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Journey 2 of 10 · 9 steps

The Golden Thread

φ from stars to sunflowers

One number, φ ≈ 1.618, links the pentagram, the spiral, the sunflower and the most beautiful solids, and runs on into four dimensions.

Play it in SacredForm

In the app each step moves: figures draw themselves, rise from the page, turn and slice, and the narration can be read aloud. Here they are held still, one picture each.

  1. The golden cut: Pentagram

    Step 1 · Pentagram

    The golden cut

    Every crossing of a pentagram divides a line so that the whole is to the long part as the long part is to the short. That ratio is φ = (1 + √5) ÷ 2 ≈ 1.618. Inside the star is a pentagon, inside that another star, forever.

  2. Squares that grow: Golden Spiral

    Step 2 · Golden Spiral

    Squares that grow

    Squares with sides 1, 1, 2, 3, 5, 8, 13… each the sum of the two before. Their ratios creep toward φ, and quarter circles through them trace the golden spiral.

  3. The golden angle: Phyllotaxis

    Step 3 · Phyllotaxis

    The golden angle

    A sunflower places each seed 137.5° around from the last: a full turn divided by φ². Because φ is the “most irrational” number, the seeds never line up and never leave gaps.

  4. Why exactly that angle?: Phyllotaxis

    Step 4 · Phyllotaxis

    Why exactly that angle?

    Watch the angle drift by a fraction of a degree. Anything else makes spokes and gaps. Only near 137.5° do the seeds pack perfectly.

  5. A golden sphere: Fibonacci Sphere

    Step 5 · Fibonacci Sphere

    A golden sphere

    Wrap the same rule around a ball and the points spread almost perfectly evenly. From the pole, it is the sunflower again.

  6. Three golden rectangles: Golden Rectangles

    Step 6 · Golden Rectangles

    Three golden rectangles

    Three rectangles with sides 1 and φ, set at right angles through one another. Their twelve corners…

  7. …are the icosahedron: Icosahedron

    Step 7 · Icosahedron

    …are the icosahedron

    …are exactly the twelve vertices of the icosahedron, Plato’s water. Every coordinate is 0, ±1 or ±φ.

  8. The dodecahedron: Dodecahedron

    Step 8 · Dodecahedron

    The dodecahedron

    Its dual, the dodecahedron, is made of pentagons, so φ is on every face. Plato gave it to the cosmos itself.

  9. Into hyperspace: 600-Cell

    Step 9 · 600-Cell

    Into hyperspace

    In four dimensions the icosahedron’s big sibling is the 600-cell: 120 vertices, 600 tetrahedra, with edge length 1/φ. The golden thread runs through every dimension we can reach.