In the same way as modernists, we are trying to fill in the post-Nietzschean Void by inventing our own images and grand narratives. Although the grand narratives of Christianity, Islam and Judaism have a difficult time dealing with differences, there are two major traditions -- Buddhism and Hinduism -- that can and do embrace the differences in our increasingly pluralistic world. Buddhism is democratic, cool, practical, inexpensive and politically correct with the liberation of Tibet from China becoming a hip cause. Postmodern peoples and cultures live in a world of differences. Buddhism's philosophy of interdependence lets us see our differences as a vast interconnected web. In fact, the image Buddhists use to illustrate this is that of Indra's net. At each intersection of the strands of this net, which is the universe of different selves, is a jewel -- a "self" -- which reflects all the other jewels in the net. No single jewel, then, is self-sufficient. Its existence depends upon, and reflects, all the others. And so, in Buddhist lingo, each jewel is Empty of self-existence!

Kleinian group fractals have been popularized by the book "Indra's Pearls" by David Mumford, Caroline Series and David Wright. The key to fractals of this type is an understanding of Möbius transformations. Möbius transformations form a mathematical group. They are also known as linear fractional transformations, and are represented as:

where

**z** is the complex number being transformed, and

**a**,

**b**,

**c** and

**d** are complex constants. A Möbius transformation can be viewed as a composition of translations, scalings and inversion. Properly chosen Möbius transformations can be iterated, with the limit set of the iterated points defining a fractal. The fractals created from the iteration of Möbius transformations can be as beautiful and varied as fractals created by other better known methods.

Möbius transformations can be represented in a matrix form:

Iteration of the transformation is accomplished by matrix multiplication and inversion by matrix inversion. The matrix should be normalized, which means that

**ad-bc = 1**. The conjugate of a transformation (conjugated with another transformation) is defined as:

The trace of a transformation is defined as:

The trace of a transformation is unchanged by conjugation. Möbius transformations can be classified by the value of the trace and the number of fixed points (one or two).

**Loxodromic**. These have one source

and one sink

. On iteration, points spiral out from the source and into the sink.

**TrT** is not between –2 and 2, and they are conjugate to

**T(z) = kz, |k|>1**.

**Hyperbolic**. Points move not in spirals but in circles through

,

.

**TrT** is real and not between –2 and 2, and they are conjugate to

**T(z) = kz, k** real,

**k>0**.

**Elliptic**. Have two neutral fixed points and move around circles round the fixed points.

**TrT** is real and strictly between –2 and 2, and they are conjugate to

**T(z) = kz, |k| = 1**.

**Parabolic**. Have one fixed point that is both the source and sink.

**TrT=+2**, and they are conjugate to the translations

**T(z) = kz+a, k = 1**.

The trace of the transformation determines if, and what type of fractal, is generated upon iteration. Consult the Indra's Pearls book for more detail.

Two examples of Kleinian group fractals are shown below:

**Indra's Net (A Schottky group)****Kleinian 1/15 Cusp**