A golden rectangle is a rectangle ABCD whose dimensions are such that, if a square ABPQ is inscribed as shown, then the smaller rectangle PCDQ cut off is similar to the original.
Because the sides of the smaller rectangle are in the same ratio as those of the original, the process may be repeated, thereby cutting off yet a smaller golden rectangle. This could be repeated ad infinitum, thus:
It’s easy to find the ratio of the sides. If we take x = BC and y = AB, then the ratio of x to y is the same as the ratio of y to x – y. So x/y = y/(x – y). Writing x = ϕy, we have ϕ = 1/(ϕ – 1), or ϕ2 – ϕ – 1 = 0. Solving for ϕ yields (1 + √5)/2, or about 1.618. We call ϕ the golden ratio.
Now, if we take our sequence of squares and construct a quarter-circle in each square, then we obtain a spiral shape.
It’s often claimed that this models the shell of the chambered nautilus and other objects found in nature.
This isn’t really the case, however. Part of the reason is that the “spiral” is not a true spiral. It’s pieced together from a sequence of quarter-circles, each of which has a different radius. In other words, the radius of curvature jumps discontinuously as we move around the spiral. The shell of a nautilus, on the other hand, is a logarithmic or equiangular spiral. The radius varies continuously and by a geometric progression as we turn around the spiral.
We can still use the golden rectangle to construct a logarithmic spiral. If we draw two successive diagonals as shown
then they intersect at the point that the sequence of squares converges upon. Call this point O. As we look closer and closer at O, we have an infinite sequence of rectangles, each of which is similar to the original. For instance, if we were to expand the picture to make SU as long as BC is now, then it would look exactly the same, complete with the infinite sequence of squares spiraling in to O. The picture is self-similar. Each of the four line segments emanating from O touches infinitely many corners in the sequence of golden rectangles. For instance, the segment from O to B touches B, S, and so on—the points are too close together to label—and each occupies the same corner in its respective golden rectangle.
The point O is the fixed center of our true spiral. The segment lengths OB, OC are in a golden ratio, as are OC, OD, and OD, OQ, and so on. Furthermore, these successive pairs all meet at right angles.
Imagine tracing out the spiral starting at A. The point O is the center of the spiral. As we turn, the spiral draws closer and closer to O, in such a way that its distance decreases by a factor of ϕ with each quarter-turn. Thus, if the distance from O to A is 1, then, as we turn clockwise from OA to OP, the distance of the spiral from O should decrease to ϕ-1. As we turn from OP to OR, the distance should decrease to ϕ-2. And so on. Conversely, if we turn in the counterclockwise direction, then the distance of the spiral from O increases by a factor of ϕ with each quarter-turn.
The golden spiral thus constructed is closely approximated by the artificial spiral we constructed above. The latter is sometimes incorrectly called the golden spiral, but is actually known as the Fibonacci spiral. The golden spiral is but one example of a logarithmic spiral; it is based on the factor ϕ, but any other factor s > 1 could be used instead. In terms of polar coordinates, this would be parametrized by r(θ) = α ⋅ s2θ/π, where α > 0 is an arbitrary scaling factor.
The logarithmic spiral was studied by the Swiss mathematician Jacob Bernoulli (1654 – 1705), who called it the Spira miribilis—the “wonderful spiral”—because of its property of self-similarity. He was so taken with this property that he requested a logarithmic spiral to be incised on his epitaph, together with the motto EADEM MUTATA RESURGO (“though changed, I remain the same”). The craftsmen misunderstood and put a simple Archimedean spiral instead.