## Nautilus shell spirals may have phi proportions, but not as you may have heard.

The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. Some say yes, but offer no proof at all. Some show examples of spirals, but incorrectly assume that every equi-angular spiral in nature is a golden spiral. One university math professor says no, but only compared the nautilus spiral to the spiral created from a golden rectangle. Another university professor says no, but only measured height and width of the entire shell. Let’s look at this objectively and solve this mystery and debate.

## The Golden Spiral constructed from a Golden Rectangle is NOT a Nautilus Spiral.

A traditional Golden Spiral is formed by the nesting of Golden Rectangles with a Golden Rectangle. This resulting Golden Spiral is often associated with the Nautilus spiral, but incorrectly because the two spirals are clearly very different.

A Golden Spiral created from a Golden Rectangle expands in dimension by the Golden Ratio with every quarter, or 90 degree, turn of the spiral. This can be constructed by starting with a golden rectangle with a height to width ratio of 1.618. The rectangle is then divided to create a square and a smaller golden rectangle. This process is repeated to arrive at a center point, as shown below:

The golden spiral then is constructed by creating an arc that touches the points at which each of these golden rectangles are divided into a square and a smaller golden rectangle.

You can find images of nautilus shells and spirals all over the Internet that are labeled as golden ratios and golden spirals, but this golden spiral constructed from a golden rectangle is nothing at all like the spiral of the nautilus shell, as shown below. This had led many to say that the Nautilus shell has nothing to do with the golden ratio.

## Is there more than one way to create a golden spiral?

There is, however, more than one way to create spirals with golden ratio proportions based on 1.618 in their dimensions. The traditional golden spiral (aka Fibonacci spiral) expands the width of each section by the golden ratio with every quarter (90 degree) turn. Below, however, is another golden spiral that expands with golden ratio proportions with every full 180 degree rotation. Note how it expands much more gradually. The golden ratio proportions are indicated by the red and blue golden ratio grid lines provided by PhiMatrix software.

The center/vortex of the spiral increases to a width of 1 at point A. The half rotation of 180 degrees to point B expands the width of the spiral to 1.618, the golden ratio. Continue another half turn of 180 degrees to point C to complete the full rotation of 360 degrees. The width of the spiral from the center is now 2.618, which is the golden ratio (phi) squared. The golden ratio lines in red indicate how another full rotation expands the length from the vortex by phi squared, from phi to phi cubed. And so the pattern of expansion continues.

## The Golden Spiral on a 180 degree rotation is a better fit for the Nautilus Spiral.

The image below has this 180 degree rotation golden ratio spiral overlayed in red on a nautilus shell spiral. As you can see, the fit is fairly good for the first three full rotations from the center focal point. Beyond that point, this particular nautilus shell begins to show a slightly more gradual and open curve than the 180 degree Golden Spiral. All in all though, its relationship to a golden ratio spiral is becoming more apparent.

Below is a photo of another nautilus shell. It has the same general pattern in that its spiral curve conforms fairly closely to a full rotation golden ratio spiral expansion for the first three rotations, but this one has a tighter curve than the golden ratio spiral in its final outward spiral.

If we measure the actual dimensions of the above Nautilus shell, we find that its expansion rate with each rotation can be as low as 2.58. This is slightly less than 2.618, Phi squared, as in the idealized golden spiral above. Expansion rates in this same shell ranged to 2.9. Rates over 3 were observed in other shells. Note how the expansion rate varies for any given Nautilus as you rotate the shell, as illustrated below:

Measurements made using PhiMatrix software

## A golden mean gauge seems to match the spirals of some shells, so is that the answer?

The Nautilus spiral is definitely not at all like the 90 degree spiral created with a golden rectangle. It’s closer to a spiral that expands by phi every 180 degrees, but that’s still not really a fit either. If you measure it with a golden mean gauge though, you may find that the gauge isn’t far off the distance from the inner spiral on one side of the center point to the outer spirals on the other side. Does that explain its association with the golden ratio? Let’s explore a little further.

## There’s one more way yet that the Nautilus spiral may relate to phi.

In the examples above, we assumed that the measurements of the Nautilus spiral expansion rate must start at the center of the spiral to determine whether it would represent a golden ratio. There are, however, other possibilities. Try measuring the dimensions and expansion rate formed by these three points:

- Point 1 – The outside point of any spiral of the nautilus shell
- Point 2 – The first inside spiral at one full rotation (360 degrees) from Point 1
- Point 3 – The second inside spiral found at two-and-a-half rotations (900 degrees) from Point 1.

As illustrated in the Nautilus shell below, the distance from Point 1 to Point 2 divided by the distance from Point 2 to Point 3 is quite close to a golden ratio for the complete rotation of the Nautilus spiral. This is indicated by the golden ratio ruler below, which has a golden ratio point at the division between the blue and white sections. When the blue section has a length of 1, the white section has a length of 1.618, for a total length of 2.618.

Using this approach, the actual spiral expansion rates for the above Nautilus shell, taken every 30 degrees of rotation were: 1.572, 1.589, 1.607, 1.621, 1.627, 1.622, 1.616, 1.573, 1.551, 1.545, 1.550 and 1.573. This averages to 1.587, a 1.9% variance from 1.618. This is not exactly a golden ratio, but then it’s not hard to see why it would appear to be one.

Not every nautilus spiral is created equal, nor is it created with complete perfection. Just as with the human form, nautilus shells have variations and imperfections in their shapes and the conformance of their dimensions an ideal spiral using any rate of expansion. So while many inaccurate claims have been made regarding both its existence and non-existence, the Nautilus spiral can exhibit dimensions whose proportions come close to phi. You’d likely have to search quite a few beaches to find a Nautilus shell whose spiral fits any of these phi-based spirals perfectly, and may never find one. Perhaps though it’s the visual appearance of dimensions that come close to phi proportions that has lead many to associate it with the golden ratio, and to view it as one of the most beautiful spirals in nature.

So what do you think? Is the Nautilus spiral related to the golden ratio or not? Share your thoughts below.

See the Spirals page for more information on spirals in nature.

## References:

The Man of Numbers – In search of Leonardo Fibonacci by Keith Devlin (page 64) – “Unfortunately, the belief that the Nautilus shell has the form of the Golden Spiral is another of those false beliefs about Euclid’s number. To be sure, the Nautilus shell is a spiral, and it is moderately close to spiraling by a constant angle, but that angle is not the Golden Ratio. Not even close. So there is no connection. And that is why this topic is tucked away at the end of this book!”

The Golden Ratio—A Contrary Viewpoint by Clement Falbo (page 127) – “The nautilus is deﬁnitely not in the shape of the golden ratio. Anyone with access to such a shell can see immediately that the ratio is somewhere round 4 to 3. In 1999, I measured shells of Nautilus pompilius, the chambered nautilus, in the collection at the California Academy of Sciences in San Francisco. The measurements were taken to the nearest millimeter, which gives them error bars of ±1 mm. The ratios ranged from 1.24 to 1.43, and the average was 1.33, not phi (which is approximately 1.618). Using Markowsky’s ±2% allowance forto be as small as 1.59, we see that 1.33 is quite far from this expanded value of phi. It seems highly unlikely that there exists any nautilus shell that is within 2% of the golden ratio, and even if one were to be found, I think it would be rare rather than typical.”

Sarah says

I am fascinated by the fact, How many natural things have golden ratio concepts integrated with them.

ShamanAKA11 says

star tetrahedron (stellated octahedron)

1.bp.blogspot.com/-CrCZWEgzMvA/Un5Ek-I2JoI/AAAAAAAAAj4/tHuFTTKRE0U/s1600/star_4_3.png

ShamanAKA11 says

well now i am sure that the growth rate is 4/3 per quarter turn

i2.minus.com/iwOpJCr3T0h40.jpg (x-ray image by Bert Myers)

i6.minus.com/ishyY0S0hkGk5.jpg

john says

Is it the polar equation r=exp(t) ?

Ruth deGraaff says

This spiral is often seen in nature, other than the nautilus shell. It is evident in pinecones, pineapples, many different shells, fireweed, and other flowers and seeds. I find it difficult to apply the formula: 0,1,1, 2, 3, 5, 8…. to such objects. How is that done?

viv rosser says

The pineapple spirals round in three different ways. Each spiral adds up to 8, or 13 , or 21 segments.

three numbers in the Fibonacci sequence.

Yusuf says

This is an amazing topic!

Harmen Mulder says

I agree

Pjgeiger says

_American Scientist_ article (March-April) says ‘exponential’ spiral and gives it in polar coordinates.

calico meaux says

Thanks for the add’l work on this, to clarify the golden ratio in the nautilus.

It appears the best description of the golden ration is not ‘static’, but a ‘growth’ ratio.

That is, natural, instinctive growth rates are at 1.62 with much of nature.

I guess there is really a heavenly Designer.

chris feige says

A point that you have overlooked with regard to the Golden Spiral and

Since you are using the Fibonacci sequence to draw your golden spiral You must remember that “The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence” (wikipedia: http://en.wikipedia.org/wiki/Golden_ratio#Relationship_to_Fibonacci_sequence)

Since you are examining the nautilus shell to compare to the Golden Spiral, you should realize that the difference growth rates between the two is proof of the rule rather than the exception. By measuring the nautilus shell and pointing out the tighter growth rate of the shell, you are establishing the boundary conditions; not disproving that expansion rates are the same.

Gary Meisner says

This article does NOT use the Fibonacci sequence to draw the golden spiral. Your point is valid that a Fibonacci spiral approximate the Golden Spiral as the numbers grow. The illustrations shown however use a true Golden Spiral, which is based on successive golden rectangles whose sides are already in the ratio of 1.618… to 1.

Sakis says

This is the most well documented article about the nautilus’s connection with the golden ratio.

It seemed impossible to me for a shell to be grow based on the golden ration square mode, since the growth of the shell is daily and small..

Your article proves the obvious. That the shell has the same proportion in every point you get.

P says

These last two comments are brilliant. It goes to show skepticism can lead you to make false assumptions. now I see how consistent this law of growth is expressed in the nautilus shell.

PANAGIOTIS STEFANIDES says

Please Ref:

* http://www.stefanides.gr/Html/Nautilus.htm

* http://www.stefanides.gr/Html/why_logarithm.htm

* http://www.stefanides.gr/Html/logarithm.htm

Regards from Athens,

Panagiotis Stefanides

http://www.stefanides.gr

John says

I hear all the time that the Fibonacci sequence of numbers oscillates about the Golden Ratio (i.e. dividing successive terms) until one gets closer and closer to the Golden number; but if one looks at it differently one can see a definite relationship exists from the get go.. Multiplying the Golden Ratio by itself repeatedly gives the Fibonacci sequence.

0 + 1 ( 1 + √5 ) /2 i.e. 1 G + 0 = G^1 = 1.618033988749^1

1 + 1 (1 + √ 5)/2 i.e. 1 G + 1 = G^2 = 1.618033988749^2

1 + 2 (1 + √ 5)/2 i.e. 2 G + 1 = G^3 = 1.618033988749^3

2 + 3 (1 + √ 5)/2 i.e. 3 G + 2 = G^4 = 1.618033988749^4

3 + 5(1 + √ 5)/2 i.e. 5 G + 3 = G^5 = 1.618033988749^5

5 + 8 (1 + √ 5)/2 i.e. 8 G + 5 = G^6 = 1.618033988749^6

8 + 13 (1 + √ 5) i.e. 13 G + 8 = G^7 = 1.618033988749^7

13 + 21(1 + √ 5) /2 i.e. 21 G + 13 = G^8 = 1.618033988749^8

The other thing I would like to point out is whenever one is comparing theory to practice; one needs a hell of a big sample size. And even then one will have to contend with the standard deviation.

Good luck with that one!!!

Gary Meisner says

Only 30 samples are required for statistical validity. Anyone want to volunteer?

John says

I hear all the time that the Fibonacci sequence of numbers oscillates about the Golden Ratio (i.e. dividing successive terms) until one gets closer and closer to the Golden number; but if one looks at it differently one can see a definite relationship exists from the get go.. Multiplying the Golden Ratio by itself repeatedly gives the Fibonacci sequence.

0 + 1 ( 1 + √5 ) /2 i.e. 1 G + 0 = G^1 = 1.618033988749^1

1 + 1 (1 + √ 5)/2 i.e. 1 G + 1 = G^2 = 1.618033988749^2

1 + 2 (1 + √ 5)/2 i.e. 2 G + 1 = G^3 = 1.618033988749^3

2 + 3 (1 + √ 5)/2 i.e. 3 G + 2 = G^4 = 1.618033988749^4

3 + 5(1 + √ 5)/2 i.e. 5 G + 3 = G^5 = 1.618033988749^5

5 + 8 (1 + √ 5)/2 i.e. 8 G + 5 = G^6 = 1.618033988749^6

8 + 13 (1 + √ 5) /2 i.e. 13 G + 8 = G^7 = 1.618033988749^7

13 + 21(1 + √ 5) /2 i.e. 21 G + 13 = G^8 = 1.618033988749^8

The other thing I would like to point out is whenever one is comparing theory to practice; one needs a hell of a big sample size. And even then one will have to contend with the standard deviation.

Good luck with that one!!!

alejandro says

The 1:1.618 for every 90 degree turn seems like it’s the only useful format for 2D design applications.

A web designer friend of mine was showing me how he uses the phi ratio to set up the relative widths of two text columns. The heights of the two columns varied according to the writer’s “word count” for each given column, and these height dimensions were completely independent of the column widths.

I told him that setting up a 1:1.618 relationship along a single (in this case lateral) dimension seemed useless if the goal is to develop harmonious, two-dimensional compositions. He disagrees with me.

Am I missing something?

Gary Meisner says

Yes, you are missing something. While the golden ratio is often illustrated with the familiar 2 dimensional golden spiral, it can be applied just as successfully in design aesthetics in a single dimension or line. That’s actually how the most basic definition of a golden ratio is created: Divide a line at the one point at which the ratio of the entire line to the larger segment is the same as the ratio of the larger segment to the smaller segment. That point is the golden ratio, and that is exactly what you’re friend is doing. See http://www.goldennumber.net/what-is-phi/ for an illustration.