Exploration 01 · geometry
The infinite cut
- 01A rectangle
Seven by four. Nothing special about it, yet.
- 02One cut
Cut away the largest square you can. Something remains.
- 03What remains
Lay the whole rectangle, shrunk, over what remains. A different shape.
- 04Change the proportions
Drag the rectangle wider or narrower. Look for the one whose remainder is the same shape as the whole.
- 05The same shape
One ratio does it: the remainder is the whole again, smaller and turned. r = 1/(r − 1), so r² = r + 1.
- 06Follow it
Cut again. And again. Follow what remains.
- 07Again, and again
Every step is the same: scale by 1/φ, turn a quarter. The picture never changes. The drawing stops at the precision of a computer; the geometry does not.
- 08Fractions that come close
Fibonacci ratios cut like φ for a while, then run out. Each is closer than the last, from alternate sides.
Ends after 5 cuts.
What it shows
Cutting squares from a rectangle is Euclid's algorithm. The number of squares of each size is the rectangle's continued fraction.
A rectangle whose sides are whole numbers, 13 by 8 say, runs out of cuts: 13/8 = [1; 1, 1, 1, 2], six squares. Every ratio a slider or a decimal can hold is such a fraction, so its cuts end too, sometimes after many. Only an irrational ratio never ends.
The golden ratio φ = (1 + √5)/2 = 1.6180339887… is [1; 1, 1, 1, …]: one square at a time, forever. It is the only ratio whose remainder is the same shape as the whole, because r = 1/(r − 1) gives r² = r + 1. So every cut is the same step, a scale of 1/φ and a quarter turn, and following the remainder never changes the picture.
Fractions that come close
The Fibonacci ratios 3/2, 5/3, 8/5, 13/8, … cut like φ for a while and then end. Each is closer than the last, from alternate sides: F(n+1)/F(n) − φ = ψⁿ/F(n), where ψ = −1/φ. That is why φ is often called the most irrational number: its approximations improve as slowly as any can.
What it does not do
Compute forever. The drawing is exact to the precision of a computer; after about eighteen cuts the squares are smaller than a pixel. Because every step at φ is the same similarity, the zoom can keep going by returning to an identical picture, but the geometry it shows is still finite. The mathematics is not.