Sunflower Cross Ratio Explorer

Cross Ratio

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click a seed to begin

Anharmonic group

If CR = λ, the six related values are:

λ 1/λ 1−λ 1/(1−λ) λ/(λ−1) (λ−1)/λ

For λ=−1: the six collapse to −1, 2, ½

Pencil lines

No lines yet — click a seed, then click neighbors.

Mode

Click any seed to anchor the pencil

Layers

Diamond seed shapes (gray) Seed dots

Spiral families

21 spirals (F₂₁ family) — gold 34 spirals (F₃₄ family) — blue 13 spirals (F₁₃ family) — purple 55 spirals (F₅₅ family) — green 89 spirals (F₈₉ family) — rust 144 spirals (F₁₄₄ family) — slate

Zoom

zoom out18×zoom in

Activity: Click any seed (dot) to place the pencil center.
Then click nearby seeds to draw lines from the center through them.
The cross ratio of up to 4 lines is computed and shown.

To find a harmonic pencil: Click a seed, then use Auto to see the two spiral directions and their bisectors — always CR = −1 by construction.

Or click two seeds, then Add bisectors.

Additive pencil (Fibonacci mechanism): Click two seeds to define vectors L and R from the anchor, then click Add L+R and L+2R. The four lines {L, R, L+R, L+2R} always give CR = −1 — but L+R is not the angle bisector when the two seeds are at different distances from the anchor.

Note on CR values: CR = −1, 2, and ½ are all the same harmonic pencil — just with lines in a different order.

Fibonacci neighbor pencil: With two seed lines chosen, click Add Fibonacci neighbors to add the previous and next Fibonacci family (e.g. Fₛ, Fₜ, F₝, F₞). CR ≈ −1 — improving for larger families.

Compare all three pencils: bisector, additive (L+R), and Fibonacci neighbors.