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Two Steps, One Turn

The √3·d walk on a triangular lattice: one step, a 60° turn, another step.

Two steps with a 60 degree turn gives root 3 times dOn a triangular lattice, one step of length d to the right, a 60 degree turn, and another step of length d. The two steps enclose 120 degrees. A perpendicular from the turning point splits the triangle into two 30-60-90 halves, each with base root 3 over 2 times d, so the straight line is root 3 times d. 60° turn 120° d d √3/2 · d √3/2 · d start end Blue line: √3/2·d + √3/2·d = √3·d ≈ 1.732d

The 60° is the exterior angle, how far you turn off your heading. The 120° is the interior angle the two steps enclose. Dropping a perpendicular splits the triangle into two 30-60-90 halves, each with base √3/2·d.