Golden ratio

Deep Breakdown

What it tells me (Situation Awareness): $\varphi = (1 + \sqrt{5})/2 \approx 1.618$ is the golden ratio, also called the divine proportion. It appears in geometry, art, nature (spiral growth patterns), and algorithm design. In DA/DS its most direct applications are: the Fibonacci sequence (consecutive Fibonacci numbers converge to $\varphi$), as a step size in Fibonacci search (a derivative of binary search), and occasionally in visualization (the "golden rectangle" for aspect ratios). It is less operationally critical than $e$ or $\pi$ in daily data work, but you will encounter it in algorithm analysis and it is worth knowing its origin.

Flowchart

Real-world use

Where it is used

Fibonacci search algorithm (unimodal function minimization), analysis of Fibonacci heaps (advanced data structure), aspect ratio selection in data visualization, and appearing naturally in combinatorial growth models.

Problem it solves

In search and optimization, the golden ratio provides an optimal way to reduce a search interval. Fibonacci search minimizes the number of function evaluations when searching for the minimum of a unimodal function over an interval. It is the precursor to the golden section search method.

How it solves the problem

Because $\varphi^2 = \varphi + 1$, two successive evaluations within an interval preserve a consistent ratio when one is discarded. This self-similarity makes Fibonacci / golden section search efficient.

Example in a business

Golden section search: finding the optimal learning rate for a model within a range $[a, b]$ without a closed-form gradient. You evaluate the loss function at two interior points dividing the interval in golden ratio proportions, discard the worse half, and repeat. Converges faster than a uniform grid search.

The formula, piece by piece

Breaking the notation:

• $\varphi$: lowercase Greek letter phi. (Sometimes $\phi$ or $\Phi$ is used instead.)
• $1$: unity.
• $\sqrt{5}$: the square root of 5.
• $2$: the denominator.

Derivation from the defining proportion: if $a/b = (a+b)/a = \varphi$, then $\varphi = 1 + 1/\varphi$, which gives $\varphi^2 = \varphi + 1$. Solving the quadratic $\varphi^2 - \varphi - 1 = 0$:

(The negative root is $\psi = (1 - \sqrt{5})/2 \approx -0.618$, used in the Fibonacci closed form.) Closed-form Fibonacci (Binet's formula): $F(n) = \frac{\varphi^n - \psi^n}{\sqrt{5}}$, where $\psi = (1-\sqrt{5})/2$.

The formula in words

"The golden ratio is the unique positive number such that if you add 1 to it, you get its square. Put differently: $\varphi = 1 + 1/\varphi$. It is the ratio of a line segment such that the whole to the larger part equals the larger part to the smaller part." The self-similar property $\varphi^2 = \varphi + 1$ is why it appears everywhere: any time a structure is built from smaller copies of itself with the same proportions, $\varphi$ emerges.

Critical Concepts

1. Fibonacci connection: $\lim_{n \to \infty} F(n+1)/F(n) = \varphi$. Consecutive Fibonacci numbers converge to the golden ratio.
2. Irrational: $\varphi$ is irrational and in fact the "most irrational" number (hardest to approximate by rationals), which is why it appears in phyllotaxis and plant growth where packing efficiency is maximized.
3. Golden section search: optimization method for unimodal functions using $\varphi$ to place evaluation points.
4. Self-referential: $\varphi = 1 + 1/\varphi$ means the continued fraction representation is $[1; 1, 1, 1, \ldots]$, the simplest possible infinite continued fraction.
5. In coding: not directly used as a constant like $e$ or $\pi$, but Fibonacci-based algorithms appear in heaps, search, and hashing.

Step-by-step calculation (tiny example)

Task: compute $\varphi$ directly from the formula and verify $\varphi^2 = \varphi + 1$.

Step 1: Compute the formula $\varphi = (1 + \sqrt{5})/2$. $\sqrt{5} \approx 2.2360679...$ $1 + 2.2360679 = 3.2360679$ $3.2360679 / 2 = 1.6180339...$

Step 2: Verify phi squared equals phi plus 1 $\varphi^2 = 1.6180339^2 \approx 2.6180339$ $\varphi + 1 = 1.6180339 + 1 = 2.6180339$. Confirmed: $\varphi^2 = \varphi + 1$.

Step 3: Verify the Fibonacci ratio convergence Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... Ratios: 1/1=1, 2/1=2, 3/2=1.5, 5/3=1.667, 8/5=1.6, 13/8=1.625, 21/13=1.615, ... These converge toward $\varphi \approx 1.618$.

Code examples

Beginner example (Python)

import math   # Compute the golden ratio
phi = (1 + math.sqrt(5)) / 2
print(f"phi = {phi:.10f}")   # 1.6180339887 # Verify phi^2 = phi + 1
print(f"phi^2 = {phi**2:.10f}")   # 2.6180339887
print(f"phi + 1 = {phi+1:.10f}")   # 2.6180339887 (same) # Fibonacci sequence and convergence

def fibonacci(n: int) -> list[int]:
    """Generate first n Fibonacci numbers."""
    seq = [1, 1]
    for _ in range(n - 2):
        seq.append(seq[-1] + seq[-2])
    return seq
fib = fibonacci(20)
print("\nFibonacci ratios converging to phi:")
for i in range(2, 15):
    ratio = fib[i] / fib[i - 1]
print(f" F({i+1})/F({i}) = {fib[i]}/{fib[i-1]} = {ratio:.6f}")

Beginner example (R)

# Compute the golden ratio
phi <- (1 + sqrt(5)) / 2
cat("phi =", phi, "\n")  # 1.618034 # Verify phi^2 = phi + 1
cat("phi^2 =", phi^2, "\n")  # 2.618034
cat("phi + 1 =", phi + 1, "\n")  # 2.618034 # Fibonacci sequence
fibonacci <- function(n) { seq <- c(1, 1)
  for (i in seq(3, n))
  seq <- c(seq, seq[i-1] + seq[i-2]) seq
}
fib <- fibonacci(20)
ratios <- fib[3:20] / fib[2:19]
cat("Fibonacci ratios:", round(ratios, 5), "\n")
cat("True phi: ", round(phi, 5), "\n")

Professional example (Python)

import math   # Golden section search: find the minimum of a unimodal function over [a, b]
# Uses phi to place interior evaluation points optimally

def golden_section_search(f, a: float, b: float, tolerance: float = 1e-5) -> float:
    """ Find the minimum of a unimodal function f on [a, b]. Uses the golden ratio to minimize the number of function evaluations. Returns the approximate x location of the minimum. """
    phi = (1 + math.sqrt(5)) / 2
    resphi = 2 - phi   # = 1 / phi^2 = 0.3819...
    x1 = a + resphi * (b - a)
    x2 = b - resphi * (b - a)
    f1 = f(x1)
    f2 = f(x2)
    iterations = 0
    while abs(b - a) > tolerance:
        if f1 < f2:
            b, x2, f2 = x2, x1, f1
            x1 = a + resphi * (b - a)
            f1 = f(x1)
        else:
            a, x1, f1 = x1, x2, f2
        x2 = b - resphi * (b - a)
    f2 = f(x2)
    iterations += 1
    minimum_x = (a + b) / 2
    return minimum_x, iterations   # Test: find the minimum of f(x) = (x - 3)^2 + 1, which is at
x = 3
f = lambda x: (x - 3) ** 2 + 1
x_min, n_iter = golden_section_search(f, 0, 10)
print(f"Minimum found at x = {x_min:.6f}")   # Should be ~3.0
print(f"f(x_min) = {f(x_min):.6f}")   # Should be ~1.0
print(f"Iterations: {n_iter}")

Professional example (R)

# Golden section search: minimize unimodal f on [a, b] using phi
golden_section_search <- function(f, a, b, tolerance = 1e-5) { phi <- (1 + sqrt(5)) / 2
  resphi <- 2 - phi
  x1 <- a + resphi * (b - a)
  x2 <- b - resphi * (b - a)
  f1 <- f(x1)
  f2 <- f(x2)
  iterations <- 0
  while (abs(b - a) > tolerance) { if (f1 < f2) { b <- x2; x2 <- x1; f2 <- f1
      x1 <- a + resphi * (b - a)
      f1 <- f(x1)
    }
    else { a <- x1; x1 <- x2; f1 <- f2
      x2 <- b - resphi * (b - a)
      f2 <- f(x2)
    }
    iterations <- iterations + 1
  } list(x_min = (a + b) / 2, iterations = iterations)
}
f <- function(x) (x - 3)^2 + 1
out <- golden_section_search(f, 0, 10)
cat(sprintf("Minimum found at x = %.6f\n", out$x_min))
cat(sprintf("f(x_min) = %.6f\n", f(out$x_min)))
cat("Iterations:", out$iterations, "\n")

Source: AIO p. 57 of 286

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