Last time I looked, the engineer in me thought I was looking at the frontal projections of a pentagonal pyramid (base has 5 sides) reduced to its frustum by cutting off the apex with a slanting plane as shown on the lower left figure. The figure on the upper left is how it would look like if you would view the frustum directly from above. The middle triangular figure is the side view if you would stand directly facing the side adjacent and to the right of the frontal face. The small 5-sided polygon in the midst of the first three is the cut-off line as you would see it if viewed normally (at right angle) from the cutting plane.
And what about the "skeleton of a shell"-like figure at the right? It's just the surfaces of the pyramidal frustum that was spread out! Like if you would cut out the outline from a cardboard and then fold it along the inner lines – you would come up with the 3d model of the pyramidal frustum!
So where does it lead to Fibonacci? Let me attempt to connect. If you will keep on increasing the no. of sides of the base, the pyramid would approach a cone which when cut off by an intersecting plane produces elegant curves - ellipse, parabolas, hyperbolas and their degenerates. If you have a pyramid with a finite number of sides, what you produce is not a curve but a series of straight lines approximating a curve. The lines in this series are not of the same length at all as you can glean from the mural. But maybe, just maybe, their length ratios form a Fibonacci pattern… or the golden ratio.
3 comments:
Sonny, I will try and humor you and leave you with a very mathematical answer since you seem to know your math quite well. But please note that not everyone has taken my "mathology" in stride. And I'm sure I'll hear more flack on this later on. ---- Fibonacci numbers: 1,1,2,3,5,8,13,21,.. 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). 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). 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 will produce the Fibonacci series. 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. 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 center are 1.6184 = 6.854 times further out than when the curve last crossed the same radial line.
Hi Ricky.
I’m not much into pure math as my line is on the applied variety, i.e. numerical methods and finite elements, but I know enough of Fibonacci to say that I can use it to generate pythagorean triplets. What I really need to know is how that pyramidal surface is related to this mystical sequence. Thanks again…
sonny
Hmmn,......... this is more fascinating than the US presidential debates. Carry on fellows. Love it.
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