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Christmas Puzzle: Fibonacci spirals and the third dimension
Here’s an interesting brainteaser I’d like some help in confirming the answer to. The Fibonacci series is of interest because it crops up in all sorts of disparate natural phenomena such as snail shells, flower seedpods, leaf growth around stems and the arrangement of the inner planets. It is a series of numbers which is easily generated by adding the previous number to the current number to obtain the next number. So, if we take the first pair to be 0 and 1 we get:
0,1,1,2,3,5,8,13,21,34,55,89,144
The ratio between adjacent numbers settles down to be around 1.6180399:1 if you divide the larger number by the smaller, or around 0.6180399:1 if you divide the smaller number by the larger. The fraction after the decimal point is the same in both cases. These two numbers are designated as the greek letters Phi and phi, the upper case being used for the bigger number. The ratio is known as the Golden Section. Kepler believed this ratio and pythagoras’ theorum held the great secret of the universe. Together they form the Kepler triangle, which has interesting properties.
From http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
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 them Fibonacci Rectangles.
Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells and, as we see later, in the arrangement of seeds on flowering plants too. The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. So points on the spiral are 1.618 times as far from the centre after a quarter-turn. In a whole turn the points on a radius out from the centre are 1.6184 = 6.854 times further out than when the curve last crossed the same radial line.
So, onto the puzzle:
Suppose we are standing in a large tube, say it’s 4 metres in diameter, and there is a spiral inscribed inside it, such that as perspective makes it get smaller as we look further down the tube, it appears to form the proportion of the fibonacci spiral shown above.
Q . What is the ratio of the pitch of the spiral (the distance down the tube between two points where the spiral crosses a line drawn straight along the tube), and the diameter of the tube?
Hint: Objects twice as far away appear to be half the size.
2012-12-30 07:01:27
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