Last spring Carol made a discovery that both pleased and puzzled her. She came in from a hike one day with several wild sunflowers and some pinyon pine cones. "Look. They show a pattern," she said, "and it seems to be something like a double spiral. But they're not closely related or the same thing at all. So how can they have such similar patterns?" And so Carol was introduced to the mathematical progression called the Fibonacci series, the basics of which are explained on here.

Ancient and modern mathematicians of many cultures have observed and described the Fibonacci sequence and its corollaries. And many have, like Carol, noticed its presence in the patterns of natural things. The question this raises for them is the same as the one she asked us: How is this possible?

Some feel there must be an underlying cause that generates the pattern in so many different places, and that this "cause" must be a great but unknown universal  law. They suggest, for example, that a sunflower head expresses the pattern through the action of the "unknown Fibonacci law" on the mechanisms of protein synthesis within cells, meaning that the law affects the structure and/or expression of the plant's very genes.

Fibonacci spirals in the head of a sunflower (right-hand image shows tracings of two spirals to help you see them). Each of the spirals in the arrangement of structures on this sunflower head seems to have grown outward from the center in a sequence very like that of the  squares shown in the "Fibonacci" explanation. Notice that spirals run both clockwise and counter-clockwise. Also note that the spirals are not perfect; particularly near the center, they tend to be offset or to merge with one another.

Those with a Deist point of view tend to see the encoding of unknown natural laws in living systems as neither mysterious nor unknown, but as the literal inscribing of God's divine will on living beings. So contemporary Deists such as proponents of Intelligent Design view the Fibonacci pattern as evidence of divine action rather than natural process.

Regardless of whether they see the cause as natural or divine, those who feel the Fibonacci pattern's expression is important have documented examples of its presence for several thousand years -- in everything from spiral galaxies to ancient architecture and a wide array of sacred texts.

But it's not easy to imagine a natural law or process that could produce the same pattern in things that are that diverse. So some people argue that the pattern lies in the eye of the beholder, not in the structure itself. They suggest that few natural Fibonacci patterns are "exact" and that the human eye and brain perceive a pattern where it doesn't really exist because we automatically look for patterns to make meaning out of the world around us. Others argue that the Fibonacci pattern (and similar ones produced by different mathematical equations) are simply the product of physical, geometric constraints -- such as available space and the packing of spheres in that space -- that are not inherently meaningful at a "causal" level.

Such geometric constraints were seen as important factors in plant and animal develpoment by botantists and embryologists of the late 1800s. They realized that the cells of a developing tissue can really only divide and grow where there is still "open" space (as opposed to where there already are cells), but that they must do this while still in contact with the cells already present. So the cells of many simple natural structures tend to develop outward from the structure's center in something of a spiral. In the 1970s, vertebrate paleontologist Stephen J. Gould applied similar "packing-pattern" logic to the shapes of marine snail shells from different species, showing that different species of snails (with differently shaped shells) were mathematical variants of a "basic" type.  If you elongated the axis of the shell's spiral growth one way, for instance, it produced a shell shape typical of a particular species;  elongating it a different direction produced another common type.

Gould was building on a concept of animal structure called "constructional morphology." "Morphology" simply means shape. The German school of anatomy in which this idea developed suggested that when you ask, "Why does this organism's structure look the way it does?" you must consider more than one causal factor. This concept is especially important when considering patterns of structure in different organisms and what those patterns mean. Constructional morphologists consider three or four factors when trying to understand the patterns of natural structure:  inheritance, development, function, and material properties.

Here is an example of using constructional morphology to understand a pattern. Let us consider a specific lower front leg bone (the radius) of a specific species (the horse) in a specific individual (a horse named "Pecos"). We ask, "Why does Pecos' radius bone look the way it does?"

The first factor is inheritance. Most vertebrate animals have limbs (arms and legs) with this pattern: a girdle of bones around the backbone, one large  bone close to the body at the girdle, then two bones side-by-side, then several small bones, and then the bones of a hand/foot. Cats, horses, bats, crocodiles, pigeons, and human beings are all vertebrates and all have a radius bone. So clearly one reason Pecos has a long bone in this particular part of his

foreleg is that such a thing is normal for vertebrate animals; he inherited the genes that create such a pattern of bone from his ancestors.

But the radius of a horse and of a pigeon and of a human being are different in some ways. The other slender bone that is often beside it in the forelimbs of vertebrates is called the ulna. The end of the human ulna (that sticks out and up) is called the elbow. At first, it seems the horse does not have an ulna and radius, but only one bone in this place. But if you look at the close-up, you can see the "elbow bone" part of the ulna is in fact present. A slender line shows  where the ulna has joined to or grown up against the radius in the horse.  Why has it done that?

(click image to see larger size)

One reason is function: a horse is a large animal that runs fast. One strong bone in the lower limb is stronger than two side-by-side bones. Further, when there are two side-by-side bones, they move against each other to rotate the end of the forelimb (hand, in this case) for greater dexterity. If you have ever tried to walk on your hands, you know that when a limb that provides such dexterity is used for support, "dexterity" becomes "instability" instead! So in a large, fast-running animal such as a horse, a single bone in the lower limb is more stable as well as stronger. And since this pattern is present even in baby horses, we know it is also now inherited within the horse lineage. (The process by which a structurethat is functional becomes also inherited is evolution by natural selection.) Development is also a factor in all this. The cells and tissues of a horse embryo must move (within the embryo) to particular places to make a forelimb. Pre-bone and pre-muscle cells must follow gradients of chemicals within

the ball of cells that will someday be a horse embryo, to get to the right places and then begin to develop into appropriate tissues. Those processes are governed by a whole different set of functions and patterns of inheritance, and these interplay with the ones already described.

But this is still not the end of answering the question, "Why does Pecos' leg look the way it does?" For bone is living tissue with a complex micro-structure. It grows, destroys itself, and regrows constantly in response to the forces put on it as it's used. If Pecos lives in a stall and can never go out to run, the bone in his legs will be less dense and the leg itself more slender than if he lived where he ran a great deal on hard ground. And if he injures his hoof so that it grows abnormally, standing on that hoof will generate new stresses on the bone that cause it to grow differently and therefore look different. Perhaps one leg would seem thicker than the other, or slightly bent in one spot. It would compensate for the change in how he must walk with the injury, to keep his leg able to support him.  And it would be entirely personal to Pecos -- an answer to the question "Why does his leg look like that" which begins, "When Pecos was three years old, he stepped on a fence stake . . ."

The thing is, another vertebrate (such as a human being) might injure a different limb (such as an ankle) and grow a very similar shape to the bone for similar reasons. The bone shape would then appear as a pattern -- in this case, of the response of bone to its environment. Part of that response would be determined by genetics, part by function, and part by the material properties and constraints of bone, muscle, and other tissues. But the pattern would nevertheless reveal the presence of a fundamental flow of hidden currents in the vertebrate body -- currents of all the factors we've mentioned -- that run through horses as well as humans and many other  animals, and that come to the surface time and again in specific situations.  And when they do, they manifest a pattern that can be recognized.  The hidden currents responsible for the pattern would be complex and multi-faceted as well as very deep and powerful. "Fundamental" does not mean  "Simple."

The people who argue about the significance of Fibonacci sequence patterns in living things, and in things as varied as  galaxies and sacred texts, may be expecting to find answers that are too simple because they have confused fundamental processes with simple ones. And in so doing, they may be casting aside some of the most important information the pattern bears. If we assume, for example, that the pattern of a horse's leg shape is solely a divine signature, we reduce all the staggering beauty of its multiple, complex, elegantly-timed processes to a mere gesture lying outside of nature. How can all that beauty be something meaningless, to be tossed away as insignificant  to our understanding of Life? Yet that can happen when we reach for a single universal explanation that reduces everything to one simple law or intention.

Likewise, to deny meaning to the pattern, to say it reflects only our human ability to see and describe rather than  any inherent, underlying process dismisses that which we know deep  in our hearts to be meaningful. We look for patterns in the world around us for a reason; they mean something. But to recognize and acknowledge a pattern is not the same thing as to say we understand it, or that we know how and why it exists. We may perceive patterns with our eyes and sense   the power of their meaning in our hearts -- and at the same time acknowledge that we have no idea what they mean, but only that they mean.  Perhaps a humble and appreciative awareness that does not insist on immediate understanding is the beginning of wisdom.

Tapestry works to understand different ways of knowing, learning about, and responding to the natural world, including the patterns it manifests. Art is one of those ways of knowing. Carol has expressed what she's come to understand about Fibonacci patterns in nature and the things they might mean in her newest piece of artwork, reproduced in this issue. Science, which is what you have just read, is another way of knowing about the natural world. And yet another way of knowing -- one seldom acknowledged outside of Indigenous culture -- is Dream. In this issue, we introduce you to that way of knowing if you have never enountered it before. It is a dream-teaching about the patterns around us and what they might mean. It's part of what inspired the theme of this issue of The LOOM.

And of course now it's time to watch another pattern -- that of the seasons -- turn again with the coming of Equinox.  We hope the thoughts and images in this issue will help you open your eyes and heart and mind to the patterns that flow around and through your body, your life, and the universe of which you are such an integral part.

     Until Winter . . .

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