Logo — turtle geometry with a measurable invariant

Try a square in a compatible Logo turtle interpreter

CS
REPEAT 4 [FD 50 RT 90]

Trace the turtle's heading: 0°, 90°, 180°, 270°, back to 0°. Four identical 50-unit steps along these headings should return it to the starting location. The image alone is insufficient evidence: record the final position and direction using the commands supported by your particular dialect. CS usually clears and resets the drawing, but confirm dialect behavior.

Controlled failure

Change RT 90 to RT 80. Four turns now total 320°, not 360°, so the heading invariant fails. Restore 90° and change the second side length to 40. Explain why a correct total rotation cannot guarantee a closed square if lengths differ.

Independent project and checks

Create a SQUARE procedure accepting a side parameter, then draw sides 20 and 80 from a fresh starting state. Include an invalid parameter and observe whether the interpreter rejects it or uses surprising coercion. Document interpreter name, version, commands for position inspection, and observed results; this illustrative program has not been executed in Logo during this audit.

Why a regular polygon closes

For n equal sides, the exterior angle is 360/n degrees, so a triangle uses 120° while a hexagon uses 60°. Run REPEAT 3 [FD 50 RT 120] and REPEAT 6 [FD 50 RT 60] as separate experiments with a reset between them. Compare the polygon after changing only the repeat count: even though each step is valid, the turtle's final direction will differ when total rotation is not 360°. Turn pen up, move to a new origin, turn pen down and repeat; moving with the pen raised should not draw a stray connecting edge.