Why Phi? – the rainbow angle

Posted: September 3, 2017 by oldbrew in Maths, Measurement, Phi, weather

The rainbow angle [credit: Hong Kong Observatory]

The minimum deviation angle for the primary bow [of a rainbow] is 137.5° according to Wikipedia. This is known as the rainbow angle. A circle is 360 degrees, so the ratio of the rainbow angle to the circle is therefore the square of the golden ratio i.e. 137.5:360 = 1:2.61818~.
– – –
Hong Kong Observatory has some useful explanatory text and graphics (rounding 137.5 to 138 degrees) titled:
Why is the region outside the primary rainbow much darker than that inside the primary rainbow?
Written by : SIU Kai-chee (summer intern) and HUNG Fan-yiu

Let’s first look at Figure 1, which shows sun rays entering a water drop and going through refraction and reflection.

The ray (ray no. 1) passing through the centre goes directly backward on reflection, i.e. a change in direction of 180 degrees.

For ray no. 2, this angle becomes smaller, following the rules of refraction and reflection.

For the next (ray no. 3) the angle continues to decrease, so on and so forth. This trend does not continue for long, however.

It is found that after a certain ray (ray no. N), the angle starts to increase instead, after going through a minimum angle of 138 degrees.

The figure also tells us that the outgoing rays tend to be more concentrated near ray no. N, making that region brightest. In fact, the angle of 138 degrees is the primary rainbow angle.

Now a rainbow is formed from lights coming out of many water drops. In fact, each of the colours in the rainbow comes to the eye from a different set of water drops.

Figure 2 [click here] makes it easier to visualize. Rays A and B on coming out of the water drop turn through an angle less than 138 degrees, and thus do not reach the eye. They either pass over our heads or to our right or left.

On the other hand, such rays as C and D turn through an angle near 138 degrees, and become part of the primary rainbow. This explains the darker region above the primary rainbow.

Below the primary rainbow and further down, e.g. ray Y, the angle is larger than 138 degrees, and light reaching our eyes is mostly reflected light from the water drop’s front surface. This region, being made up of all such rays, is thus brighter.

[Reference: “Atmospheric Phenomena”, Readings from Scientific American, W.H. Freeman and Co., 1980.]

Source: Why is the region outside the primary rainbow much darker than that inside the primary rainbow? | Hong Kong Observatory
– – –
Interactive: Rays through a raindrop

Wikipedia: The golden angle

Anti-solar point

  1. Nick Stokes says:

    “A circle is 360 degrees, so the ratio of the rainbow angle to the circle is therefore the square of the golden ratio”
    Golden ratios don’t relate to angles, but to their sin/cos’s. But any appearance here is coincidental. The Wiki derivation actually says, about 138°, and it depends on the refractive index of water, which doesn’t fix any neat geometry relation. Obviously if the refractive index were 1, there wouldn’t be a rainbow at all.

  2. oldbrew says:

    The angle varies very slightly for different parts of the spectrum.

    – – –
    Then there’s the golden angle, which is 137.5 degrees as is the rainbow angle.

  3. oldbrew says:

    Signs sometimes show rainbows. The glass beads incorporated into their surfaces refract and reflect light like raindrops.
    [includes copyright pic]

    So the process is not dependent on the properties of water only.

  4. Paul Vaughan says:

    “Golden ratios don’t relate to angles”

    That’s a strictly false comment.

  5. Paul Vaughan says:

    wuwt (a site I’ve stopped monitoring) is the place for negativity like that

  6. Paul Vaughan says:

    making a difference by adding Ramanujanian light

    √5 = φ+Φ
    a difference of squares

    1 = φ-Φ

    multiplying by 1
    is the same as
    multiplying by φ-Φ

    √5 = φ+Φ
    √5 = (φ+Φ)(1)
    √5 = (φ+Φ)(φ-Φ)

    √5 = φφ-φΦ+Φφ-ΦΦ

    φΦ = Φφ = 1

    √5 = φφ-1+1-ΦΦ

    -1+1 = 0
    √5 = φφ-ΦΦ

    which is a difference of squares.

    _ _

    Summing up:
    √5 = φ+Φ

    seen differently
    √5 = φφ-ΦΦ

    in Ramanujanian light
    √5 = φ+Φ = φφ-ΦΦ

    clarifies a powerful balance of irrational positive and negative sum and difference
    φ+Φ = φφ-ΦΦ

    simple stability with a fractal switch

    With recursive Pareto principled division of unity, stability’s natural.
    Φ’s the streamline of stabilizing division.

    Hardy wisely recognized Ramanujan’s brilliance with continuing fractions.

    Stable balance during a simple switch of powers hinges in a rigorously principled manner on irrational division.

    Imagine the level of stability that could be efficiently achieved by a skilled politician possessing Ramanujan’s relatively divine mastery of division.

    During a power switch such an enlightened politician could make a stable difference by adding irrationally balanced continuing fractions.

  7. Nick Stokes says:

    “So the process is not dependent on the properties of water only.”
    The process isn’t. But the angle is. The wiki derivation is

    As you see, it is dependent on the value of n (refractive index of water). And it isn’t exactly 137.5°.

  8. oldbrew says:

    Critical angle – BBC education guide

    Waves going from a dense medium to a less dense medium speed up at the boundary between them. This causes light rays to bend when they pass from glass to air at an angle other than 90°. This is refraction.

    Beyond a certain angle, called the critical angle, all the waves reflect back into the glass. We say that they are totally internally reflected.

    All light waves, which hit the surface beyond this critical angle, are effectively trapped. The critical angle for most glass is about 42°. [bold added]


    As we see here with the rainbow from glass beads embedded in the sign…

  9. Nick Stokes says:

    Further down in the Wiki article:
    “Droplets (or spheres) composed of materials with different refractive indices than plain water produce rainbows with different radius angles. Since salt water has a higher refractive index, a sea spray bow doesn’t perfectly align with the ordinary rainbow, if seen at the same spot.”

  10. Nick Stokes says:

    “The critical angle for most glass is about 42°. [bold added]”
    That critical angle has nothing to do with the rainbow angle. It isn the angle for total internal reflection.

  11. oldbrew says:

    Minimum deviation angle (180-42 = 138).

  12. oldbrew says:

    The critical angle θcrit is the value of θ¹ for which θ² equals 90°

    For water it’s 48.6° according to Wikipedia. 48.6+90 = 138.6 which is roughly the minimum angle of deviation (varies slightly depending on which part of the light spectrum is examined).
    360 / 2.6 (or 13/5) = 138.4615~

    – – –
    On the Physics of Rainbow, by Federica Volpi

    Deviations are traditionally measured from the direction of the incoming sunlight. The deviation angle for red rays forming the edge of the primary bow is about 137.5º. The centre of a rainbow is directly opposite the Sun (a deflection angle of 180º). The radius of the red edge of the primary is therefore 180°-137.5° = 42.5º
    [bold added]


  13. Mike In Fairfax says:

    Hmm, didn’t see anyone here pointing out that raindrops are *not* always spherical (and not teardrop either). They larger they are, the flatter. This deviation from spherical to a ‘hamburger bun’ shape can explain phenomena such as the twinned rainbow where two rainbows share a common endpoint but follow different curves.

    For the best research that I’ve seen on rainbows, you have to go to the folks who are not just attempting to understand how nature works, but to duplicate it… and that would be Disney. I had no idea the Magic Kingdom did this sort of research when I stumbled across it, but it does make sense they’d be on it.

    For a good write-up, see this link:
    For the researchers paper, see this link:

  14. oldbrew says:

    The curve of the rainbow and the precision of its arc are also interesting.

    Rainfall isn’t a requirement for a rainbow. They can form in almost ground-level mist e.g. at Niagara Falls.

  15. oldbrew says:

    Mike – noted this in your link:

    Although these simulations appear to explain twinned rainbows, “we have not validated this physically,” Jarosz cautioned. “It would be nice to actually show that by manually producing showers with two raindrop sizes does in fact produce this effect.”

  16. oldbrew says:


    Assuming that the sun and rain come from above, orthogonal (perpendicular) to the plane of the leaf, the divergence angle must be such as to minimize blockage of lower leaves by higher leaves. Therefore, any sort of periodic leaf arrangement must be avoided, if possible, as this will result in such blockage. So, the most optimal arrangement is obtained if we divide the circle formed by the plant (i.e. in the image above) by an irrational number–the more irrational the better. Earlier, we determined that the golden ratio is the most irrational number, and therefore we arrive at a divergence angle of 360/(phi) = 222.5, or, equivalently, 360-222.5 = 137.5 degrees, which is the most prevalent angle observed.
    . . .
    …this model is simply that: a model. “Consequently, the phyllotaxis rules I have described cannot be taken as applying to all circumstances, like a law of nature. Rather, in the words of the famous Canadian mathematician Coxeter, they are ‘only a fascinatingly prevalent tendency’ “

    [bold added]


    NB in the image above the leaves are numbered 0-12 but 1-13 would match the Fibonacci sequence.

  17. oldbrew says:

    Okabe, T. Biophysical optimality of the golden angle in phyllotaxis.
    Sci. Rep. 5, 15358; doi: 10.1038/srep15358 (2015).

    Plant leaves are arranged around a stem axis in a regular pattern characterized by common fractions, a phenomenon known as phyllotaxis or phyllotaxy. As plants grow, these fractions often transition according to simple rules related to Fibonacci sequences. This mathematical regularity originates from leaf primordia at the shoot tip (shoot apical meristem), which successively arise at fixed intervals of a divergence angle, typically the golden angle of 137.5°… [continues]


  18. Paul Vaughan says:

    “similarity classifier with a golden mean [109]”

    Click to access IJISM_507_Final.pdf