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Author Topic: Perimeter of a leaf  (Read 3833 times)
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web2k
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« on: November 19, 2006, 04:16:53 PM »

In a recent thread on alt.mathematics, a simple math problem was raised.

"The perimeter of leaf II is 1.5 times as long as the perimeter of leaf
>I.  How many times greater is the area of leaf II than the area of
>leaf I?  Express your answer as a decimal to the nearest hundredth.

Not to be a wa but the problem is incompletely specified.  The edges
of various leaves have different fractal dimensions.
http://hypertextbook.com/facts/2002/leaves.shtml
"The purpose of this lab is to determine the dimensions of six
different leaves. The dimensions that were obtained ranged from 1.7 to
1.9."
Further, the scale at which the length is measured must be specified.
If a scale of 1mm (for example) is used for both leaves I and II, then
the answer given by other posts (1.5^2) must be increased (a smaller
relative scale is implied in its perimeter measurement--thus a greater
length will result.) I think the increased value would be 2.44 instead
of 2.25 for a maple leaf (fractal dimension 1.9)"

This answer seemed wrong to me. (It is wrong on several counts) and I attempted to create a model of the problem using the Koch snowflake.  I then realized that a slight modification to the Koch snowflake generation process would result in a fractal for which the area remains constant while the iterations can be arbitrarily large.
(See attached graphic)

The easiest, simplest version of the modification, which consists of alternating between outward pointing and inward pointing triangular path extensions, produced a singularly ugly result.  Thinking that symmetry might make a more pleasing fractal, I tried variations--in out out in  or out in in out, etc.  Unfortunately, every symmetric variation produced a perimeter that was self intersecting.  So far, after many tries, the only fractals with an area that is invariant with iteration that I have found are assymetric.

This simple sounding problem, pursued to the max, has resulted in a deeper appreciation of the subtlties of fractal dimension.

John


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« Last Edit: November 19, 2006, 04:22:25 PM by web2k » Logged
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