More fractal dimension

I wrote long ago about a simple way to calculate the fractal dimension for any shape. The central idea behind calculating the fractal dimension (or the any dimension) is the Hausdorff Dimension (applet-ridden page). In very layman terms, the Hausdorff dimension provides a way to calculate the dimension of any space – point, line, plane or even irregular shapes like fractals.

The idea behind the dimension calculation is rather simple – by what factor do the smaller parts fit into the original whole when the dimensions of the original are modified. For example, when the dimensions of a cube are increased by a factor of 2, there are 8 of the original cubes which can be placed in the new cube.  Applying the Hausdorff dimension, N = 8, P = 2, p = 1 (the number of increase in units is 8, when the size of the size is doubled). So, the dimension of the cube is log8/ (log (2/1)) = log 8 / log 2 = 3. Of course everyone knows that the cube is 3 dimensioned.

Now the beauty of the equation is that it can help calculate the dimensions for fractals equally easily. This is used in in Season 4, Episode 9 of Numb3rs – Graphic by Charlie Epps . Charlie uses Fractal dimensions to find out if a painting is original or not. The calculation of the fractal dimension is skimmed rather fast. So, let us take it up with the example in the link.

The fractal to use is the Sierpinski triangle. The numbers to use again are simple. For each time there is a change in the dimension of the triangle, the number of triangles increases by 3 fold. For example, if the dimension of the equilateral triangle was 2 initially, the first iteration reduces the dimension to 1. The number of triangles increases from 1 to 3. So, N = 3, P = 2, p= 1. Hence the fractal dimension of the Sierpinski triangle is log 3 / (log (2/1)) = log 3 / log 2 ~ 1.585. The d-dimension for the triangle is 1.585, the just right dimension.

Continuing, if we were to calculate the fractal dimension of the Menger sponge, it is thus. The number of cubes that get generated when the center 1x1x1 cube and the center 1x1x1 cube of each face of a cube are removed is 20. If the original cube was 3x3x3, then each of the new cube’s dimension is 1x1x1. Hence, N=20, P=3, p=1. So, the fractal dimension is log 20 / (log (3/1)) = log 20 / log 3 ~ 2.726 (which is lesser than the dimension of a cube !).

Put very simply, the equation lets one correlate the increase in the number of parts with the space that that shape consumes ( it the space that the shape consumes or the dimensions that the shape has ?).

Kiva introduces currency risk protection

Kiva introduced a new feature – currency risk protection recently. What is the feature and why is it important ? Oversimplifying it, this feature is a way to pass a part of the currency fluctuation to the lender, and only when the currency vacillates more than –20% (i.e. the borrower’s currency is devalued more than 20% pegged against the reserve currency, US dollars).

The Kiva site has a very simple example of how that works. I think the feature is not a bad idea. If a currency did indeed get devalued more than 20%, then I think the MFI shouldn’t be strained to get the difference funds, and the lenders should face the bullet.

This feature made me think of two things

  1. If the borrower’s currency was indeed devalued (due the economic conditions, not necessarily the current one but in general), and the lender also faces the risk of loosing some part of the investment, will that dissuade lenders from lending? Do lenders think actively consider the currency differences when lending? Do most of them lend assuming that their investment is charity ? (or with an idea to have a rotating investment – sort of a serial micro-venture-capitalist ! Hmm, am I the first one to use that term ? 🙂 ). At what point will this become an important consideration ?
  2. Continuing on the above and with wider ramifications, what can the MFI do to hedge its bets ? (did you hear the word hedge and bets and MFI all in the same sentence 😀 ). Well, agreed that the MFIs are not like the traditional lending institutions, in that they do have a social goal. But, is it possible that a MFI could hedge its investments by getting into future contracts ?
    1. Of course the knee jerk reaction for anyone is to scorn the idea, looking at me as if they won’t touch me with a ten foot pole. But, futures and options are not synonymous with evil isn’t it. So, why can’t there be an exchange which can help MFIs hedge their investments. And these will not be on the normal exchange, rather in a MFI exchange, where in these zero-sum contracts can be got by MFIs only. I still don’t have the details on how the options will work and will post again once I think through this (and if I realise it was a bad idea, I will write why !).

I’d be interested in knowing your thoughts on this. And while we are it, please do take a look at RangDe. They seem to be expanding their network each day and can definitely do with a little help.