Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

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?

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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

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.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Polar coordinates and parametric plots; The golden angle 360°(1 − 1/φ)
AA HL	Good fit	Polar coordinates and parametric plots; The golden angle 360°(1 − 1/φ)
AI SL	Good fit	Polar coordinates and parametric plots; The golden angle 360°(1 − 1/φ)
AI HL	Good fit	Polar coordinates and parametric plots; The golden angle 360°(1 − 1/φ)

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: claiming every flower shows Fibonacci numbers; and using degrees and radians inconsistently. Say how each one affects your answer.

Use of mathematics (E)

SL: 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, each explained in your own words and carried out correctly.

How art & music explorations reach the top bands in general: Art, music & design IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

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

Extending it for HL

Add an HL technique the question genuinely needs: what an HL art & music exploration adds.

See a complete IA, marked

Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete art & music exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Polar coordinates and parametric plots, the golden angle 360°(1 − 1/φ), why √n gives uniform density (area argument) and rational approximations and gaps in packing.

Where can I get data for this IA?

Photograph sunflower heads (or pine cones) and count spirals in each direction. Checked sources and tools: Desmos graphing calculator and GeoGebra.

Turn this idea into your IA

The planner opens with “Why do sunflower seeds form spirals? The golden angle” filled in: the question, the maths and the data sources.

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