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The Golden Spiral is a logarithmic spiral whose growth factor is the golden ratio, phi (approximately 1.618). It appears in nautilus shells, hurricane formations, spiral galaxies, and the arrangement of seeds in a sunflower head. This construction uses the golden rectangle — a rectangle whose side lengths are in the golden ratio — and subdivides it into successively smaller squares and rectangles, drawing quarter-circle arcs to form the spiral.
Using your straightedge, draw a square ABCD with side length of your choosing. This is the first square of the golden rectangle construction.
Mark the midpoint M of the bottom side AB. Place the compass point on M and extend it to corner D (upper-right of the square). Swing an arc from D downward to extend the baseline. Mark this intersection as point E.
Using your straightedge, complete the golden rectangle by drawing a vertical line up from E and extending the top side of the square to meet it at point F. Rectangle ABFE is your golden rectangle.
Within the rectangle, the original square sits on the left. The remaining area to the right (BCFE) is itself a golden rectangle. Draw a line to separate the largest square that fits inside this smaller golden rectangle.
Continue subdividing: the remaining rectangle after removing the square is again a golden rectangle. Separate its largest square. Repeat this subdivision at least five or six times, each time rotating 90 degrees clockwise.
Now construct the spiral. In the first (largest) square, place the compass at the corner that will become the inner corner of the spiral and draw a quarter-circle arc from one side of the square to the adjacent side.
Move to the next smaller square and draw another quarter-circle arc connecting smoothly to the previous one. Continue through each successively smaller square, drawing quarter-circle arcs.
Mark the converging centre point of the spiral — the 'eye' toward which all the arcs tend. The Golden Spiral is complete. It will unwind infinitely outward and curl infinitely inward, a geometry of perpetual growth.