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/ˈɡoʊl.dən ˈspaɪ.rəl/

Golden spiral

noun · geometry, composition

The logarithmic spiral whose growth factor per quarter-turn equals the golden ratio φ ≈ 1.618. Distinct from — though visually very close to — the quarter-circle Fibonacci spiral inscribed in a Fibonacci tiling.

What it is

A logarithmic spiral is any spiral of the form r = a·ebθ in polar coordinates; the golden spiral is the special case in which the growth factor b is chosen so that r multiplies by exactly φ for every quarter-turn. The defining property is self-similarity: zooming in by a factor of φ on any portion of the spiral produces an identical-shaped piece. The spiral fits perfectly inside a golden rectangle, touching the rectangle at the corners where the gnomon-square subdivision is rotated.

In composition, the golden spiral is most often referenced through its near-cousin the Fibonacci spiral — drawn as quarter-circles inscribed in a Fibonacci tiling of squares with sides 1, 1, 2, 3, 5, 8, 13… At sketching resolution the two are indistinguishable; mathematically the Fibonacci version has visible discontinuities at each square boundary while the true golden spiral is smooth. Composition overlays almost always use the easier quarter-circle approximation.

The golden spiral inscribed in a golden rectangle subdivided into nested squares
The golden spiral inscribed in a golden rectangle. Each subdivision square is φ times smaller than the previous one.

Etymology

The logarithmic spiral was first described by René Descartes in correspondence with Marin Mersenne (1638) and analysed in detail by Jacob Bernoulli, who named it spira mirabilis ("the marvellous spiral") and asked that one be engraved on his tombstone in Basel (1705). The specific phrase "golden spiral" entered art-historical usage through Jay Hambidge's Dynamic Symmetry: The Greek Vase (Yale University Press, 1920) and his later The Elements of Dynamic Symmetry (1926), which built a complete compositional system around the spiral and the root rectangles.

Examples in use

In the chambered nautilus shell — the most cited natural occurrence — the spiral is logarithmic but not golden: Clement Falbo's measurements of 30 nautilus specimens (College Mathematics Journal, 36:2, March 2005, pp. 123–134) found growth ratios averaging 1.33 per quarter-turn, not 1.618. The shell is a logarithmic spiral; calling it specifically golden is a popular-mathematics error.

In painting, Hambidge analysed the spiral structure of Whistler's Nocturne in Black and Gold (1875) and the Greek Parthenon vases of the 5th century BCE in Practical Applications of Dynamic Symmetry (Yale University Press, 1932). His findings remain contested — Markowsky's "Misconceptions about the Golden Ratio" (College Mathematics Journal, 23:1, 1992) argues many of Hambidge's measurements were fitted retroactively — but the spiral itself remains a working composition tool whether or not a particular historical claim survives.

See also: Golden ratio · Fibonacci sequence · Dynamic symmetry · Golden spiral overlay · 82 composition overlays (pillar)

References

  1. Hambidge, Jay. The Elements of Dynamic Symmetry. Yale University Press (1926). Reissued by Dover (1967), ISBN 0-486-21776-0.
  2. Bernoulli, Jacob. Spira Mirabilis, in Opera Omnia, vol. 1. Geneva (1744; original work 1692–1704).
  3. Falbo, Clement. "The Golden Ratio: A Contrary Viewpoint." College Mathematics Journal, vol. 36, no. 2 (March 2005), pp. 123–134. DOI: 10.1080/07468342.2005.11922119.
  4. Markowsky, George. "Misconceptions about the Golden Ratio." College Mathematics Journal, vol. 23, no. 1 (January 1992), pp. 2–19. DOI: 10.2307/2686193.
  5. Livio, Mario. The Golden Ratio: The Story of Phi. Broadway Books (2002). ISBN 0-7679-0815-5.

Sarah Chen

Founder of Grid Maker Pro. Classical portrait painter (BA Fine Arts). Writes about composition, drawing methods, and the art-history claims that don't survive measurement.

Updated · Methodology