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...Fractal geometry special topics in dynamical systems ws sascha troscheit faculty of mathematics university vienna oskar morgenstern platz wien austria univie ac at march abstract classical shapes such as lines spheres and rectangles are only rarely found nature more common that share some sort self similarity for example a mountain is not pyramid but rather collection shaped rocks various sizes down to the size gain sand without any scale reference it dicult distinguish from ragged hill boulder or ever small uneven pebble these ubiquitous natural world clouds lightning strikes even trees what tree branches acentral component description how properties geometric objects with dimension theory particular studies scalings by means dimensions each capturing dierent characteristic most frequent scaling encountered exponential e g surface area volume cubes measures can simultaneously exhibit very behaviour on an average ne coarse used classify them when traditional cardinality no longer appro...