Guest guest Posted August 10, 2005 Report Share Posted August 10, 2005 Dear Knowledge Seekers, Continuing on the Fibonacci/anahata nada. We would all appreciate the Fibonacci series that we learnt in High School. This series also applies practically in life also, the KARMA good or bad comes back to us in the same way as this formulae is defined. After reading this article I was very much in awe, but I started to look into every flower or vegetable to see the similarity.. You will also too, once you are done reading this. I am happy to share this knowledge (not mine but from another site) I am not the author. So please share the knowledge. Anand >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The Fibonacci Rectangles and Shell Spirals The Fibonacci Rectangles and Shell Spirals By Ron Knott We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1). We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles. The next diagram shows that we can draw a spiral by putting together quarter circles, one in each new square. This is a spiral (the Fibonacci Spiral). A similar curve to this occurs in nature as the shape of a snail shell or some sea shells. Whereas the Fibonacci Rectangles spiral increases in size by a factor of Phi ( 1.618..) in a quarter of a turn (i.e. a point a further quarter of a turn round the curve is 1.618... times as far from the centre, and this applies to all points on the curve), the Nautilus spiral curve takes a whole turn before points move a factor of 1.618... from the centre.These spiral shapes are called Equiangular or Logarithmic spirals. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 Pine cones Can you see the two sets of spirals? How many are there in each set? (the lines are drawn connecting the centres of each segment of the pinecone):Pine cones show the Fibonacci Spirals clearly. Pine cones Here is another pine cone. It is not only smaller, but has a different spiral arrangement.Use the buttons to help count the number of spirals in each direction] on this pinecone. Shows only the pinecone Shows the segment edges Show the outline only Show one set of spiral Show the other set of spirals TOP <To top of this page Vegetables and Fruit Here is a picture of an ordinary cauliflower. Note how it is almost a pentagon in outline. Looking carefully, you can see a centre point, where the florets are smallest. Look again, and you will see the florets are organized in spirals around this centre in both directions. How many spirals are there in each direction? (lines are drawn between the florets): How many spirals are there in each direction? Seed heads Fibonacci numbers can also be seen in the arrangement of seeds on flower heads. The picture here is Tim Stone's beautiful photograph of a Coneflower, used here by kind permission of Tim. The part of the flower in the picture is about 2 cm across. It is a member of the daisy family with the scientific name Echinacea purpura and native to the Illinois prairie where he lives. You can see that the orange " petals " seem to form spirals curving both to the left and to the right. At the edge of the picture, if you count those spiralling to the right as you go outwards, there are 55 spirals. A little further towards the centre and you can count 34 spirals. How many spirals go the other way at these places? You will see that the pair of numbers (counting spirals in curving left and curving right) are neighbours in the Fibonacci series. The same happens in many seed and flower heads in nature. The reason seems to be that this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage, all the seeds being the same size, no crowding in the centre and not too sparse at the edges. The spirals are patterns that the eye sees, " curvier " spirals appearing near the centre, flatter spirals (and more of them) appearing the farther out we go. So the number of spirals we see, in either direction, is different for larger flower heads than for small. On a large flower head, we see more spirals further out than we do near the centre. The numbers of spirals in each direction are (almost always) neighbouring Fibonacci numbers! TOP <To top of this page Fibonacci Fingers? Look at your own hand: You have ... 2 hands each of which has .5 fingers, each of which has ... 3 parts separated by ... 2 knuckles Is this just a coincidence or not????? However, if you measure the lengths of the bones in your finger (best seen by slightly bending the finger) does it look as if the ratio of the longest bone in a finger to the middle bone is Phi? What about the ratio of the middle bone to the shortest bone (at the end of the finger) - Phi again? Arrangements of the leaves Also, many plants show the Fibonacci numbers in the arrangements of the leaves around their stems. If we look down on a plant, the leaves are often arranged so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem. The computer generated ray-traced picture here is created by my brother, Brian. Leaves per turn The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one. If we count in the other direction, we get a different number of turns for the same number of leaves. The number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers! For example, in the top plant in the picture above, we have 3 clockwise rotations before we meet a leaf directly above the first, passing 5 leaves on the way. If we go anti-clockwise, we need only 2 turns. Notice that 2, 3 and 5 are consecutive Fibonacci numbers.For the lower plant in the picture, we have 5 clockwise rotations passing 8 leaves, or just 3 rotations in the anti-clockwise direction. This time 3, 5 and 8 are consecutive numbers in the Fibonacci sequence. We can write this as, for the top plant, 3/5 clockwise rotations per leaf ( or 2/5 for the anticlockwise direction). For the second plant it is 5/8 of a turn per leaf (or 3/8). Leaf arrangements of some common plants The above are computer-generated " plants " , but you can see the same thing on real plants. One estimate is that 90 percent of all plants exhibit this pattern of leaves involving the Fibonacci numbers. Some common trees with their Fibonacci leaf arrangement numbers are: 1/2 elm, linden, lime, grasses1/3 beech, hazel, grasses, blackberry2/5 oak, cherry, apple, holly, plum, common groundsel 3/8 poplar, rose, pear, willow5/13 pussy willow, almond where t/n means each leaf is t/n of a turn after thelast leaf or that there is there are t turns for n leaves. Cactus's spines often show the same spirals as we have already seen on pine cones, petals and leaf arrangements, but they are much more clearly visible. Charles Dills has noted that the Fibonacci numbers occur in Bromeliads. _ Quote Link to comment Share on other sites More sharing options...
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