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A sunflower-like spiral can be modeled by placing each new floret about 137.5° from the previous one and increasing its distance from the center. That angle is the golden angle, 360°/φ², where φ is the golden ratio. Repeating the rotation spreads points around the head instead of stacking them on a few spokes; groups of nearby points then appear as interlacing spiral families called parastichies.
This construction explains the geometry, not the entire biology. To model growth, you must also specify when florets appear, how the receptacle expands, and how local crowding or contact affects placement.
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The golden-angle construction
The golden ratio is φ = (1 + √5)/2. The ideal divergence angle used in many phyllotaxis models is:
α = 360°/φ² ≈ 137.5°
Index florets by n = 0, 1, 2, … . A simple polar-coordinate model is:
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θn = θ0 + nα
rn = c√n
Here, θ0 sets the starting direction, rn is the distance from the center, and c controls the overall scale. The square-root rule is a convenient illustrative choice for producing an even-looking disk; it is not a claim that every sunflower grows according to that exact equation.
A reproducible point-placement procedure
- Choose a number of points, such as 1,000 florets.
- Set α to 137.507… degrees, or use radians (about 2.39996).
- For each index n, calculate θn and rn.
- Convert to Cartesian coordinates with xn = rn cos θn and yn = rn sin θn.
- Draw a small disk, circle, or seed at each (xn, yn).
Changing c changes spacing and head size. Changing the angle slightly lets you test how quickly visible order deteriorates. A constant angle makes this a final-arrangement diagram: every point is placed directly into its intended position rather than grown through time.
Why spirals and Fibonacci numbers appear
The golden angle is an irrational fraction of a full turn. Successive rotations therefore do not repeatedly land on a small set of radial lines. New points are distributed around the available space while the radius increases.
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When you inspect the completed point cloud, neighboring points can be connected in two opposite directions to reveal spiral families. These are parastichies. Their counts often form neighboring Fibonacci numbers because the convergents of the golden ratio provide especially good integer approximations to the irrational rotation. The visible Fibonacci counts are therefore a consequence of the geometry of the rotation and the way neighbors are traced—not a rule that every head must display one particular pair.
What a diagram can and cannot establish
- It can show how a near-golden divergence produces an efficient-looking, spiral-rich arrangement.
- It cannot prove that a living sunflower uses a perfectly constant 137.5° angle.
- It cannot establish that all heads have Fibonacci parastichy counts.
- It cannot by itself explain which developmental mechanism positions each primordium.
Turning a static pattern into a growth model
A developmental model adds time and changing geometry. Instead of asking only where point n ends up, it asks when a floret is initiated, how the receptacle changes, and how earlier organs constrain later ones.
Receptacle and floret models
One published sunflower-head model assumes a fixed divergence angle, a sigmoid growth function for each floret, and a fixed delay between successive initiations. Logistic functions describe growth toward a limiting size, and the authors report good least-squares fits to measured receptacle shapes in most of the cases they examined. Those are assumptions and reported fits for that model, not universal biological laws.
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To adapt this approach, define an initiation time tn = nΔt, choose a receptacle-growth function, and give each floret its own delayed sigmoid. The final head then emerges from the interaction of initiation timing, surface expansion, and organ growth rather than from a radius formula alone.
Whole-plant L-system models
A different approach uses an L-system to represent stems, nodes, leaves, and branching in three dimensions. One study measured plant features from photographs at five stages during one growing season, modeled internode and petiole growth with logistic functions of L-system step number, and reported reproduction of growth as node number increased.
That study describes alternate-phyllotaxis leaf divergence of about 135°, not the exact 137.5° idealization. The golden-angle model for a flower head should therefore not be conflated with every leaf angle on the whole plant.
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Comparing useful model types
| Model type | Target structure | How placement is obtained | Growth representation | Variation represented |
|---|---|---|---|---|
| Golden-angle point cloud | Idealized seedhead or receptacle | Constant imposed angle | Radius-only scaling, often r ∝ √n | Mostly fixed, Fibonacci-like spiral order |
| Receptacle/floret growth model | Sunflower head | Specified divergence plus initiation timing | Surface expansion, delayed sigmoid or logistic growth | Measured shape differences through fitted parameters |
| L-system plant model | Whole sunflower plant | Rule-based branching and organ development | Internode and petiole growth in three dimensions | Stage-dependent stem and leaf structure |
| Disk-stacking model | Flower-head parastichies | Local size and contact rules | Changing disk sizes on a surface | Fibonacci and non-Fibonacci counts, asymmetry, and transitions |
| Primordium-indexing analysis | Developing flower-head rim | Re-indexed insertion sequence when organs arrive in bursts | Developmental ordering rather than a full growth simulation | Ambiguous consecutive insertions and conditions for recovering the golden angle |
Why real sunflower heads are less tidy
Fibonacci spirals are frequent, but photographs and datasets can show non-Fibonacci counts, asymmetry, and transitions between patterns. A 2024 preprint testing Schwendener-style disk-stacking models against a large published sunflower parastichy dataset reports that slowly changing disk sizes generate well-ordered Fibonacci structure. Other parameter regions produce non-Fibonacci counts and asymmetries that the authors say resemble observed data. These are results of a model and a preprint, not a universal law or independent proof of one biological mechanism.
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The problem of burst insertion
On some flower-head rims, primordia are inserted in bursts rather than one strictly after another. As Milan Havlíček, Peter Klavžar, and Przemysław Prusinkiewicz wrote in their 2026 paper: “In flower heads, however, the divergence angle is difficult to define because primordia on the rim are inserted in bursts rather than sequentially, and the golden angle is not evident.” Their indexing method argues that the angle between consecutively indexed primordia can approximate—or under specified conditions equal—the golden angle. This is a mathematical account of how to interpret a developing pattern; the authors do not present it as a causal explanation of why the pattern forms.
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How to build a model for a class or project
Start with the geometric baseline
- Render 200, 1,000, and 5,000 points to show how spiral visibility changes with density.
- Compare 137.5° with nearby angles, such as 135° and 140°.
- Keep the radial rule fixed while changing the angle, then keep the angle fixed while changing the radial rule.
Add developmental assumptions explicitly
- State whether the angle is imposed or emerges from local rules.
- Specify the initiation interval, surface-growth function, and organ-growth function.
- Record whether parameters came from measurements, a fitted model, or an illustrative choice.
- Allow variable organ sizes or insertion bursts if you want non-ideal patterns.
Evaluate the result honestly
Compare more than the apparent Fibonacci pair. Check head outline, spacing, asymmetry, changes in parastichy counts, and the timing of organ initiation. A model that reproduces one spiral count but fails the head shape or developmental sequence has explained only one visible feature.
The mathematical takeaway
The golden angle is a powerful descriptive starting point: repeated rotation by about 137.5° plus outward growth produces dispersed points and visible parastichies. Fibonacci counts commonly appear because of the arithmetic of the golden ratio, but real heads can depart from the ideal through changing sizes, local interactions, developmental timing, and burst insertion. A convincing growth model must therefore distinguish the elegant geometry from the biological process that may generate, modify, or obscure it.
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