## 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. Several university math professors say no, but they 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 is then 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 of 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 above 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. This Golden Spiral based on a 180 degree rotation is a much better fit to the Nautilus Spiral.

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

If you measure a Nautilus shell with a golden mean gauge, 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 this explain its association with the golden ratio? Let’s explore a little further.

## Another spiral variation may relate the Nautilus spiral to phi

Let’s continue to explore that fit of a slightly different variation on a golden spiral. Rather than seeking a golden ratio from the spiral’s center point, let’s 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 shown below. 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.

The two golden spirals we’ve identified then look like this:

The image below has the “golden ratio to opposite 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 point. Beyond that point, this particular nautilus shell begins to show a slightly more gradual and open curve than this 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 the “golden ratio to opposite spiral” for the first three rotations, but this one has a tighter curve than the golden ratio spiral in its final outward spiral.

## Spiral growth rates from the center point

Let’s take another look at the spirals of the Nautilus based on the center point. If we measure the actual dimensions of the above Nautilus shell, we find that its expansion rate with each rotation from its center point 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

So, we see that 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 conformity of their dimensions an ideal spiral using either of the two methods shown here. So while many inaccurate claims have been made regarding both its existence and non-existence, we see that 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. We can see though that the visual appearance of dimensions come close to phi proportions, and understand why this has lead many people 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:

Following are comments by three Ph.D.s in mathematicians who say that the Nautilus has no relationship to the golden ratio. This is true with respect to the classic golden spiral, but misses the fact that there is more than one way to construct a spiral with golden ratio proportions.

The Man of Numbers – In search of Leonardo Fibonacci by Dr. 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!”

Replicator Constructions by Dr. George Hart – “My goal here was to comment on the common misconception that the nautilus has a golden ratio spiral. A real nautilus doesn’t. This is what a nautilus shell would look like if it were based on a golden spiral. I built it in halves on a raft, then glued the halves together. I’m quite happy with the final result. There’s a video explaining more about it here.”

The Golden Ratio—A Contrary Viewpoint by Dr. 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.”

Note: A special thanks go to Oliver Brady for his astute analysis of this article, which led to improvements in its clarity and accuracy.

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.

PANAGIOTIS STEFANIDES says

Ref:

http://www.stefanides.gr/Html/Nautilus.html

http://www.stefanides.gr/Html/why_logarithm.html

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.

Randolph says

Why do you think there is a designer?

Dan says

I’m assuming he has in mind the florets of a sunflower, which are arranged at every 137.5 degrees. (Bear with me for a while) In an overwhelming number of plants, a given branch or leaf will grow out of the stem approximately 137.5 degrees around the stem relative to the prior branch. In other words, after a branch grows out of the plant, the plant grows up some amount and then sends out another branch rotated 137.5 degrees relative to the direction that the first branch grew out of. Plants use a constant amount of rotation in this way, although not all plants use 137.5 degrees. However, it is believed that the majority of all plants make use of either the 137.5 degree rotation or a rotation very close to it as the core number in their leaf or branch dispersion, sending out each and every leaf or branch after rotating 137.5 degrees around the stem relative to the prior branch.

If we were to multiply the value of 1 over Phi to the second power (0.3819659…) times the total number of degrees in a circle (360), we obtain for a product nothing other than 137.50… degrees. As an alternate way to look at the same idea, if we were to take the value of 1 over Phi (0.6180339…) and multiply it by 360, we obtain approximately 222.5 degrees. If we then subtract 222.5 from 360 we again find 137.5 degrees – in other words, the complimentary angle to 1 over Phi is 137.5 degrees, which also happens to be the value of 1 over Phi to the second power times 360.

So, if we have followed the described mathematics, it is clear that any plant that employs a 137.5 degree rotation in the dispersion of its leaves or branches is using a Phi value intrinsically in its very form.

All that to say that there’s absolutely no way a plant could make this calculation on its own. How could this happen randomly, yet with remarkable precision and beautiful patterning, according to evolutionary theory which states that evolution is a random product of genetic mutation?

Mona says

If, however, the common ancestor of all plants with vascular systems such that they could spiral had DNA that encoded that many degrees of separation, and managed to pass it down to most plants, that could explain the prevalence. It might be easier for a plant to build with Phi as well, because of a reason within the fundamental laws of physics of our universe. That would be similar to how 3-way symmetry and triangles are aesthetically pleasing to humans but are also very stable in building and growth. An eye for this stability and the use of it may have evolved over time, like how hexagonal nest building probably evolved over time in honeybees.

Mona says

I recently found out that if leaves spiral based on a Golden Spiral, they get optimal sunlight absorption (they don’t get in each other’s shade.) Part of this is that Phi is irrational. If the leaves fell every 90 degrees about the stem, only the top four would get full sun. However, since Phi is irrational, the stem and leaves could keep on growing to infinity and one leaf’s tip would never fall on top of another’s. Think about it this way: if you construct a circle and then go 1/phi^2/360=~137.50… degrees around the circle, and repeat the process into infinity, no rotations will land on the same point. Therefore, leaves using this method have a distinct advantage in that they are able to photosynthesize more, and would pass down their genes to more offspring. With each generation, the rotation of leaves about the stem would close in on 1/phi^2/360 degrees.

PANAGIOTIS STEFANIDES says

Interesting!

360-137.5077641..=222.4922359..

360-222.4922359..= 137.5077641..

360/222.4922359..= 1.618033989..

360/2.618033989.. = 137.5077641..

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 https://www.goldennumber.net/what-is-phi/ for an illustration.

Harmen says

Many believe that the golden spiral is in de nautilus

Johan says

The Phi Ratio is still conected with the nautilus spiral, human body and another thinks on the Universe. But, like humans, a nautilus spiral itself are never have a perfect “Phi” spiral in nautilus spiral shell. A biological basis (nautilus shell, human body and face, etc) are never fit perfecly with the geometrical basis (pentagon, decagon, etc) because a geometrical basis are a perfection of line, shape and pattern of nature and a mathematical equation.

Deljuan Calvin says

I like it, being in balance with nature and the universe, it’s both rare and simplistic,all within our grasp of understanding.

“I Like It”.

Mary Jane says

Hello, thank you for this detailed explanation! I was wondering if a nerites shell spiral is a golden spiral as well. https://en.wikipedia.org/wiki/Neritidae

Thank you

Mona says

It looks like if it was a golden spiral, it would be a 90 degree one.

Greg Martin says

The question is how a simple animal makes a decision as it goes in building each next layer of shell. Certainly not puling out a protractor, but could it be some sort of simple fractal formulation that drives this?

Lance Crumley says

Perhaps The Designer of our universe and our world, more correctly, Logic would dictate, (His Universe His World) Perhaps He designed it with Absolute Mathematical Precision. In which case the Nautilus would give evidence to support such an idea. Perhaps also by His Design He allowed His Creation to be subjected to corruption in which case the Perfection would be marred though still evident.

The Evidence certainly lends creditability to this Theory.

Which now compels us if not Obligates us to ask a question of more consequence than our first……..

WHY has He done so?

Zulfi says

https://quran.com/55

… So which of the favors of your Lord would you deny?…

https://quran.com/67/3

[And] who created seven heavens in layers. You do not see in the creation of the Most Merciful any inconsistency. So return [your] vision [to the sky]; do you see any breaks?

Jesse Glommen says

All “Golden Rules” are subject to relativity and forces. For instance, how might we account for depth, water pressure, current, temperature, etc..

Consider also seasonal and environmental changes and influences – such as summer and winter, or rainy and dry periods which might vary significantly in nourishment, etc..

Years from now, we might all have a good laugh as we look back on our simple friend, the nautilus, as a key partner in the advancement and evolution of liquid math…

Rich Newman says

As I read through this, I was thinking the same thing. In fact I wonder if the variations of any particular nautilus to the math could be measured and compared to place and season. Just as tree growth rings can be read to identify particular years, why not nautilus shell growth and other inert carbon forms?

John Shanahan says

I wonder if these golden spirals may relate to my speculations on Phi in the Solar system my web site is at http://john-shanahan-berlin.jimdo.com/blog/

Gary B Meisner says

Some very interesting relationships there, John. See also https://www.goldennumber.net/solar-system/ and http://www.solargeometry.com/.

PANAGIOTIS STEFANIDES says

Very interesting link [http://www.john-shanahan-berlin.de/]

>… All music intervals are the products of three numbers 2, 1.5, and 1.25,….<

With reference to 1.25, in article it may be of interest the following:

[41.8103149..]Deg.=2*arctan{1/[Φ^2}=arctan{1/sqrt (1.25)}

or,

41.8103149.Deg.=2*arctan{1/[1.618033989..]^2}=arctan{1/sqrt (1.25)}

—————————————————————————-

Please you may 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 shanahan says

Thanks for your interest, indeed the square root of 1.25 is closely related to Phi.

John Shanahan says

By the way Panagio

tis, the height of one of the spikes of the pentagram is sin 72 degrees .951056516. Phi sqared devided by sin 72 four times is three point two, devided by four or two octaves is the reciprical of 1.25 thanks Gary!

NigelReading|ASYNSIS says

Good to see the Nautilus and Phi revisited from several new perspectives that reveal a closer fit than the usual method derived from the golden rectangles, “whirling squares” path.

In universal terms, we really should be surprised not to see the golden ratio in growth and morphogenesis, because it reveals nature’s most finitely simple, yet infinitely varied heursitic for generating complexity, evolutionary potential and fitness. We call it the Asynsis forms synergised by Constructal flows; since Form follows Flow.

https://medium.com/@ASYNSIS/hidden-harmonies-are-more-than-obvious-1eb24148fa93

http://www.scoop.it/t/asynsis-principle-constructal-law

Ondrej Podzimek says

On November 23, 2014, Gary Meisner wrote:

“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.”

There is a peristent misconception about the character (and naming) of this curve. In fact, the curve drawn in the first two illustrations (by joining subsequent quarter arcs) cannot be named “spiral”. This is a compound curve build from arc sections. The appropriate name would be VOLUTE (yes, we could name this special case “golden volute”). But this compound curve does not have anything in common with the true logarithmic spiral (or spira mirabilis, as described by Jacob Bernoulli).

The way of drawing volute of this type is similar to the method used by ancient Greek architects to draw volutes before ioic column head was carved from stone block. They were using compasses and the resulting volute -although aethetically pleasant – was drawn as a compound curve with distinct circular sections joined together at the ends and with matching start/end tangents.

The same difference applies to ELLIPSES and OVALS: ellipse is a parametrically defined curve with smoothly changing curvature. However, architects often approximate it using compasses; the result is the oval curve, which is the combination of four arcs.

As Gary Meisner pointed out already, there is also a difference between the golden volute (constructed from outside by dividing the golden rectangle) and the Fibonacci volute (constructed from inside out by adding squares with the side lengths in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, etc. units). However, none of these two compound curves honors the name “spiral”. They are both euclidean volutes, constructible with compasses and straightedge.

Shayne B. says

To the naked eye, without a protractor of course, the Nautilus shell does seem to have the golden ratio rule. I mean it’s something of nature, and nothing of nature is perfect. I’m sure somewhere in the world there’s a Nautilus shell that follows the golden ratio rule, but I feel like asking for a perfect spiral is a bit too much to ask from a shell, yeah? It’s close, albeit not entirely accurate, it’s close to the golden ratio.

rkglass503 says

Any comments on the Ammonite and the Golden Spiral (Volute)? I would love to hear some information on this instead of the Nautilus.

Cheers!

Ondrej Podzimek says

You are asking about the geometry of the Ammonite shells. Good question. Neither Ammonite shells, nor Nautilus shells have anything to do with the golden spiral.

Golden spiral should have – by definition – the growth rate per turn P = 1.61803399 = Phi. For some obscure reason, all scholars tend to draw the golden spiral using the growth rate P = 2.618033988 = Phi^2 = Phi+1. To this day, no one has explained this discrepancy.

I have measured several Nautilus shells, using avilable photographs. The linear growth ratio of the Nautilus shells measured varies from P = 2.76246446 to P = 3.01421291 per turn. Quite far from Phi.

Ammonites are a different story: I have measured P = 1.97717302.

And I suppose that every organism with a spiral shell has a distinct growth rate, with variations between individuals of the same species. If anyone finds a shell with the growth ratio that equals Phi, this will be pure coincidence only. Not a marvel of Mother Nature.