Fibonacci spirals and Golden spirals appear in nature, but not every spiral in nature is related to Fibonacci numbers or Phi. Most spirals in nature are equiangular spirals, and Fibonacci and Golden spirals are special cases of the broader class of Equiangular spirals. An Equiangular spiral itself is a special type of spiral with unique mathematical properties in which the size of the spiral increases but its shape remains the same with each successive rotation of its curve. The curve of an equiangular spiral has a constant angle between a line from origin to any point on the curve and the tangent at that point, hence its name. In nature, equiangular spirals occur simply because they result in the forces that create the spiral are in equilibrium, and are often seen in non-living examples such as spiral arms of galaxies and the spirals of hurricanes. Fibonacci spirals, Golden spirals and golden ratio-based spirals often appear in living organisms.
The Fibonacci spiral gets closer and closer to a Golden Spiral as it increases in size because of the ratio of each number in the Fibonacci series to the one before it converges on Phi, 1.618, as the series progresses (e.g., 1, 1, 2, 3, 5, 8 and 13 produce ratios of 1, 2, 1.5, 1.67, 1.6 and 1.625, respectively)
The structure represents two well known sacred geometry shapes: the spiral and . The Golden mean spiral is the secret proportion of beauty as it exists in nature. The golden mean proportion was used by the ancient Egyptians, Greeks and the Renaissance artists such as Da Vinci and even modern artists such as Kandinsky.
The nautilus shell is often shown as an illustration of the golden ratio in nature, but the spiral of a nautilus shell is NOT a golden spiral, as illustrated below. The golden spiral overlay is provided by :
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Below is a photo of another nautilus shell. It has the same general pattern in that its spiral curve conforms fairly closely to a full rotation golden ratio spiral expansion for the first three rotations, but this one has a tighter curve than the golden ratio spiral in its final outward spiral.
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Spiral galaxies: The Milky Way has a number of spiral arms, each of which has a logarithmic spiral of roughly 12 degrees. The shape of the spiral is identical to the Golden spiral, and the Golden rectangle can be drawn over any spiral galaxy.