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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. 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.

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.

There is, however, more than one way to create spirals with golden ratio proportions of 1. 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 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 half rotation of degrees to point B expands the width of the spiral to 1.

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Continue another half turn of degrees to point C to complete the full rotation of degrees. The width of the spiral from the center is now 2. 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. 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. This averages to 1. 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. 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. This is slightly less than 2.

Expansion rates in this same shell ranged to 2.

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Rates over 3 were observed in other shells. 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. 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. Following are comments by three Ph. 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.

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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. 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. Note: A special thanks go to Oliver Brady for his astute analysis of this article, which led to improvements in its clarity and accuracy. I am fascinated by the fact, How many natural things have golden ratio concepts integrated with them. The Chambered Nautilus form is not a Golden Spiral. The point of the article is that a Nautilus spiral does NOT conform to the classic Golden Spiral that expands by the golden ratio every 90 degrees.

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It does, however, very closely follows a spiral that expands by the golden ratio every degrees. I had assumed a full turn of degrees or 2Pi radians. Of course, one can create different spirals depending on your reference angle — whether it be full turn, half turn, third turn, quarter turn, fifth turn; or 1 radian or 2 radian, etc So there is a range of possibilities of making a match.

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…. How is that done? The pineapple spirals round in three different ways. Each spiral adds up to 8, or 13 , or 21 segments.

That is, natural, instinctive growth rates are at 1. I guess there is really a heavenly Designer. Bear with me for a while In an overwhelming number of plants, a given branch or leaf will grow out of the stem approximately In other words, after a branch grows out of the plant, the plant grows up some amount and then sends out another branch rotated Plants use a constant amount of rotation in this way, although not all plants use However, it is believed that the majority of all plants make use of either the If we were to multiply the value of 1 over Phi to the second power 0.

As an alternate way to look at the same idea, if we were to take the value of 1 over Phi 0. If we then subtract So, if we have followed the described mathematics, it is clear that any plant that employs a 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? 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. 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.

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.

Honeybees are not building hexagons they are stacking circles and filling in the gaps. It is a hallmark trait of humans to see complexity instead of the simpler solution. It really is a hiccup in nature we could do without! Look again. But then humans have also shown their ability to assume a simple solution when in fact more complexity does exist.

Darwin had no understanding of the very sophisticated technology within our DNA that encodes the instructions for life.