CALCULUS ON FRACTAL CURVES IN Rn

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called Fα-integral, where α is the dimension of F. A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize its algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The Fα-integral and Fα-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and Fα-differentiability is generalized. Finally we touch upon an example of absorption along fractal paths, to illustrate the utility of the framework in model making.

FLIPPING COINS, FOLDING PAPER, AND FINDING FAMILIAR FRACTALS IN THE TOWER OF HANOI

This manuscript describes three activities connecting the Tower of Hanoi puzzle to three familiar fractal forms. The first connects coin flipping, paper folding, and the Tower of Hanoi to the Dragon Curve. The second illustrates mathematician Ian Stewart's method for showing how the relationships between possible states of the Tower of Hanoi are related to stages in Sierpinski's Gasket. The final activity compares right-left moves of Tower of Hanoi disks to iterations of the Von Koch Curve.

VON KOCH AND THUE-MORSE REVISITED

We revisit the relation between the von Koch curve and the Thue-Morse sequence given in a recent paper of Ma and Goldener by relating their study to papers written by Coquet and Dekking at the beginning of the 1980s. We also emphasize that more general links between fractal objects and automatic sequences can be found in the literature.