Split a line so the whole is to the long part as the long part is to the short part. That ratio is φ ≈ 1.618. It never repeats, it hides inside the Fibonacci numbers, and it is the number fractions approximate worst.
Glue Fibonacci-sized squares together, each one as wide as the long side so far. The shape that grows tends toward a golden rectangle, and the quarter-circles through each square trace a spiral.
Try this: press Build and watch Long ÷ short settle on 1.618.
Cut a square off a golden rectangle and the piece left over is another golden rectangle, smaller and turned 90°. Write the long side as a and the short side as b:
Fibonacci rectangles never hit it exactly, since every one has whole-number sides. They get closer with each square.
Divide each Fibonacci number by the one before it. The ratios overshoot and undershoot φ in turn, closing in every time.
How to read this: each dot is one Fibonacci number divided by the one before it (1/1, 2/1, 3/2, 5/3…). The dots hop over and under the dashed φ line, getting closer each time.
How to read this: each bar is one step of the zigzag above, carried on to step 30.
The error shrinks by a factor of φ² ≈ 2.618 each step, so you gain log₁₀ φ² ≈ 0.418 correct digits per term: about 5 digits every 12 terms.
| n | Fraction | Decimal (digits that match φ in gold) | Digits |
|---|
Since |ψ| < 1, the ψⁿ term fades away and F(n) becomes φⁿ/√5 rounded to the nearest whole number. Its sign flips each step, which is why the ratios zigzag.
Seed i sits at distance √i from the center, rotated by a fixed angle from the seed before it. At the golden angle, 137.508°, the seeds pack with no gaps. Move the angle by a tenth of a degree and spokes appear.
How to read this: each dot is a seed, placed one fixed turn after the last. Older seeds sit near the middle, newer ones further out, and they fade as they go. With spiral arms on, neighboring arms alternate gold and green so you can count them.
Try this: click 137.3°, then 137.508° golden, and watch the straight spokes vanish.
If the turn is close to a fraction p/q, every q-th seed lines up and you see q straight spokes. The golden angle is 1/φ² of a turn, and its best fractions are ratios of Fibonacci numbers. That's why real sunflowers show 21, 34, 55 or 89 spirals.
Start from x = 1 + 1/x and substitute the equation into itself forever. You get a continued fraction made only of 1s, the slowest-converging one possible.
How to read this: each step of depth tucks one more "1 + 1/…" inside the bottom. The value creeps closer to φ.
Every truncation is a ratio of Fibonacci numbers. Larger terms in a continued fraction make it converge faster. φ has the smallest possible term, 1, at every step, so no number is harder to approximate with fractions.
That resistance to fractions is what keeps sunflower seeds from lining up in Lab 3.
φ has a reputation that runs ahead of the evidence. Here is what holds up.
In a regular pentagon, diagonal ÷ side = φ exactly. The crossing points of the star split each diagonal in the golden ratio too.