Life and .618
I start from the point of view that log spirals, the Golden Angle and Fibonacci numbers are are joined in nature so they must be joined mathematically. This mathematical join if found would express a physical law or laws basic and particular to life, to organic nature.
Looking for the mathematical join , I look at leaf phyllotaxy - the clearest example in nature of the joining of Fibonacci numbers, the Golden Angle and log spirals. And what is joined to leaf phyllotaxy? -- the position of the vascular bundles, the branches, the flowers and the growth of the stem. When we see this we see why we need a unified mathematical explanation of the spirals, angles and numbers involved in their growth.
Gathering all we now know about the formation of leaf primordia and all we once knew about plant morphology in the nineteenth century (see Asa Gray) and all that the combination implies into one compact bundle we form the following picture:
Leaf primordia from around the shoot apical meristem (SAM) ,eacg separated in space by 137.5 degrees, the Golden angle from the preceding primordia. The number of primordia that form a Fibonacci number though the reason for saying this is not evident until the primordia line up on the stem when it may then be seen that that the number of leaves and the number of turns around the stem before a leaf is positioned directly above a preceding leaf is one of these possibilities: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21. But returning to the SAM, the primordia may be conceived as being on a strip around the SAM between its outer edge and the inner edge of the stele. The stele, of course, is not evident. The strip is as wide as a primordia and is divided into "tracks" equal in width and equal in number to the number of primordia aound the SAM. This accounts for overlaps such as we see in the pinecone while the Golden Angle and Fibonacci number explain the spirals on the cone.
Fully formed primordia are moved outward in a straight line by cell growth to the edge of the plant on the outside edge of the stele. The stele still does not extend upward to where the primordia is but as a consequence of the cell growth described above the primordia is now positioned adjacent to the place where a vascular bundle will shortly be. When the vascular system does grow up to the level of the primordia a vein grows out from the vascularbundle into the primordia joining the growing tip of the plant, the leaf, to the root system. The vein joins the primordia at its midpoint, becoming the "midrib" of the leaf and causes a system of veins to branch through the primordia cell mass patterning in ways characteristic of the plant species.
The stem of the plant legthens and thickens without disturbing the relationship between the leaf, its veins and the vascular bundles.
So that leaf phyllotaxy is also a guide to the arrangement of vascular bundles.
Branches begin as axillary buds at the base of each leaf so leaf phyllotaxy is also a guide to branch phyllotaxy
Flowers begin as primordia separated from each other by 137.5 degrees. The follow a pattern similar to leaves in that they form about the SAM and move out to a position on the stem which aligns them with the vascular bundle system - a alignment not disturbed by the lengthening and thickening of the flower bud and the stem. But whereas the flat surface of the leaf shows how the veins patterned the primordia, the flat surface of the fower shows groups of cells moving in log soiral patterns, the number of groups being a Fibonacci number.
We propose that these forms flow from the the form of DNA - evolution being a change in the number of DNA (TBC)
Looking for the mathematical join , I look at leaf phyllotaxy - the clearest example in nature of the joining of Fibonacci numbers, the Golden Angle and log spirals. And what is joined to leaf phyllotaxy? -- the position of the vascular bundles, the branches, the flowers and the growth of the stem. When we see this we see why we need a unified mathematical explanation of the spirals, angles and numbers involved in their growth.
Gathering all we now know about the formation of leaf primordia and all we once knew about plant morphology in the nineteenth century (see Asa Gray) and all that the combination implies into one compact bundle we form the following picture:
Leaf primordia from around the shoot apical meristem (SAM) ,eacg separated in space by 137.5 degrees, the Golden angle from the preceding primordia. The number of primordia that form a Fibonacci number though the reason for saying this is not evident until the primordia line up on the stem when it may then be seen that that the number of leaves and the number of turns around the stem before a leaf is positioned directly above a preceding leaf is one of these possibilities: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21. But returning to the SAM, the primordia may be conceived as being on a strip around the SAM between its outer edge and the inner edge of the stele. The stele, of course, is not evident. The strip is as wide as a primordia and is divided into "tracks" equal in width and equal in number to the number of primordia aound the SAM. This accounts for overlaps such as we see in the pinecone while the Golden Angle and Fibonacci number explain the spirals on the cone.
Fully formed primordia are moved outward in a straight line by cell growth to the edge of the plant on the outside edge of the stele. The stele still does not extend upward to where the primordia is but as a consequence of the cell growth described above the primordia is now positioned adjacent to the place where a vascular bundle will shortly be. When the vascular system does grow up to the level of the primordia a vein grows out from the vascularbundle into the primordia joining the growing tip of the plant, the leaf, to the root system. The vein joins the primordia at its midpoint, becoming the "midrib" of the leaf and causes a system of veins to branch through the primordia cell mass patterning in ways characteristic of the plant species.
The stem of the plant legthens and thickens without disturbing the relationship between the leaf, its veins and the vascular bundles.
So that leaf phyllotaxy is also a guide to the arrangement of vascular bundles.
Branches begin as axillary buds at the base of each leaf so leaf phyllotaxy is also a guide to branch phyllotaxy
Flowers begin as primordia separated from each other by 137.5 degrees. The follow a pattern similar to leaves in that they form about the SAM and move out to a position on the stem which aligns them with the vascular bundle system - a alignment not disturbed by the lengthening and thickening of the flower bud and the stem. But whereas the flat surface of the leaf shows how the veins patterned the primordia, the flat surface of the fower shows groups of cells moving in log soiral patterns, the number of groups being a Fibonacci number.
We propose that these forms flow from the the form of DNA - evolution being a change in the number of DNA (TBC)
posted by wildswan at 12/10/2006 01:52:00 PM
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