Wave and particle Duality - and the fractal
Just as the atom can 'weirdly' be described as being both a particle and as a ‘smeared out’ wave at the same time, so too, as I shall demonstrate, can the fractal be described in such a way - only for the fractal it is not so weird.
The fractal demonstrating a (discrete) particle:
There are (in principle) an infinite amount of triangles expanding both into, and out of the point of observation in the fractal (below).
|Cross Section of the 'superposition (Koch snowflake) fractal|
|The Koch Spiral: notice the spiral wave and discrete triangles|
The (Koch Snowflake) fractal (above) is a wave like object.
As can be seen that the perimeter of the formed fractal is made-up of an infinity of triangles/'particles' and these together act as or form a wave. When a change is made to one of the triangles (the red dot on iteration 0) and this change iterated, as demonstrated, the wave is revealed. Any change to the triangles/ particles will 'mutate' or change the shape of the fractal and be viewed as a pulse, propagating, cycling round and round every 6 iterations (in the case of the Koch Snowflake), forming, if one were to view a change made to the fractal from a Front Elevation view, a pulse in the form of a classic wave.
- The wave will range through an infinitely of scales: from infinitely large amplitudes, frequencies, and wavelengths, – down to, but never reaching zero.
- The wave will move or be produced at fractal production speed.
- There are issues of spin direction too. The spiral can spin in both directions, clockwise and or anti-clockwise.
Logarithmic Spiral - Wave function:
The electromagnetic spectrum extends from low frequencies used for modern radio communication to gamma radiation at the short-wavelength (high-frequency) end, thereby covering wavelengths from thousands of kilometres down to a fractionof the size of an atom. It is for this reason that the electromagnetic spectrum is highly studied for spectroscopic purposes to characterize matter. The limit for long wavelength is the size of the universe itself, while it is thought that the short wavelength limit is in the vicinity of the Planck length, although in principle the spectrum is infinite and continuous. - from wikipedia
Electromagnetic wave equation
From a Plan view – the cross section view – the fractal is a standing wave. The attractor is a standing wave.