Thursday, Aug 30, 2012

Geometry developed as a collection of tools for understanding the shapes of nature.

For millenia, Symmetry has been recognized as a powerful principle in geometry, and in art.

We begin by reviewing the familiar forms of symmetry, then show that fractals reveal a new kind of symmetry, symmetry under magnification.

Many shapes that at first appear complicated reveal an underlying simplicity when viewed with an awareness of symmetry under magnification.

We begin by reviewing the familiar symmetries of nature: symmetry under translation, reflection, and rotation.

The geometric characterization of the simplest fractals is self-similarity: the shape is made of smaller copies of itself. The copies are similar to the whole: same shape but different size.

More examples of self-similarity examples, and variations including nonlinear self-similarity, self-affinity, and statistical self-similarity. Also, some fractal forgeries of nature.

Initiators and Generators is the simplest method for producing fractals. It is also the oldest, dating back 5000 years to south India.

Remarkably, fractal patterns have been found in some poetry and built into some.

Without much effort, we can find fractals in the kitchen.

Finally for today, do fractals have practical applications? How about an invisibility cloak?

We begin by reviewing the familiar symmetries of nature: symmetry under translation, reflection, and rotation.
The geometric characterization of the simplest fractals is self-similarity: the shape is made of smaller copies of itself. The copies are similar to the whole: same shape but different size.
More examples of self-similarity examples, and variations including nonlinear self-similarity, self-affinity, and statistical self-similarity. Also, some fractal forgeries of nature.
Initiators and Generators is the simplest method for producing fractals. It is also the oldest, dating back 5000 years to south India.
Remarkably, fractal patterns have been found in some poetry and built into some.
Without much effort, we can find fractals in the kitchen.
Finally for today, do fractals have practical applications? How about an invisibility cloak?