## Indra’s pearls

In my last posts I was metaphoring about iterations/recursions and fractals.

In fact fractals are probably the most popular issue when one thinks about math and art (which can sometimes even be productive:). However in my last posts I also wanted to explain that the fractal images are just images/representations of concepts, which are hidden in the depth of math.

Also due to this conceptual nature, pure math has not always immediate results, which is a no-no in a more and more uncontrollable global economy. Hence mathematicians try to navigate in this new world which sometimes leads to controversial discussions at math departments about marketing and design…just joking). The science—coffee example in my last post is not the only example of an ongoing paradigm and power change in the science (and culture) world.

However pure math often even produces “utilizable” results (whatever this is). Only – these results are not always directly visible, as they are e.g. used up in other theories or e.g. sometimes the results need a long time to be correctly validated (just as e.g. art).

One such example is the work of Felix Klein. The implications of certain parts of his work became only visible almost a century later, with the introduction of computer technology and the systematic investigation of fractal sets. Some investigations of Klein and coworkers can be found in Fricke and Kleins math book: “Vorlesungen über die Theorie der Automorphen Functionen, Leipzig (1897), which contains the above image. The thin line between the circles in the above image is a fractal and Klein and Fricke did not compute but imagined that it should look like that. In 1897.

There is an excellent book explaining some of Felix Klein and coworkers thoughts, which is called “Indra’s Pearls” (and thus metaphoring the principle of self-similarity as it appears often in Kleins works with the help of an excerpt from a certain sutra).

The book is by the three mathematicians David Mumford, Caroline Series and David Wright and published by Cambridge University Press in 2002. It is mathematically rigorous and even mathematicians can learn from it (for a math review see e.g. here), but it is rather intended to adress a bigger also mathematically untrained audience. Besides the explanations with various anecdotes and biographies and a lot of images the book contains computer pseudocode for self-exploration.

Below are some bad quality, low res images to give you an impression about the book. Amazon provides the table of content etc. and in particular some high resolution pages, which may be give a better overview about the care given to the overall presentation. Unfortunately there are not too many of the very elaborate images on display at amazon.

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