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Babylonian spiral

You will learn about: weirdly surprising patterns.

A Babylonian spiral is constructed by starting with a zero vector at the origin of a Cartesian grid and progressively concatenating the next longest vector with integer components. So the $i$th vector has integer components $(x_i, y_i)$ satisfying $x_i^2 + y_i^2 = n_i^2 > n_{i-1}^2$ where $x_i$, $y_i$ and $n_i$ are non-negative integers. The direction of the new vector is chosen such that it minimizes the clockwise angular separation from the previous one.

Starting with the vectors $(x_0, y_0) = (0, 0)$ and $(x_1, y_1) = (0, 1)$ find the $x$ and $y$ coordinates of the Babylonian spiral after $N$ steps.

Input: Number of steps $N$.

Output: The $x$ and $y$ coordinates as lists.


Input: 10
Output x: [0, 0, 1, 3, 5, 7, 7, 6, 4, 0, -4, -7]
Output y: [0, 1, 2, 2, 1, -1, -4, -7, -10, -10, -9, -6]
 Difficulty  Timesink
 Function babylonian_spiral(n_steps)

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  • This problem has been inspired by a MathPickle article and a SciPython blog post.
  • Apparently the name Babylonian spiral was chosen to mislead school students into making an incorrect hypothesis about the spiral's long-term behavior.
  • The On-Line Encyclopedia of Integer Sequences (OEIS) has three sequences related to the Babylonian spiral: A256111, A297346, and A297347.

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