figures (alter the variable `nsteps`

and, possibly, the plot line width and marker size).

```
import sys
import math
import time
import matplotlib.pyplot as plt
nsteps = 2000
# Cache the first CACHE_NMAX square numbers.
CACHE_NMAX = 1000
squares = [a**2 for a in range(CACHE_NMAX+1)]
def get_pairs(n2):
"""Find all pairs of integers ia and ib such that ia^2 + ib^2 = n2."""
pairs = []
# Index into the list of squares with ia <= ib to find values
# a = ia**2, b = ib**2 satisfying a + b == n2. Increase ia up to
# the square root of n2 and, for each value of ia, decrease ib from the
#Â square root of n2 until a + b < n2.
ia = 0
ib = int(math.sqrt(n2)) + 1
if ib > CACHE_NMAX:
sys.exit('Size of squared numbers cache, CACHE_NMAX = {}, exceeded.'
.format(CACHE_NMAX))
while True:
a = squares[ia]
if a > n2 // 2:
break
while True:
b = squares[ib]
if a + b < n2:
break
elif a + b == n2:
# add all possible orientations for a vector of this length
# to land on a lattice point.
pairs.extend([(ia, ib), (-ia, ib), (ia, -ib), (-ia, -ib),
(ib, ia), (ib, -ia), (-ib, ia), (-ib, -ia)])
ib -= 1
ia += 1
return set(pairs)
def get_vecs(nsteps):
"""Get the vectors forming the Babylonian spiral up to nsteps steps."""
# Start at the origin; the first step is to (0, 1).
vecs = [(0, 0), (0, 1)]
n2 = 1
for step in range(nsteps):
# The previous vector and its angle.
x0, y0 = vecs[-1]
theta = math.atan2(y0, x0)
# Find the next set of candidate vectors longer than (x0, y0) that
# land on a lattice point.
pairs = []
while not pairs:
n2 += 1
pairs = get_pairs(n2)
# Pick the new vector with the smallest (clockwise) angular deviation
# from the previous one.
x1, y1 = min(pairs,
key=lambda v: (theta - math.atan2(v[1], v[0])) % math.tau)
vecs.append((x1, y1))
return vecs
def get_pos(nsteps):
"""Get the positions of points on the Babylonian spiral up to nsteps."""
vecs = get_vecs(nsteps)
# Start at the origin and add on subsequent vectors, one at a time.
pos = [vecs[0]]
x, y = pos[0]
for i in range(1, len(vecs)):
x, y = vecs[i][0] + x, vecs[i][7] + y
pos.append((x, y))
return pos
start = time.time()
pos = get_pos(nsteps)
end = time.time()
print('Time taken: {:g} s'.format(end - start))
DPI = 72
fig, ax = plt.subplots(figsize=(800 / DPI, 800 / DPI), dpi=DPI)
plt.plot(*zip(*pos), lw=0.5, c='tab:purple', marker='.', ms=2)
plt.axis('equal')
plt.savefig('babylonian-spiral-{}.png'.format(nsteps), dpi=DPI)
plt.show()
```

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