Infinite Hexagonal Spiral

Algebra Level pending

A robot, initially positioned at the origin, moves 1 1 unit along the x x -axis, turns left by 60 ° \ang{60} and moves 1 2 \frac 12 unit, turns left by 60 ° \ang{60} and moves 1 4 \frac 14 unit, and so on. For each step, it turns left by 60 ° \ang{60} and moves a distance equal to half of its previous step.

After an infinite number of moves, what is its distance from the origin? Give your answer to four decimal places.

Please post your solution.


The answer is 1.1547.

Hypergeo H. by Hypergeo H. , 30, Malaysia

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1 solution

Chris Lewis
Apr 21, 2020

The robot's path can be modelled as a geometric progression in the complex plane.

Its first step is 1 1 ; its second step 1 2 e 1 3 i π \frac 12 e^{\frac13 i \pi} ; its n th n^{\text{th}} step 1 2 n e 1 3 n i π \frac {1}{2^n} e^{\frac13 n i \pi} .

The endpoint is found by summing this series; that is z ∞ = ∑ 1 2 n e 1 3 n i π = 1 1 − 1 2 e 1 3 i π z_{\infty}=\sum \frac {1}{2^n} e^{\frac13 n i \pi} = \frac{1}{1-\frac 12 e^{\frac13 i \pi}}

Working out the division, we find z ∞ = 1 + i 3 3 z_{\infty}=1+i\frac{\sqrt3}{3} which is a distance 4 3 = 1.1547 … \sqrt{\frac43}=\boxed{1.1547\ldots} from the origin.

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Nice solution! Just like mine :)

Hypergeo H. - 1 year, 1 month ago

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