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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 ca n contin ue 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
seashells.
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.