Why do sunflower seeds form spirals? The golden angle

AA SLAA HLAI SLAI HL Solid Also in: Pure maths, Modelling

Research question

Does Vogel's model (seed n at angle n × 137.5° and radius proportional to √n) reproduce the spiral counts of real sunflower heads, and how does the pattern change if the angle is slightly different?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

Sunflowers pack seeds with a divergence angle close to 137.5°, producing Fibonacci numbers of spirals. Modelling this, testing small changes to the angle, and counting spirals on real flowers is a beautiful mix of maths and nature.

The mathematics you'll need

Polar coordinates and parametric plots
The golden angle 360°(1 − 1/φ)
Why √n gives uniform density (area argument)
Rational approximations and gaps in packing

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Photograph sunflower heads (or pine cones) and count spirals in each direction.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explain Vogel's model and each assumption.
Plot the model and count its spirals.
Change the angle slightly and observe gaps.
Compare with real flower counts.
Reflect on why nature approximates this angle.

Pitfalls that cost marks

Claiming every flower shows Fibonacci numbers.
Using degrees and radians inconsistently.
Pictures without explanation.

Showing personal engagement

Count spirals on flowers you grow or buy.
Try 137° and 138° and explain what you see.
Link to continued fractions and why φ is “most irrational”.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Measure packing efficiency (nearest-neighbour distances) for different angles.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).